DATA-DRIVENSINGLE-/MULTI-DOMAINSPECTRALMETHODSFORSTOCHASTIC FRACTIONALPDES By EhsanKharazmi ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof MechanicalEngineeringDoctorofPhilosophy ComputationalMathematics,ScienceandEngineeringDualMajor 2018 ABSTRACT DATA-DRIVENSINGLE-/MULTI-DOMAINSPECTRALMETHODSFORSTOCHASTIC FRACTIONALPDES By EhsanKharazmi Fractionalderivativesareintegro-di˙erentialconvolutiontypeoperatorswithpowerlawkernels, whichseamlesslygeneralizethenotionofstandardintegerorderdi˙erentiationtotheirfractional counterparts.Insuchoperators,theorderofdi˙erentiationisanon-integernumber,whichin thelimitingcasesofintegernumbersrecoversthestandardderivatives.Thefractionaldi˙erential equations(FDEs)particularly,havebeenshownintheliteraturetoprovidearigorousmathematical toolthatcanbeusedtodescribetheanomalousbehaviorinawiderangeofphysicalphenomenon. Theyintroducehowever,theorderoffractionalderivativesasanadditionalsetofmodelparameter, whosevaluesareessentiallyobtainedfromexperimentalobservations.Theinherentrandomnessin measurements,incompletesetsofdata,signi˝cantapproximationsandassumptionsuponwhichthe modelisbuilt,andtherandomnatureofquantitiesbeingmodeledpervadeuncertaintyinthecorre- spondingmathematicalformulations.Thisrendersthemodelparameters,includingthefractional orders,randomandthus,thefractionalmodelstochastic.Wedevelopproperdata-drivenmath- ematicalframeworkstoe˚cientlyinfuseexperimentalobservations/dataintothecorresponding mathematicalmodelsinthecontextofstochasticfractionalpartialdi˙erentialequations. Weextendthefractionalorderderivativestothedistributedorderones,wherethedi˙erential ordersaredistributedoverarangeofvaluesratherthanbeingjusta˝xedinteger/fractionasitisin standard/fractionalODEs/PDEs.Suchdistributedoperatorscanalsobeconsideredasexpectation offractionalderivativewithrandomordersinthecontextofstochasticmodeling.Wedeveloptwo spectrally-accurateschemes,namelyaPetrov-Galerkinspectralmethodandaspectralcollocation methodfordistributedorderfractionaldi˙erentialequations.Inbothmethods,weemploythe fractional(non-polynomial)functions,called Jacobipoly-fractonomials ,whicharetheanalytical eigenfunctionsofthefractionalStrum-Liouvilleeigenvalueproblemof˝rstandsecondkind.We alsode˝netheunderlying distributedSobolevspace andtheassociatednorms,wherewecarryout thecorrespondingdiscretestabilityanderroranalysesoftheproposedscheme. Wedevelopa fractionalsensitivityequationmethod ,whereweobtainthenewsetofadjoint fractionalsensitivityequations ,inwhichweintroduceanotherfractionalintegor-di˙erentialop- erator,associatedwithlogarithmic-powerlawkernel,forthe˝rsttimeinthecontextoffractional sensitivityanalysis.Weshowthatthedevelopedsensitivityanalysisprovidesamachinelearning tool,whichbuildabridgebetweenexperimentsandmathematicalmodelstogearobservabledata viaproperoptimizationtechniques.Wealsodevelopanoperator-baseduncertaintyquanti˝cation frameworkinthecontextofstochasticfractionalpartialdi˙erentialequations,inwhichwechar- acterizedi˙erentsourcesofuncertaintiesandfurtherpropagatetheassociatedrandomnesstothe fractionalmodeloutputquantityofinterest. Wefurtherapplythedevelopedmathematicaltoolstoinvestigatethenonlinearvibrationofa viscoelasticcantileverbeam.Intheabsenceofexternalexcitation,theresponseamplitudeoffree vibrationrevealsasuper-sensitivitywithrespecttothefractionalorder.Primaryresonanceofthe beamsubjecttobaseexcitationalsodisclosesasofteningbehaviorinthefrequencyresponseof thebeam.Theseuniquefeaturescanbeusedfurthertobuildavibration-basedhealthmonitoring platform. Copyrightby EHSANKHARAZMI 2018 Tomywife,Fatemeh v ACKNOWLEDGEMENTS Iwouldliketothankmydearadviser,ProfessorMohsenZayernouri,whohasbeenagreatmentor andadearfriendtomeoverthecourseofmyPhDprogramatMSU.Hispatientguidance, enthusiasticencouragement,passioninlearning,trust,andveryconstructivecritiqueswereendless sourcesofmotivationtome.IamindebtedtoMohsenformanygreatopportunitieshegaveme andalsoforsharingvaluableexperienceswithme. IwouldliketothankProfessorBrianFeenyforhissupportandhelpduringmystudyatMSU.I havethehonoroftakingallofBrian'sclassesandlearningfromhim.Hispositiveandconsiderate attitudewillremainasavaluablereferenceformeintherestofmycareer.Iwasalsofortunateto learnfromProfessorChangWangandamverygratefulforhispreciousinputstomyresearchtoo. IwouldliketoacknowledgeProfessorShankerBalasubramaniamforhissupportivecommentsand questionsthathelpedimprovemyworkbythinkingoutofthebox. Ihavelearnedalotaboutthenonlinearmechanicsinthecontextofdynamics,vibration,and controlfromtheexcellentlectures,givenbyProfessorSteveShawandProfessorHassanKhalil, towhommygratitudegoes.IwouldalsoliketothankProfessorMarkMMeerschaertforthe greatconversationduringhisclassandgroupmeeting.Moreover,manythanksduetoourprevious andcurrentwonderfulsta˙bothinMEandCMSEdepartments,especiallyAidaMontalvo,Stacy Hollon,SuzanneKroll,JamieLake,LindsayBurns,HeatherJohnson,MelindaMcEwan,andLisa Royforbeingalwaystheretowillinglyhelp. IwouldliketoexpressmygratitudetomydearpeersandfriendsintheFMATHgroup,Mehdi Samiee,Dr.JorgeLSuzuki,EduardoBarrosDemoraes,AliAkhavanSafayi,PegahVarghaei, HadisNouri,andYongtaoZhou.Thankyouallfortheveryinterestingdiscussions,beingso supportive,andopentoshareideas. Iwouldlovetoespeciallythankmydearwife,FatemehAfzali,towhomthisworkisdedicated. Herunconditionallove,emotionalsupport,andencouragementmadememuchstrongerandmore faithfulthroughoutmystudyatMSU.Atlastbutcertainlynottheleast,Iwouldliketoexpressmy vi gratitudetomydearparents,lovelybrothers,andwonderfulfriendsfortheirconstantlove,support, andfriendship. Thisworkwassupportedbythedepartmentofmechanicalengineeringanddepartmentof computationalmathematics,scienceandengineeringinthecollegeofengineeringatMichigan StateUniversity. vii TABLEOFCONTENTS LISTOFTABLES ....................................... xii LISTOFFIGURES ....................................... xiv CHAPTER1INTRODUCTION ............................... 1 1.1FractionalModels...................................1 1.1.1FractionalRheology:ViscoelasticModeling.................2 1.1.2AnomalousDi˙usion.............................6 1.1.3FractionalCalculus:De˝nitionsandUsefulProperties...........7 1.2Data-DrivenFractionalModeling...........................9 1.2.1DistributedOrderDi˙erentialEquations...................10 1.2.2FractionalSensitivityEquationMethod:ApplicationtoModelConstruction11 1.2.3OperatorBasedUncertaintyQuanti˝cation(UQ)inStochasticFPDEs...13 1.3Single/Multi-DomainNumericalMethods......................14 1.3.1Petrov-GalerkinSpectralMethodandSpectralCollocationMethod.....15 1.3.2Petrov-GalerkinSpectralElementMethod..................17 CHAPTER2APETROV-GALERKINSPECTRALELEMENTMETHODFORFRAC- TIONALELLIPTICPROBLEMS ....................... 19 2.1Background......................................19 2.2De˝nitions.......................................23 2.2.1ProblemDe˝nition..............................23 2.2.2LocalBasisFunctions............................23 2.2.3TestFunctions:Localvs.Global.......................24 2.3Petrov-GalerkinMethodwithLocalTestFunctions.................25 2.3.1Elemental(Local)Operations:theconstructionoflocalmatrices S ¹ " º and M ¹ " º ,andvector f ¹ " º ...........................27 2.3.2Non-LocalOperation:theconstructionofhistorymatrix ^ S ¹ e ;" º .......30 2.3.3AssemblingtheGlobalSystemwithLocalTestFunctions..........32 2.3.4O˙-LineComputationofHistoryMatricesandHistoryRetrieval......33 2.3.5Non-UniformKernel-BasedGrids......................34 2.3.6Non-UniformGeometricallyProgressiveGrids...............37 2.4Petrov-GalerkinMethodwithGlobalTestFunctions.................39 2.4.1Elemental(Local)Operations:theconstructionof f ¹ " º ............40 2.4.2GlobalOperations:theconstructionof ^ S ¹ "; e º and ^ M ¹ "; e º ...........40 2.4.3AssemblingtheGlobalSystemwithGlobalTestFunctions.........41 2.5NumericalExamples.................................41 2.5.1SmoothProblems...............................42 2.5.2HistoryRetrieval...............................43 2.5.3SingularProblems..............................44 2.5.4Non-UniformGrids..............................47 viii 2.5.5ASystematicMemoryFadingAnalysis...................48 CHAPTER3DISTRIBUTED-ORDERFRACTIONALODES:PETROV-GALERKIN ANDSPECTRALCOLLOCATIONMETHOD ............... 51 3.1Background......................................51 3.2Preliminaries.....................................53 3.2.1FractionalSobolevSpaces..........................54 3.2.2DistributedFractionalSobolevSpaces....................55 3.3Distributed-OrderDi˙erentialEquations:ProblemDe˝nition............57 3.3.1VariationalFormulation............................57 3.4Petrov-GalerkinMethod:ModalExpansion.....................59 3.4.1DiscreteStabilityAnalysis..........................62 3.4.2ProjectionErrorAnalysis...........................65 3.5FractionalCollocationMethod:NodalExpansion..................69 3.5.1Di˙erentiationMatrices D and D 1 + , 2¹ 0 ; 1 º ..............69 3.6NumericalSimulations................................71 3.6.1SmoothSolutions...............................71 3.6.2Non-SmoothSolutions............................72 3.6.3ConditionNumber..............................74 3.7ProofofLemmasandTheorems...........................76 3.7.1ProofofLemma(3.2.1)............................76 3.7.2ProofofTheorem(3.2.3)...........................77 3.7.3ProofofLemma(7.3.4)............................80 3.7.4ProofofTheorem(4.3.2)...........................81 CHAPTER4DISTRIBUTED-ORDERFRACTIONALPDES:FRACTIONALPSEUDO- SPECTRALMETHODS ............................ 86 4.1Background......................................86 4.2De˝nitions.......................................87 4.2.1DistributedFractionalSobolevSpaces....................87 4.2.2ProblemDe˝nition:Initial/BoundaryValueProblem............89 4.2.3WeakFormulation..............................90 4.3FractionalPseudo-SpectralMethod..........................91 4.3.1InitialValueProblem.............................91 4.3.2BoundaryValueProblem...........................92 4.3.3WeakDistributedDi˙erentiationMatrix...................93 4.3.4ConstructionofLinearSystem........................96 4.3.5ConditionNumberofLinearSystem.....................97 4.3.5.1InterpolationPoints........................97 4.3.5.2Pre-Conditioning..........................99 4.3.6WeakDistributedDi˙erentiationMatrix:Two-SidedDistributed-Order BVPs.....................................101 4.4NumericalSimulations................................102 4.4.1Distributed-OrderIVP............................103 4.4.2(1+1)-DimensionsSpaceDistributed-OrderBurgersEquation........104 ix 4.4.3(1+2)-DimensionsTwo-SidedSpaceDistributed-OrderDi˙usionReac- tionEquation.................................106 4.5DetailedDerivations.................................109 4.5.1PolynomialsExpansionsInTermsOfJacobiPolynomials..........109 4.5.2E˚cientComputationofNon-linearTerms.................110 CHAPTER5TEMPORALLY-DISTRIBUTEDFRACTIONALPDES:PETROV-GALERKIN SPECTRALMETHOD ............................ 112 5.1Background......................................112 5.2De˝nitions.......................................112 5.2.1DistributedFractionalSobolevSpaces....................113 5.2.2ProblemDe˝nition..............................115 5.3PetrovGalerkinMathematicalFormulation......................116 5.3.1SpaceofBasis( U N )andTest( V N )Functions................116 5.3.2ImplementationofPGSpectralMethod...................117 5.3.3Uni˝edFastFPDESolver...........................118 5.3.4StabilityAnalysis...............................119 5.3.5ErrorAnalysis.................................119 5.4NumericalSimulations................................120 5.5ProofofLemmasandTheorems...........................121 5.5.1ProofofTheorem(5.3.2)...........................121 CHAPTER6FRACTIONALSENSITIVITYEQUATIONMETHOD:APPLICATIONS TOFRACTIONALMODELCONSTRUCTION ............... 126 6.1Background......................................126 6.2De˝nitions.......................................129 6.2.1ProblemDe˝nition..............................130 6.2.2MathematicalFramework:FractionalSobolevSpaces............130 6.2.3WeakFormulation..............................134 6.3Petrov-GalerkinSpectralMethod...........................134 6.3.1StabilityAnalysis...............................137 6.4FractionalSensitivityEquationMethod(FSEM)...................138 6.4.1FSEM(FIVP).................................140 6.4.2FSEM(FPDE)................................141 6.4.3MathematicalFramework:CoupledSystemofTheFPDEandDerivedFSEs142 6.5FractionalModelConstruction............................143 6.5.1ModelError..................................143 6.5.1.1ModelError:Type-I........................144 6.5.1.2ModelError:type-II........................145 6.5.2ModelErrorMinimization:IterativeAlgorithm...............145 6.5.3FractionalModelConstruction:FSEM-basedIterativeAlgorithm.....147 6.6NumericalResults...................................148 6.7ProofofLemmasandTheorems...........................153 6.7.1ProofofLemma6.2.2.............................153 6.7.2ProofofLemma7.3.1.............................154 x 6.7.3ProofofLemma7.3.2.............................155 6.7.4ProofofLemma7.3.5.............................157 6.7.5ProofofLemma7.3.7.............................157 6.7.6ProofofTheStabilityTheorem7.3.9.....................159 CHAPTER7OPERATOR-BASEDUNCERTAINTYQUANTIFICATIONFORSTOCHAS- TICFRACTIONALPDES ........................... 161 7.1Background......................................161 7.2ForwardUncertaintyFramework...........................165 7.2.1FormulationofStochasticFPDE.......................165 7.2.2RepresentationoftheNoise:DimensionReduction.............166 7.2.3InputParametrization.............................168 7.2.4StochasticSampling..............................168 7.3ForwardDeterministicSolver.............................171 7.3.1MathematicalFramework...........................172 7.3.2WeakFormulation..............................175 7.3.3Petrov-GalerkinSpectralMethod.......................176 7.3.4StabilityAnalysis...............................179 7.4NumericalResults...................................180 7.4.1Low-DimensionalRandomInputs......................180 7.4.2Moderate-toHigh-DimensionalRandomInputs...............183 CHAPTER8NONLINEARVIBRATIONOFFRACTIONALVISCOELASTICCAN- TILEVERBEAM:APPLICATIONTOSTRUCTURALHEALTHMON- ITORING .................................... 185 8.1Background......................................185 8.2MathematicalFormulation..............................185 8.2.1NonlinearIn-PlaneVibrationofaVisco-ElasticCantileverBeam......186 8.2.2Viscoelasticity:BoltzmannSuperpositionPrinciple.............188 8.2.3ExtendedHamilton'sPrinciple........................192 8.2.4Nondimensionalization............................198 8.2.5WeakFormulation..............................200 8.2.6AssumedMode:ASpectralGalerkinApproximationInSpace.......201 8.2.7SingleModeApproximation.........................201 8.3LinearizedEquation:DirectNumericalTimeIntegration..............202 8.4PerturbationAnalysisofNonlinearEquation.....................204 8.4.1MethodofMultipleScales..........................204 8.4.1.1Case1:NoLumpedMassAtTheTip...............206 8.4.1.2Case2:LumpedMassAtTheTip.................211 8.5EigenvalueProblemofLinearModel.........................213 CHAPTER9SUMMARYANDFUTUREWORKS .................... 216 9.0.1FutureWorks.................................221 BIBLIOGRAPHY ........................................ 224 xi LISTOFTABLES Table2.1:Conditionnumberoftheresultingassembledglobalmatrixforthetwochoices oflocalbases/testfunctions(left)andlocalbaseswithglobaltestfunctions (right)fordi˙erentnumberofelementsandmodes.................43 Table2.2:CPUtimeofconstructingandsolvingthelinearsystembasedono˙-line retrievalandon-linecalculationofhistorymatrices.................44 Table2.3:Single-BoundarySingularity: L 2 -normerrorintheboundaryandinterior elementsusingPGSEMwithlocalbasis/testfunctions.Here, L b represents thesizeofleftboundaryelement, P b and P I denotethenumberofmodesin theboundaryandinteriorelementsrespectively..................45 Table2.4:Full-BoundarySingularity: L 2 -normerrorintheboundaryelement(BE)and interiorelement(IE)byPGSEMwithlocalbasis/testfunctions.Here, u ext = ¹ 1 x º 3 + 1 x 3 + 2 with 1 = 1 4 ; 2 = 2 3 , L b representsthesizeofleftandright boundaryelements, P b and P I denotethenumberofmodesintheboundary andinteriorelementsrespectively..........................46 Table2.5: L 2 -normerror,usinguniformandnon-uniformgrids.Theexactsingular solutionis u ext = ¹ 1 x º x 1 + with = 1 š 10 ....................48 Table2.6:Fullhistoryfading: L 2 -normerrorusingPGSEMwithlocalbasis/testfunc- tions,where u ext = x 7 x 6 , N el = 19 , P = 6 .The˝rstcolumninthetable showsthenumberoffullyfadedhistorymatrices..................50 Table2.7:Partialhistoryfading: L 2 -normerrorusingPGSEMwithlocalbasis/test functions,where u ext = x 7 x 6 , N el = 19 , P = 6 .The˝rstcolumninthe tablesshowsnumberofpartiallyfadedhistorymatrices..............50 Table3.1:Case-I;PGschemeconvergencestudyin L 1 -norm, L 2 -norm, H 1 -normand ˚ H -norm,where T = 2 ...............................72 Table3.2:Case-II;PGschemeconvergencestudyin L 1 -norm, L 2 -norm, H 1 -normand ˚ H -norm,where T = 2 ...............................73 Table3.3:Case-IandII;collocationschemeconvergencestudyin L 1 -norm,where T = 2 ..73 Table3.4:Multi-termcase;collocationschemeconvergencestudyin L 1 -norm,where T = 2 .........................................73 Table3.5:Case-IV;PGschemeconvergencestudyin L 2 -norm,where T = 2 .........74 xii Table3.6:Case-IandII;PGschemeconditionnumberoftheconstructedlinearsystem, where T = 2 .....................................74 Table3.7:Case-IandII;collocationschemeconditionnumberoftheconstructedlinear system,where T = 2 .................................74 Table3.8:Leftbiaseddistributionfunction;PG(top)andcollocation(bottom)scheme conditionnumberoftheconstructedlinearsystem,where T = 2 ..........75 Table3.9:Symmetricdistributionfunction;PG(left)andcollocation(right)scheme conditionnumberoftheconstructedlinearsystem,where T = 2 ..........76 Table3.10:Rightbiaseddistributionfunction;PG(left)andcollocation(right)scheme conditionnumberoftheconstructedlinearsystem,where T = 2 ..........76 Table4.1:Pseudo-spectralmethod:conditionnumberoftheresultinglinearsystem, Case-I(left),Case-II(right).............................103 Table4.2:Conditionnumberoftheresultinglinearsystem.Thecomparisonbetween fractionalcollocationmethod(employingfractionalinterpolantsinthestrong sense[87]),andpseudo-spectralmethod(employingfractionalinterpolantsin theweaksense)...................................104 Table5.1:PGspectralmethod,CPUtime(inmin)and L 2 -normerrorformulti-dimensional problems......................................121 Table6.1:FractionalmodelconstructionforthetwocasesoffractionalIVP.........151 Table6.2:FractionalmodelconstructionforthetwocasesoffractionalBVP.........151 Table7.1:Thetotalnumberofnodalpointsinrandomspacesampling,usingSmolyak sparsegridgeneratorandfulltensorproductwith10pointsineachdirection...171 xiii LISTOFFIGURES Figure1.1:Classicalvisco-elasticmodelsascombinationofaspring(pureelastic)anda dash-pot(pureviscous)elements.Kelvin-Voigt(top)andMaxwell(bottom) rheologicalmodel..................................4 Figure1.2:Schematicofafractionalmodelelement(ScottBlairelement):inthelimiting casesof = 0 and = 1 ,convergestospringanddash-potelement,respectively.5 Figure1.3:Normalandanomalousdi˙usionprocess:themeanssquaredisplacement ofparticlesinadi˙usiveprocessisnonlinearlyproportionaltotime,i.e. r 2 / D ˝ . = 1 :standarddi˙usion(blueline), > 1 :super-di˙usion(red curve),and < 1 :sub-di˙usion(greencurve)...................7 Figure1.4:Schematicoutlineofthisthesiswithmainfocusondevelopingproperdata- drivencomputationalframeworksforstochasticfractionalPDEs.Distributed- Orderoperatorsareconsideredasanextensionoffractionaloperatorstothe casewherethekerneliscomprisedofadistributionofpower-laws;alsoas expectationoffractionalderivativewithrandomorders.Fractionalsensitivity analysisprovidesatooltostudythesensitivityofmodeloutputwithrespect tofractionalderivativeorders;thus,leadstoamachinelearningtooltocon- structthefractionalmodelsfromavailablesetsofdata.Theyalsointroduce newtypeofoperatorswithlogarithmic-powerlawkernelsinthiscontext. Therandomnessoffractionalordersispropagatedtomodeloutputviaan operator-basedUQframework.Fastnumericalmethodswithspectralrateof convergencearedevelopedtobackupthesimulationsineachcase,wherethe stabilityandconvergencearemathematicallyproveninthediscretefunction spaces.......................................10 Figure1.5:Uncertaintypropagationtowardthemodeloutputquantityofinterestdueto inherentrandomnessofmeasurements,incompletesetsofdata,signi˝cant approximationinmodels,andnumericalerrors.Thegraycloudsshowthe associateuncertaintieswitheachsourceandarrowsshow˛owofinformation betweenthem.Experimentalobservationsfeedintotheconstructionofform ofmathematicalmodelandestimationofitsparameters.Themathematical modelsarethennumericallysolvedandthesimulationresultsareveri˝edand validatedagainbytheexperimentalobservation,makingthetwo-way˛ow backtotheexperiments...............................14 Figure2.1:Domainpartitioning................................24 xiv Figure2.2:Locationofthe(dummy)elementnumber, e ,withrespecttothecurrent element, " .If e >" ,(top),then RL x D L v " k ¹ x º = 0 .If e = " ,(mid- dle),then RL x D L v " k ¹ x º = RL x D x " h ¹ 2 º P k + 1 ¹ x º i .If e <" ,(bottom),then RL x D L v " k ¹ x º = H ¹ " º k ¹ x º ...............................26 Figure2.3:Sparsityoflocalsti˙nessmatrix.........................29 Figure2.4:Theassembledglobalmatrixcorrespondingtoauniformgridwith N el = 9 . Inthisglobalmatrix, M ¹ " º = S ¹ " º M ¹ " º , " = 1 ; 2 ; ; N el ,represents thelocalmatrix,associatedwiththeelement " .To˝llthelower-triangular blockmatrices,weconstructonly ¹ N el 1 º historymatrices ^ S ¹ " º ,where " = 1 ; 2 ;::; N el 1 ,ratherthan N el ¹ N el 1 º 2 matrices...............34 Figure2.5:Historycomputationandretrieval.........................35 Figure2.6:Kernel-basednon-uniformgridintheboundarylayer; L b and N b arethe lengthofandthenumberofelementsintheboundarylayer,respectively....35 Figure2.7:Theassembledglobalmatrixcorrespondingto N el = 11 with N b = 4 non- uniformboundaryelementsand 7 uniforminteriorelements.Inthisglobal matrix, M ¹ " º = S ¹ " º M ¹ " º , " = 1 ; 2 ; ; N el ,representsthelocalmatrix, associatedwiththeelement " .Thelower-triangleconsistsofthreeparts: 1)Thesmallsquare N b ¹ N b 1 º 2 historymatrices(interactionofboundaryele- ments, " = 1 ; 2 ; ; N b ).2)Thebigsquarehistorymatrices(interactionofin- teriorelements, " = N b + 1 ; ; N el ).3)Theskinnyrectangular ¹ N el N b º N b historymatrices(interactionofboundaryelementswithinteriorelements)...37 Figure2.8:Non-uniformgeometricallyprogressivegrid...................38 Figure2.9:Theassembledglobalmatrixcorrespondingto N el = 5 elementswhenglobal testfunctionsareemployed.Inthisglobalmatrix, M ¹ " º = ^ S ¹ " º ^ M ¹ " º , " = 1 ; 2 ; ; N el ,representsthelocalmatrix,associatedwiththeelement " .To˝llthelower-triangularblockmatrices,wemustconstruct N el ¹ N el 1 º 2 historymatrices ^ S ¹ "; e º ...............................42 Figure2.10:PGSEMwithlocalbasis/testfunctions.Plottedistheerrorwithrespectto thepolynomialdegreeofeachelement(spectralorder)..............43 Figure2.11:Schematicofglobalmatricescorrespondingtothecaseofsingularsolutions. (left):leftboundarysingularity,(right):leftandrightboundarysingular- ities. ^ S ¹ bI º , ^ S ¹ Ib º ,and ^ S ¹ bb º denotetheinteractionofboundary/interior, interior/boundaryandboundary/boundaryelements,respectively.........46 xv Figure2.12:InteriorSingularity.(left):exactsolutions,(right):thecorrespondingforce functions......................................47 Figure2.13:InteriorSingularity:PGSEMwithlocalbasis/testfunctions.Plottedisthe errorwithrespecttospectralorderineachelement................47 Figure2.14: Memoryfading:(a)B-Binteraction,thecornerentries(b)B-BandB-Iinteraction, theboundaryentries(c)B-B,B-IandS-Iinteraction,boundaryanddiagonalentries ..49 Figure3.1: SchematicofdistributedfractionalSobolevspace ˚ H¹ R º :(left) ˚ = ¹ max º hence ˚ H¹ R º = H max ¹ R º ;(middle) ˚ de˝nedonacompactsupportin » min ; max ¼ , hence, ˚ H¹ R º˙ H max ¹ R º ;(right) ˚ = ¹ min º ,where ˚ H¹ R º = H min ¹ R º . ..56 Figure3.2: Distributionfunctions:(a) Leftbiased (b) Symmetric (c) Rightbiased .........75 Figure4.1: Initial/Boundaryvalueproblem:conditionnumberoftheresultinglinearsystemfor left-andright-biasednormaldistributions(left),where = 0 : 1 (middle), = 0 : 9 (right),andchoice ¹ ii º ofinterpolationpointsisused. ................98 Figure4.2: Initial/Boundaryvalueproblem:conditionnumberoftheresultinglinearsystemfor di˙erentchoicesofinterpolationpointsfordi˙erentinterpolationparameter,(leftto right) = 1 10 4 , 0 : 1 , 0 : 5 ,and 0 : 9 ,where ˚ ¹ º = ¹ 6 º ¹ 6 º and 2¹ 0 ; 2 º . ......100 Figure4.3: Conditionnumberoftheoriginallinearsystemwith ˚ ¹ º = ¹ 6 º ¹ 6 º (left),andthe pre-conditionedonefor = 1 š 10 (middle),and = 9 š 10 (right). ..........101 Figure4.4: Conditionnumberofthelinearsystemforright-biasednormaldistribution.The originallinearsystem(blacklines)andthepre-conditionedone(redandbluelines) for = 1 š 10 (left), = 5 š 10 (middle),and = 9 š 10 (right). .............101 Figure4.5: Pseudo-spectralmethod: L 2 -normerroroftheapproximatesolution,(left)CaseI (right)CaseII ...................................104 Figure4.6: (1+1)-Dspacedistributed-orderBurgersequation,pseudo-spectralinspaceand 2 nd - orderAdamsBashforthintime.(left):Theleft-andright-biasednormaldistribution functions(mean 1 : 3 and 1 : 7 ,respectivelywithvariance 0 : 1 )inthedistributed-order operator.(middle): L 1 -normerrorv.s. N for = 0 and = 1 .(right): L 1 -norm errorv.s. N for = 1 and = 0 . ...........................107 Figure4.7: ¹ 1 + 2 º -Dtwo-sidedspacedistributed-orderdi˙usionreactionequation,pseudo- spectralinspaceand 2 nd -orderAdamsBashforthintime.(left): L 1 -normerrorv.s. N .(right):Timeevolutionofthesolution. ......................108 Figure5.1:PGspectralmethod,temporalandspatial p -re˝nementfor(1+2)-Dproblem..121 xvi Figure6.1:SchematicofstrategiesinderivingtheweakformofFSEs.(I-1):˝rsttake @ @ q andthenobtaintheweakformulation,fedbystrongsolution u s .(I-2): ˝rsttake @ @ q andthenobtaintheweakformulation,fedbyweaksolution u w . (II):˝rstobtaintheweakformulationandthentake @ @ q ,fedbyweaksolution u w .139 Figure6.2:Schematicoffractionalmodel...........................144 Figure6.3:Schematicofvariationoffractionalmodelbasedon(a)modelerrortype-I and(b)modelerrortype-II.............................144 Figure6.4:Iterativealgorithm:coarsegridsearchingfor ¹ 1 + 1 º -Dparameterspace, where = 0 : 3 and = 0 : 8 ............................146 Figure6.5:PlotofexactfunctionsforcaseIwith š 2 = 0 : 25 and š 2 = 0 : 75 :exact solution u ext (left),exactsensitivity˝eld S u ext ; = @ u ext @ (middle),exact sensitivity˝eld S u ext ; = @ u ext @ (right).......................149 Figure6.6:PlotofexactfunctionsforcaseIIwith š 2 = 0 : 25 and š 2 = 0 : 75 :exact solution u ext (left),exactsensitivity˝eld S u ext ; (middle),exactsensitivity ˝eld S u ext ; (right).................................149 Figure6.7:PGspectralmethod, L 2 -normconvergencestudy: ¹ 1 + 1 º -dFPDEadjointto correspondingFSEswithone-sidedfractionalderivative, k = 1 , š 2 = 0 : 25 , and š 2 = 0 : 75 ,forCaseI(left)andCaseII(right),where N = M ........150 Figure6.8:FractionalmodelconstructionforthecaseFPDE,usingFSEMbasediterative algorithm.Thetruevaluesoffractionalindicesare f ; g = f 0 : 1 ; 1 : 64 g ....152 Figure7.1:Illustrationofsamplingnodalpointsintwo-dimensionalrandomspace,using Smolyaksparsegridgenerator(a) A ¹ 2 ; 2 º ,(b) A ¹ 4 ; 2 º ,(c) A ¹ 6 ; 2 º ;and(d) fulltensorproductrulewith50pointsineachdirection.Thetotalnumberof pointsineachcaseis,25,161,837,and2500,respectively............171 Figure7.2: L 2 -normconvergencerateofMCMandPCMforstochasticfractionalIVP Eqn.(7.48).....................................181 Figure7.3:ExpectationofsolutiontoEqn.(7.49)withuncertainty(standarddeviation) bounds,employingMCSandPCMfor(left) h ¹ t º = t 2 and(right) h ¹ t º = sin ¹ ˇ t º .......................................182 Figure7.4: L 2 -normconvergencerateofMCMandPCMforSFPDEEqn.(7.50)......182 xvii Figure7.5:ExpectationofsolutiontoEqn.(7.51),employingMCSandPCMat t = 0 : 125 ; 0 : 625 ; 1 ...................................183 Figure8.1:In-planelateraldeformationofaslenderisotropiccantileverbeam. u ¹ s ; t º and v ¹ s ; t º aretheaxialandlateraldisplacements,and ¹ s ; t º istherotationangle about z axis.....................................187 Figure8.2:Detailedin-planelateraldeformationofaslenderisotropiccantileverbeam. The˝gureshowstotaldeformationofanarbitrarypoint(theredpoint)as thebeamundergoesdeformation.Thisdeformationiscomprisedoftheaxial displacementofthebeam u ,thelateraldisplacementofbeaminadditionto thebasemotion v + V b ,andthedisplacementduetorotation ..........187 Figure8.3:Deformationofanarbitraryelementofthebeam. CD extends,traverses,and rotatesto C D ...................................188 Figure8.4:Classicalvisco-elasticmodelsasacombinationofspring(purelyelastic)and dash-pot(purelyviscous)elements.Kelvin-Voigt(top)andMaxwell(bottom) rheologicalmodels.................................190 Figure8.5:Power-LawDecay:Timeresponseoflinearfractionallydampedoscillator usingNewmarkand L 1 scheme.Thefractionaldamperhastwoconstants E r c l and asthecoe˚cientandderivativeorderoffractionaloperator.....204 Figure8.6:Freevibrationoftheviscoelasticcantileverbeamwithnolumpedmassat thetip.Therateofdecayofamplitudestronglydependsonthefractional derivativeorder andthecoe˚cient E r .Theleft˝gure(log-linearscale) showstherapidincreaseinamplitudedecayingas isincreasedand E r = 0 : 1 . Theright˝gure(linearscale)showsthephaselag ' ¹ t º ,whereitsincrease ratedecreasesas isincreased...........................207 Figure8.7:Freevibrationoftheviscoelasticcantileverbeamwithnolumpedmassatthe tip.Thisgraphshowssensitivityofthedecayrate ˝ d withrespecttochange of .Increasing when < cr leadstohigherdissipationanddecayrate. Thereversee˙ectisobservedwhen > cr .Bysofteningandhardeningwe re˛ecttotheregionswhereincreasing (introducingextraviscosity)leads tohigherandlowerdecayrate,respectively....................208 Figure8.8:Primaryresonanceoftheviscoelasticcantileverbeamwithnolumpedmass atthetip.Steadystateamplitude(right)anditsbifurcationdiagram(left)by changingthedetuningparameter fordi˙erentvaluesof and E r = 0 : 3 ; f = 1 ..........................................210 xviii Figure8.9:Frequency-Responsecurveforthecaseofprimaryresonanceinthevis- coelasticcantileverbeamwithnolumpedmassatthetip.Eachsub-˝gure correspondstoa˝xedvalueof and f when E r = f 0 : 1 ; 0 : 2 ; ; 1 g .Ase˙ect offractionalelementbecomesmorepronounced,i.e. and E r increase,the curvemovesdownanddrifttoleft.........................211 Figure8.10:The˝rsteigenfunctions, X 1 ¹ s º ,oftheundampedlinearcounterpartofour model.Itisusedasthespatialfunctionsinthesinglemodeapproximation....214 Figure8.11:The˝rsteigenfunctions, X 1 ¹ s º ,oftheundampedlinearcounterpartofour modelwithnolumpedmassatthetip.Itisusedasthespatialfunctionsin thesinglemodeapproximation...........................215 xix CHAPTER1 INTRODUCTION Thescientistdoesnotstudynaturebecauseitisuseful; hestudiesitbecausehedelightsinit,andhedelightsinitbecauseitisbeautiful. HenriPoincaré 1.1FractionalModels Understandingandpredictingbehaviorofnaturalphenomena,andinnovativedesigninten- tionshavealwaysbeenanencouragingmotivationforscientistsprogress.Physicalphenomena, ingeneral,arecharacterizedasasystem,whosebehaviorisunderstoodviacoupleofstatevari- ables,suchasposition/velocityofparticlesthatcollectivelyconstitutethesystem.Mathematical modelsconstructpropertoolstostudytime/spaceevolutionofthestatesofsystem-of-interest, wherethevalidityrangeofsuchmodelsareessentiallyinvestigatedbycomparisonoftheiroutput (simulations)withobservedexperiments.Avastrangeofexperimentalobservations,conducted oncomplexsystems,however,demonstratesthediscrepancyofexistingmathematicalmodelsin properlydescribingandpredictingtheubiquitousanomaliesinbehaviorofsuchsystems,raising thedemandtoaccretemorecapablemodels.Fractionalpartialdi˙erentialequations(FPDEs), asaseamlessextensionoftheirstandardintegerordercounterparts[90,134,146],openupnew possibilitiesfordevelopingrobustmathematicaltoolswiththeabilitytomoreaccuratelypredictthe anomalousbehaviors.TheyputtheexistingPDEsintoasubsetofthislargerfamilyofmathematical models,andarerecentlybeingextensivelystudiedandrecognizedasmosttractablemathematical frameworkfordescriptionofanomalousprocesseswithnonlocalinteractions,self-similarstruc- tures,longmemorydependence,andpower-lawbehavior.Thesecriticalinterpretativefeatures ofcomplexphysicalsystemsareconsolidatedintheorderoffractionalderivatives,resultingina morecompliantsimulationswithexperimentaltests.Examplesofemployingfractionalmodeling stretchovertherangeofapplicationsincluding:viscoelasticityinstructuralvibrations[8],tissue 1 mechanicsandbiologicalphenomena[72,113],non-Newtonian˛uidsandrheology[74,126], non-Browniantransportphenomenainporousanddisorderedmaterials[19,120],non-Markovian processesinmulti-scalecomplex˛uidsandmulti-phaseapplications[76],non-Gaussian(Lévy ˛ights)processesinturbulent˛ows[36,78,153],andearthsystemdynamics[198]. 1.1.1FractionalRheology:ViscoelasticModeling Viscoelasticbodiesisaclassofmaterialswithpropertiesthatexhibitsbothviscousandelastic characteristicswhenundergoingdeformationorloading.Theenergydissipationfeatureofviscous partresultsinnonfully-responsivebehaviorofsuchmaterial.Ingeneral,theresponsecharacteristics ofviscoelasticbodiesareunderstoodbystudyingtheirresponsetostressandstraininputsin andationtest.Wedenoteby J ¹ t º ,creepcompliance,i.e.thestrainresponsetotheunit stepofstress,andby G ¹ t º ,relaxationmodulus,i.e.thestressresponsetoaunitstepofstrain. Thetwofunctions J ¹ t º and G ¹ t º areusuallyreferredtoasthematerialfunctionsofthebody.The constitutiverelationoflinearviscoelasticmodels,i.e.stress-strainrelation,canberepresentedby aVolterraequationthroughBoltzmannsuperpositionprinciple[116].Whenthespecimenisunder loading,thematerialinstantaneouslyreactelasticallyandthen,immediatelystartstorelax,where dissipationtakesplace[59].Thus,asastepincreaseinelongation(fromthestretch = 1 tosome )isimposed,thedevelopedstressinthematerialwillbeafunctionoftimeandthestretch: K ¹ ; t º = G ¹ t º ˙ ¹ e º ¹ º ; (1.1) where G ¹ t º isthereducedrelaxationfunctionand ˙ ¹ e º istheelasticresponse(inabsenceof anyviscosity). ˙ ¹ e º canbealsointerpretedastensilestressresponseinasu˚cientlyhighrate loadingexperiment.TheBoltzmannsuperpositionprinciplestatesthatthestressesfromdi˙erent smalldeformationsareadditive,meaningthatthetotaltensilestressofthespecimenattime t is obtainedfromthesuperpositionofin˝nitesimalchangesinstretchatsomepriortime ˝ j ,givenas 2 G ¹ t ˝ j º @˙ ¹ e º » ¹ ˝ j º¼ @ ¹ ˝ j º .Therefore, ˙ ¹ t º = Õ ˝ j < t G ¹ t ˝ j º @˙ ¹ e º » ¹ ˝ j º¼ @ ¹ ˝ j º ˝ j ˝ j ; (1.2) whereinthelimitingcase ˝ j ! 0 givestheintegralformoftheequationas ˙ ¹ t º = ¹ t G ¹ t ˝ º @˙ ¹ e º » ¹ ˝ º¼ @ @ @˝ d ˝ = ¹ t G ¹ t ˝ º Û ˙ ¹ e º d ˝: (1.3) ExponentialRelaxation,ClassicalModels: Therelaxationfunction G ¹ t º istraditionallyana- lyzedintothesummationofexponentialfunctionswithdi˙erentexponentsandconstantsas G ¹ t º = Í C i e t š ˝ i Í C i : (1.4) Forthesimplecaseofsingleexponentialterm(Maxwellmodel),wehave G ¹ t º = e t š ˝ .Thus,in thecaseofzeroinitialstrainwehave ˙ ¹ t º = ¹ t 0 e t ~ t ºš ˝ E Û " d ~ t ; (1.5) whichsolvestheinteger-orderdi˙erentialequation Û " = 1 E Û ˙ + 1 ˙ ,whererelaxationtimeconstant ˝ = š E ,obtainedfromexperimentalobservations.TheMaxwellmodelisinfactacombinationof pureelasticandviscouselementinseries,seeFig.8.4.Otherdi˙erentcombinationsofpureelastic andviscouselementsinseriesandparallelgiverisetovariousrheologicalmodelswithdistinctive properties,eachofwhichcanbeusedtomodeldi˙erenttypesofmaterial.Thekeyissueisthatthe complexhereditarybehaviorofmaterialrequirescomplicatedcombinationsofelasticandviscous elements,yettheycannotbefullycapturedasthebuildingblockelementsdonotre˛ectany memorydependenceinthematerialresponse.Moreover,thecomplicatedcombinationsintroduce relativelybignumberofmodelparameter,whichadversetheconditionofill-posedinverseproblem ofmodel˝tting(parameterestimation). Power-LawRelaxation,FractionalModels: Themechanicalstressappearedatthedeformation ofviscoelasticmaterialsdecreasesaspower-typefunctionsintime,suggestingthatrelaxationof stressobeysapowerlawbehaviorandtherelaxationtimecannotbedescribedwithsingletime 3 Figure1.1:Classicalvisco-elasticmodelsascombinationofaspring(pureelastic)andadash-pot (pureviscous)elements.Kelvin-Voigt(top)andMaxwell(bottom)rheologicalmodel. scaleanymore[116].Therefore,bylettingthekernelin(8.13)tohavepower-typeform,thetensile stresstakestheformof ˙ ¹ t º = ¹ t g ¹ º ¹ t ˝ º E Û " d ˝ = E g ¹ º ¹ t Û " ¹ t ˝ º d ˝; (1.6) wheretheelasticresponse ˙ ¹ e º = E " .Ifwechoose g ¹ º = 1 ¹ 1 º ,thentheintegro-di˙erential operator(8.16)givestheLiouville-Weylfractionalderivative.Underthehypothesisofcausal histories,statingthattheviscoelasticbodyisquiescentforalltimepriortosomestartinginstant t = 0 ,theequation(8.16)canbewrittenas ˙ ¹ t º = " ¹ 0 + º E g ¹ º t + E g ¹ º ¹ t 0 Û " ¹ t ˝ º d ˝; (1.7) = " ¹ 0 + º E g ¹ º t + E C 0 D t "; = RL 0 D t "; where C 0 D t and RL 0 D t aretheCaputoandRiemann-Liouvillefractionalderivatives,de˝nedlater. Theconstitutiveequation(8.17)istheScottBlairelement[75,138],whichcanbethoughofasan interpolationbetweenapureelastic(spring)andapureviscous(dashpot)elements[115,116,160], seeFig.1.2. Distributed-OrderFractionalModels: Inamoregeneralsense,wherethematerialcontaina spectrumofpower-typerelaxation,thesingleorderfractionalconstitutivemodelcanbeextended tothedistributed-orderone[8,87].Thus,welettherelaxationfunction G ¹ t º in(8.13)notbeonlya singleorderpower-lawasin(8.16),butratherbedistributedoverarange.Thisleadstoadistributed 4 Figure1.2:Schematicofafractionalmodelelement(ScottBlairelement):inthelimitingcasesof = 0 and = 1 ,convergestospringanddash-potelement,respectively. formofconstitutiveequationsexpressedas ¹ max min ¹ º 0 D t ˙ ¹ t º d = ¹ max min ¹ º 0 D t " ¹ t º d ; (1.8) where ¹ º and ¹ º aredistributionfunctionsthatcancon˝nethetheoreticalterminals min , max , min ,and max accordingtothephysicalrealizationofproblem(see[87]andthereferencestherein). Weseethatindistributedorderfractionaloperators,thedi˙erentialorderisdistributedoverarange ofvaluesratherthanbeingjusta˝xedfractionasitisinstandard/fractionalODEs/PDEs.Thus, theyo˙erarigoroustoolformathematicalmodelingofmulti-ratesmulti-physicsphenomena,such asultraslowtosuperdi˙usiveprocesses[85,114,143],anddistributedorderformofviscoelastic models[7,20,21,34]. Ifweletthedistributionfunctionsbedeltafunctions,thedistributedordermodelbecomesthe followingmulti-termmodel: 1 + p ˙ Õ k = 1 a k 0 D k t ! ˙ ¹ t º = c + p " Õ k = 1 b k 0 D k t ! " ¹ t º : (1.9) Wenotethatinviscoelasticmodeling,onecandesigndistinctiverheologicalmodel(constitutive equations)togetdi˙erenttypesofbehaviorbychoosingdi˙erentdistributionfunctions ¹ º and ¹ º .Forexample,byconsidering ¹ º = ¹ º and ¹ º = E 1 ¹ º + E ¹ 0 º ,weshowthat werecoverthefractionalKelvin-Voigtmodelas ˙ ¹ t º = E 1 " ¹ t º + E 0 RL 0 D 0 t " ¹ t º ; (1.10) 5 where 0 2¹ 0 ; 1 º .Otherfractionalviscoelasticmodelssuchasstandardlinearsolidmodeland generalizedMaxwellmodel,shownin(1.11)canalsobeobtainedbychoosingdi˙erentdistribution functionsinthedistributed-ordermodels[10,22,23,97,140,164,195]. J 0 ¹ ˙ + ˝ D ˙ º = + ˝ ˙ D ; (1.11) J 1 ¹ ˙ + ˝ D ˙ º = ˝ ˙ D : StructuralDissipation: Theinternaldissipativemechanismsinstructuraldeformationsare usuallymodeledbyviscoelasticity.Theinteractionofsystemswithanenergysink(e.g.non- Newtonianviscousfrictionandrobbermadefoundations)alsoinducesadissipativebehaviorinthe overallresponseofinteractivesystem.Acommonexampleistheproblemof˛uidsolidinteraction, whichmainlyhappeninstructuresimmersedinviscousmediasuchaswindturbineblades andpipeconveying˛uids[182].Inthecaseofviscoelasticmaterialsinnon-Newtonian˛uid media,theinternalandexternaldissipationmechanismscanbemodeledbyfractionalconstitutive equations.Thus,thecorrespondinggoverningequationtakesthegeneralformof @ 2 @ t 2 + L q t ; x ! u ¹ t ; x º = f ¹ t ; x º ; t ; x 2 ; (1.12) where u ¹ t ; x º : ! R denotesthedisplacement, isthephysicaldomain, L q t ; x isa(nonlinear) operatorwithsetofparameters q ,andf ¹ t ; x º istheexternalforce,whichincludesthedissipative couplingterms.Extensivederivationandinvestigationofnonlinearvibrationofaviscoelastic cantileverbeamisgiveninthefollowingchapters. 1.1.2AnomalousDi˙usion Severalexperimentalobservationshaverevealedthatthestandarddi˙usionprocess,inwhichthe meansquaredisplacementoftheparticlesisproportionaltoelapsedtime,isonlyasubsetof alagerclassi˝cationofphenomena,calledanomalousdi˙usion.Inthiscase,themeansquare displacementchangesnonlinearlyintime(seeFig.1.3)andthus,theanomalousdi˙usiveprocess 6 Figure1.3:Normalandanomalousdi˙usionprocess:themeanssquaredisplacementofparticles inadi˙usiveprocessisnonlinearlyproportionaltotime,i.e. r 2 / D ˝ . = 1 :standarddi˙usion (blueline), > 1 :super-di˙usion(redcurve),and < 1 :sub-di˙usion(greencurve). isdescribedbyapowerlaw,i.e. r 2 / D ˝ ,where D isthedi˙usioncoe˚cientand ˝ istheelapsed time.Thenormaldi˙usionprocesstakesplacefor = 1 .If > 1 ( < 1 ),therateofdi˙usion increases(decreases),resultinginasuper-(sub-)di˙usiveprocess(seee.g.,[91]andreferences therein).Forsuchanomalousprocesses,thenon-Markovianand/ornon-Gaussianjumpdistribution oftheparticles,ismodeledbycontinuoustimerandomwalk(CTRW),wherethecontinuouslimit forsuchmodelsleadstoatemporaland/orspatialfractionaldi˙usionequation[89,123,135], givenasfollows.Let = » 0 ; T ¼» a 1 ; b 1 ¼» a 2 ; b 2 ¼» a d ; b d ¼ bethecomputationaldomain forsomepositiveinteger d ,and u ¹ t ; x ; q º : Q ! R ,where q isthevectorofmodelparameters containingthefractionalindicesandmodelcoe˚cients.Then,thelineartwo-sidedFPDE,subject toDirichletinitialandboundaryconditions,takestheform 0 D t u ¹ t ; x ; q º d Õ j = 1 k j a j D j x j + x j D j b j u ¹ t ; x ; q º = f ¹ t ; x ; q º ; (1.13) inwhich D isafractionaloperator, 2¹ 0 ; 1 º , j 2¹ 1 ; 2 º , k j arerealpositiveconstantcoe˚cients. 1.1.3FractionalCalculus:De˝nitionsandUsefulProperties Webrie˛ygivede˝nitionoffractionalintegralandderivativeofdi˙erentsensesandsomeoftheir usefulproperties.Let ˘ 2 1 ; 1 ¼ .Then,theleft-sidedandright-sidedRiemann-Liouvilleintegral 7 oforder ˙ , n 1 <˙ n , n 2 N ,arede˝ned(seee.g.,[122,134])respectivelyas ¹ RL 1 I ˙ ˘ º u ¹ ˘ º = 1 ¹ ˙ º ¹ ˘ 1 u ¹ s º ds ¹ ˘ s º n ˙ ;˘> 1 ; (1.14) ¹ RL ˘ I ˙ 1 º u ¹ ˘ º = 1 ¹ ˙ º ¹ 1 ˘ u ¹ s º ds ¹ s ˘ º n ˙ ;˘< 1 : (1.15) Thecorrespondingleft-sidedandright-sidedfractionalderivativeoforder ˙ arethende˝ned,as ¹ RL 1 D ˙ ˘ º u ¹ ˘ º = d n d ˘ n ¹ RL 1 I n ˙ ˘ u º¹ ˘ º = 1 ¹ n ˙ º d n d ˘ n ¹ ˘ 1 u ¹ s º ds ¹ ˘ s º ˙ + 1 n ;˘> 1 ; (1.16) ¹ RL ˘ D ˙ 1 º u ¹ ˘ º = d º n d ˘ n ¹ RL ˘ I n ˙ 1 u º¹ ˘ º = 1 ¹ n ˙ º d º n d ˘ n ¹ 1 ˘ u ¹ s º ds ¹ s ˘ º ˙ + 1 n ;˘< 1 ; (1.17) respectively.WerecallausefulpropertyoftheRiemann-Liouvillefractionalderivatives[134]. Assumethat 0 < p < 1 and 0 < q < 1 and g ¹ x L º = 0 x > x L ,then x L D p + q x g ¹ x º = x L D p x x L D q x g ¹ x º = x L D q x x L D p x g ¹ x º : (1.18) Analternativeapproachinde˝ningthefractionalderivativesistobeginwiththeleft-sidedCaputo derivativesoforder ˙ , n 1 <˙ n , n 2 N ,de˝ned,as ¹ C 1 D ˙ ˘ u º¹ ˘ º = ¹ 1 I n ˙ ˘ d n u d ˘ n º¹ ˘ º = 1 ¹ n ˙ º ¹ ˘ 1 u ¹ n º ¹ s º ds ¹ ˘ s º ˙ + 1 n ;˘> 1 ; (1.19) ¹ C ˘ D ˙ 1 u º¹ ˘ º = ¹ ˘ I n ˙ 1 d n u d ˘ n º¹ ˘ º = 1 ¹ n ˙ º ¹ 1 ˘ u ¹ n º ¹ s º ds ¹ s ˘ º ˙ + 1 n ;˘< 1 : (1.20) Byperformingana˚nemappingfromthestandarddomain 1 ; 1 ¼ totheinterval t 2» a ; b ¼ ,we obtain RL a D ˙ t u = ¹ 2 b a º ˙ ¹ RL 1 D ˙ ˘ u º¹ ˘ º ; C a D ˙ t u = ¹ 2 b a º ˙ ¹ C 1 D ˙ ˘ u º¹ ˘ º : (1.21) Hence,indevelopingnumericalmethods,wecanperformtheoperationsinthestandarddomain onlyonceforanygiven ˙ ande˚cientlyutilizethemonanyarbitraryintervalwithoutresorting torepeatingthecalculations.Moreover,thecorrespondingrelationshipbetweentheRiemann- LiouvilleandCaputofractionalderivativesin » a ; b ¼ forany ˙ 2¹ 0 ; 1 º isgivenby ¹ RL a D ˙ t u º¹ t º = u ¹ a º ¹ 1 ˙ º¹ t a º ˙ + ¹ C a D ˙ t u º¹ t º : (1.22) 8 1.2Data-DrivenFractionalModeling Theinherentnon-localnatureoffractionaloperatorsmakesthemexcellentchoiceinaccurately predictingthenon-localityandmemorye˙ectinanomalousbehaviorofcomplexphysicalsystems. Theexponentofsingularpower-lawkernelintheseoperatorsde˝nesthefractionalderivativeorder orfractionalindex,whichisanessentialparameterincharacterizingtheunderlyinganomaly(e.g., theregionofsub-/super-di˙usioninananomaloustransportprocessischaracterizedbytheorders offractionalderivatives,seeFig.1.3).Thevaluesoffractionalordersareintroducedasnewset ofparametersinphysicalsystemsmodeledbyfractionaloperators,andtheirvaluesarestrongly tiedtotheexperimentaldatainpractice.Thesensitivityassessmentoffractionalmodelswith respecttothefractionalindeciscanbuildabridgebetweenexperimentsandmathematicalmodels togearobservabledataviaproperoptimizationtechniques.Thesensitivityanalysisprovides amachinelearningtoolwhichexploitgivenexperimental/syntheticdatatoiterativelyestimate theparametersandthusconstructthefractionalmodel.However,theinherentrandomnessin measuredexperimentaldata,lackofinformationabouttruevaluesofparameters(incompletedata), signi˝cantapproximationsaspartofassumptionsuponwhichthemodelisbuilt,andrandomnature ofquantitiesbeingmodeledpervadeuncertaintyinthecorrespondingmathematicalformulations. Thisrendersthemodelparameters,includingthefractionalindices,randomandthus,thefractional modelstochastic,inwhichthenoveltyistointroducetheorderoffractionalderivativeasrandom variable. WeconsiderstochasticfractionalPDEsanddevelopthefollowingusefulmathematicaland computationalframeworksforthem.Theyaremainlylistedasfollows(seealsoFig.1.4)andare brie˛yintroducedafterwards: ‹ Distributed-Orderfractionaldi˙erentialequations ‹ Fractionalsensitivityanalysiswithapplicationtofractionalmodelconstruction ‹ Forwarduncertaintyquanti˝cationofstochasticfractionalPDEssubjecttoadditivenoise 9 Figure1.4:Schematicoutlineofthisthesiswithmainfocusondevelopingproperdata-drivencom- putationalframeworksforstochasticfractionalPDEs.Distributed-Orderoperatorsareconsidered asanextensionoffractionaloperatorstothecasewherethekerneliscomprisedofadistributionof power-laws;alsoasexpectationoffractionalderivativewithrandomorders.Fractionalsensitivity analysisprovidesatooltostudythesensitivityofmodeloutputwithrespecttofractionalderivative orders;thus,leadstoamachinelearningtooltoconstructthefractionalmodelsfromavailable setsofdata.Theyalsointroducenewtypeofoperatorswithlogarithmic-powerlawkernelsinthis context.Therandomnessoffractionalordersispropagatedtomodeloutputviaanoperator-based UQframework.Fastnumericalmethodswithspectralrateofconvergencearedevelopedtoback upthesimulationsineachcase,wherethestabilityandconvergencearemathematicallyprovenin thediscretefunctionspaces. ‹ Applicationtononlinearvibrationofviscoelasticcantileverbeams Theextensivediscussionondevelopingandimplementationofeachframeworkisgivenlaterinthe correspondingchapters. 1.2.1DistributedOrderDi˙erentialEquations Thereisarapidlygrowinginterestintheuseoffractionalderivativesintheconstructionof mathematicalmodels,whichcontaindistributedordertermsoftheform ¹ 2 1 ˚ ¹ º a D t u ¹ t º d = f ¹ t º ; t > a ; (1.23) inthe˝eldofuncertaintyquanti˝cationastheinherentuncertaintyofexperimentaldatacanbe directlyincorporatedintothedi˙erentialoperators.Inthissetting,distributedorderderivative isconsideredastheexpectationoffractionalderivateofrandomorder ¹ ! º withrespecttothe 10 probabilitydensityfunction ˚ ¹ º ,where ! isthenotionofrandomnessintheparameter .This providesanewoperatorinassessingtheuncertaintiesassociatedwiththerandomnessofderivative orderinthecontextofstochasticfractionalmodeling;see[6,12,47,87,114,154]forsomework onnumericalmethods.Wenotethatwhilenumericaltreatmentoffractionaloperatorarecostly duetonon-localkernel,thedistributedorderoperatorswillexcessivelyincreasethecomputational expensebyrequiringadditionaldiscretizationofintegralinthedistributedorder.Mostofthe numericalstudieshavefollowedatwo-stagesapproach,whereinthe˝rststage,thedistributed orderdi˙erentiationtermwasapproximatedusingaquadraturerule,andinthesecondstage,a suitablemulti-termnumericalmethodwasemployed.Wedeveloptwospectrally-accurateschemes totreatlinear/nonlineardistributedorderfractionaldi˙erentialequations,whichareconstructed basedontherecentlydevelopedspectraltheoryforfractionalSturm-Liouvilleproblemsin[186]. Thelistofmaincontributionsarelistedbelow: ‹ Introducingdistributedsobolevspacesforthe˝rsttimeintheliterature ‹ Developingspectrally/exponentiallyaccuratequadraturerulefordistributed-orderderivatives ‹ DevelopingPetrov-Galerkingspectralmethodandfractionalcollocationmethod ‹ PerformingstabilityanderroranalysisofPGscheme ‹ Extensiontotemporally-distributedorderPDEs 1.2.2FractionalSensitivityEquationMethod:ApplicationtoModelConstruction Weextendthecontinuumderivativetechniquetodevelopafractionalsensitivityequationmethod (FSEM)inthecontextoffractionalpartialdi˙erentialequations(FPDEs).Wethen,constructan iterativealgorithminordertoexploittheobtainedsensitivity˝eldinparameterestimation.Inthis setting,weintroduceanewfractionaloperator,associatedwiththelogarithmic-powerlawkernel, whichtobestofourknowledgehasbeenpresentedforthe˝rsttimehereinthecontextoffractional sensitivityanalysis.Thekeypropertyofderivedfractionalsensitivityequations(FSEs)isthatthey 11 preservethestructureoforiginalFPDE.Thus,similardiscretizationschemeandforwardsolvercan bereadilyappliedwithminimalrequiredchanges.Byextendingthemathematicalframeworkin [145]andaccommodatingextrarequiredregularityintheunderlyingfunctionspaces,weformulate anumericalschemeinsolvingcoupledsystemofFPDEandadjointFSEs.Weprovethatthe coupledsystemismathematicallywell-posed,furtherdevelopafastsolvertoe˚cientlysolve theequationsandperformthestabilityanderroranalysisoftheproposednumericalscheme. Moreover,webuildourmachinelearningtoolbasedonthedevelopedFSEM.Weestimatethe fractionalindicesandthus,constructthefractionalmodelfromavailableexperimentaldatainan inverseproblemsetting.Theoptimizationproblemisformulatedbyde˝ningobjectivefunctions astwotypesofmodelerrorthatmeasuresthedi˙erenceincomputedoutput/inputoffractional modelwithtrueoutput/inputinanL2-normsense.Theparametersareobtainedbyminimizing theerrorsviaagradient-basedminimizer,whichusesthedevelopedFSEM.Wenotethatgenerally theinverseproblemofparameterestimationisanill-posedproblem[25].Bynumericalevaluation ofintroducedmodelerrorsonacoarsegridovertheparameterspace,weshowthatthereexistsa uniqueminimumintheobjectivefunction,whichleadstouniquevaluesoffractionalorders and .Thelistofmaincontributionsarelistedbelow: ‹ Sensitivityanalysisoffractionalmodelswithrespecttoderivativeorders ‹ Anewfractionaloperatorwithlogarithmic-powerlawkernel ‹ Accommodatingextraregularityinfunctionspacesduetonewoperator ‹ PGspectralmethodtosolvecoupledsystemofFPDEandadjointFSEs ‹ Modelerrorsasobjectivefunctionsinparameterestimation ‹ Fractionalmodelconstruction 12 1.2.3OperatorBasedUncertaintyQuanti˝cation(UQ)inStochasticFPDEs Inordertoassesstheuncertaintyinoutputoffractionalmodelassociatedwiththerandomness ofmodelparametersincludingfractionalorderderivatives,wedevelopanoperator-basedcom- putationalforwardUQframeworkinthecontextofstochasticfractionalPDEs.Assumingthe mathematicalmodelunderconsiderationiswell-posedandaccountsinprincipleforallfeaturesof underlyingphenomena,weidentifythreemainsourcesofuncertainty,i)parametricuncertainty, includingfractionalindicesasnewsetofrandomparametersappearedintheoperator,ii)addi- tivenoises,whichincorporatesallintrinsic/extrinsicunknownforcingsourcesaslumpedrandom inputs,andiii)numericalapproximations(seealsoFig.1.5).Unliketheclassicalapproachin modelingrandominputs,whichconsidersthemassomeidealizeduncorrelatedprocesses(white noises),wemodeltherandominputsasmore/fullycorrelatedrandomprocesses(colorednoises), andparametrizethemviaKarhunen-Loève(KL)expansionby˝nite-dimensionalnoiseassump- tion.Thisyieldstheproblemin˝nitedimensionalrandomspace.Topropagatetheparametric uncertaintiesintothesystemresponse,weemployMonteCarlosampling(MCS)andahigh-order probabilisticcollocationmethods(PCM).MCSenjoysfrombeingembarrassinglyparallelizable andcanbeimplementquitereadilyonhighdimensionalrandomspacesbuthasslowrateofconver- gence.PCMusestheideaofinterpolation/collocationintherandomspacesandlimitsthesample pointstoane˚cientsubsetofrandomspace.ComparetoMCS,PCMhasthebene˝tofeasily samplingatnodalpointsandtherateofconvergenceisrelativelyhigher.Ineachsimulationof stochasticmodel,weneedtosolvethedeterministiccounterpartinthephysicaldomain,forwhich weformulateafastandstableforwardsolverbydevelopingahigh-orderPetrov-Galerkin(PG) spectralmethod.Thelistofmaincontributionsarelistedbelow: ‹ Renderingthefractionalderivativesasrandomoperator ‹ Operator-BasedUQframework ‹ ProbabilisticcollocationmethodandMonteCarlosimulation 13 Figure1.5:Uncertaintypropagationtowardthemodeloutputquantityofinterestduetoinherent randomnessofmeasurements,incompletesetsofdata,signi˝cantapproximationinmodels,and numericalerrors.Thegraycloudsshowtheassociateuncertaintieswitheachsourceandarrows show˛owofinformationbetweenthem.Experimentalobservationsfeedintotheconstructionof formofmathematicalmodelandestimationofitsparameters.Themathematicalmodelsarethen numericallysolvedandthesimulationresultsareveri˝edandvalidatedagainbytheexperimental observation,makingthetwo-way˛owbacktotheexperiments. ‹ PGspectralmethodasdeterministicforwardsolver 1.3Single/Multi-DomainNumericalMethods Theanalyticalsolutionoffractionalmodelsaremainlylimitedtoveryspeci˝ccases,andthus, inalmostallpracticalproblems,weneedtoformulateapropernumericalscheme.Akeychallenge indevelopingsuchscheme,however,istheinherentnon-localityoffractionaloperators,which imposesextracomputationalcosts.Thisbecomesevenmoreimportantasdata-infusedinverse problemofmodelconstructionanduncertaintyframeworksusuallyinstructseveralnumerical simulationoffractionalmodels,predominatingtheurgeofe˚cientnumericalmethodsforfractional di˙erentialequations(FDEs).Theimmanentpower-lawsingularityofthekernel,whichessentially transmitstotheoutputofmodel,alsodemandanumericalmethodthatcancapturetheexisting singularityofsolutionwhileyieldingafast,accurate,andstablescheme.Overthepasttwo decades,anextensiveamountofworkhasbeendonedevelopingnumericalschemesforfractional di˙erentialequationssuchasvariationaliterationmethod[71],homotopyperturbationmethod [162],Adomian'sdecompositionmethod[73],homotopyanalysismethod[68]andcollocation 14 method[136].Whilemostoftheattentionhasbeendevotedtothe˝nitedi˙erencemethods (FDMs),[33,48,65,70,93,103,111,112,121,148,158,159,167,172,192,196]with˝xed algebraicaccuracyandsigni˝cantmemoryallocationandhistorycalculation,lesse˙orthasbeen putindevelopingglobalmethodssuchasspectralschemesfordiscretizingFPDEs,seee.g., [28,39,83,84,99,100,103,136,158,170]. 1.3.1Petrov-GalerkinSpectralMethodandSpectralCollocationMethod TwonewspectraltheoriesonfractionalSturm-Liouvilleproblems(FSLPs)havebeenrecently developedbyZayernourietal.in[183,186].Thisapproachfractionalizesthewell-known theoryofSturm-Liouvilleeigen-problems,wheretheexpliciteigenfunctionsofFSLPsareana- lyticallyobtainedintermsof Jacobipoly-fractonomials .Recently,in[185,187,188],Jacobi poly-fractonomialsaresuccessfullyemployedindevelopingaseriesofhigh-orderande˚cient Petrov-GalerkinspectralanddiscontinuousspectralelementmethodsofGalerkinandPetrov- GalerkinprojectiontypeforfractionalODEs. Wedevelopdi˙erenthigh-orderaccuratespectralschemestonumericallysolvelinear/non-linear single-orderanddistributedorderFPDEs.Petrov-Galerkin(PG)spectralmethodsapproximatethe solutionbyusingamodalexpansionintheweakformofproblem,wherethemodesarecalled basis/trialfunctionsandweakformisobtainedviainnerproductionbypropertestfunctions.We developaPGspectralmethodbyfollowing[186]andemployingJacobipoly-fractonomialsof˝rst andsecondkindastemporalbasisandtestfunctions,giveninthestandarddomain 1 ; 1 ¼ as ¹ 1 º P n ¹ x º = ¹ 1 + x º P ; + n 1 ¹ x º ; n = 1 ; 2 ; (1.24) ¹ 2 º P k ¹ x º = ¹ 1 x º P ; k 1 ¹ x º ; k = 1 ; 2 ; (1.25) respectively,where > 0 ,and P ; n 1 ¹ x º and P ; k 1 ¹ x º denoteJacobipolynomials.Weseethat theyhavethepropertytovanishattheleftandrightboundaries,respectively,i.e. ¹ 1 º P n 1 º = 15 ¹ 2 º P k ¹ 1 º = 0 .WelaterprovethattheirRL-fractionalderivativespreservestructure,i.e. RL x D ˙ 1 ¹ 2 º P n ¹ x º = ¹ n + º ¹ n + ˙ º ¹ 2 º P ˙ n ¹ x º ; (1.26) RL 1 D ˙ x ¹ 1 º P n ¹ x º = ¹ n + º ¹ n + ˙ º ¹ 1 º P ˙ n ¹ x º where ˙> 0 .ExtensivepropertiesofJacobipoly-fractonomialscanbefoundin[186].Wealso notethatJacobipoly-fractonomialsarenon-polynomialfunctions,comprisedofafractionalpart multipliedbyapolynomial,wherethefractionalexponentintheformercanplaytheroleofa tunningknobtoaccuratelycapturethesolutionsingularity,whilethelatterapproximatessmooth partofsolution.IndevelopingPGspectralmethodforproblemsinvolvingderivativesinspace direction,weadditionallyconsiderLegendrepolynomialsasspatialbasis/testfunctions.Whilethe nonlocalnatureofthefractionaloperatorsgenerallyleadstonon-symmetricfulllinearsystem,we symmetrizethecorrespondingmass/sti˙nessmatricesinourmethodbyembeddingsmartchoices ofcoe˚cientsintheconstructionsoffunctionspaces.Thisfurtherhelpsusformulateafast solvertoobtainthesolutionofresultinglinearsystem.TheexcellenceofdevelopedPGmethod becomemorepronouncedwhenusedininverseproblemofdata-drivenmodelconstructionand alsouncertaintyquanti˝cationframeworks.Thenumericalanalysisandcomputerimplementation ofeachPGspectralmethodisdiscussedindetailinthecorrespondingchapter. Spectralmethodsgenerallyleadtohigharithmeticcomplexityintreatingnonlinearitydueto crosstermsofmodalexpansion.Forfasttreatmentofnonlinearandmulti-termfractionalPDEs, anewspectralmethod,called fractionalspectralcollocationmethod ,isdevelopedin[189].This newclassofcollocationschemesintroducesanewfamilyoffractionalLagrangeinterpolants, mimickingthestructureofJacobipoly-fractonomials.Theyaregivenas h j ¹ ˘ º = ˘ x 1 x j x 1 N Ö k = 1 k , j ˘ x k x j x k ; j = 2 ; 3 ; ; N ; (1.27) where x j 'saretheinterpolationpointsand istheparametertobeadjustedtocapturesolution singularity.WenotethatfractionalLagrangeinterpolantssatisfytheKroneckerdeltaproperty,i.e., h j ¹ ˘ k º = jk ,atinterpolationpoints,howevertheyvaryasapoly-fractonomialbetweeninterpola- 16 tionpoints.Thismakesitpossibletoe˚cientlytreatthealgebraicanddi˙erentialnonlinearities suchasinnonlinearreactiondi˙usionandBurger'sequations.Weextendtheexistingsettingin [189]todevelopaspectralcollocationmethodfordistributed-orderdi˙erentialequations.We properlymodifythenumberofinterpolationpointsinconstructionofinterpolantstoaccommodate enoughregularityintheapproximationsolutionandderivethefractionaldi˙erentiationmatrices fordi˙erentrangesofderivativeorder. Torelaxtherequiredhighregularityofsolutioninthestrongformofproblem,yetpreserving spectralrateofconvergence,wecombinethetwomodalandnodalexpansionmethods,i.e.PG spectralandcollocationmethodsandthus,developapseudo-spectralmethod.Weconstruct twoseparatesetsoffractionalLagrangeinterpolantsof˝rstandsecondkindasbasisandtest functions,respectively,andplugintotheweakformoftheproblemtoobtainthecorresponding weakdistributeddi˙erentiationmatrices .Wefurtherstudythee˙ectofdistributionfunction andinterpolationpointsontheconditionnumberoftheresultinglinearsystemandalsodesign distributedpre-conditioners,basedonthedistributionfunction.Weshowthebetterconditioning oftheresultinglinearsystemcomparetothecasethatwesolelyusefractionalspectralcollocation method,whichemployssimilarexpansionsbutinthestrongformofproblem. 1.3.2Petrov-GalerkinSpectralElementMethod Almostallnumericaltimeintegratorsthatmarchintime(e.g.˝nitedi˙erencemethod)instruct solvingtheresultingboundaryvalueproblemineachtimestep.Theoverallconvergenceoftime integrationandcomputationalcostarethenboundedbythespatialsolver,requiringaccurateandfast schemesinspatialdirections.Theexistingcomplexityingeometryofspatialdomainoverwhich high-dimensionalboundaryvalueproblemsarede˝nedusuallyimposesspeci˝cregularity/property overtheunderlyingfunctionspaces,whichmakesexpansionoversingledomainapproximation impracticaltouse.Acommonexampleisthee˙ectofshocksingularitiesinthreedimensional˛uid ˛ows.Suchcomplexitycanbeovercomeviadomaindecompositionintosub-domainswithsimpler geometries,andthuslessregularities,wheretheuseofhigh-orderaccuratemethodssuchasspectral 17 elementmethod(SEM)isfeasible.TheSEMdiscretizationhasthebene˝tofdomaindecomposition intonon-overlappingelements,wherehigh-orderapproximationswithineachelementyielda fastrateofconvergenceeveninthecasesofnon-smoothand/orrapidtransientsinthesolution. Therefore,atractablecomputationalcostofthemethodcanbeachievedbyasuccessfulcombination of h-re˝nement ,wherethesolutionisrough,and p-re˝nement ,wherethesolutionissmooth.We notethattheapproximatesolutioninsideeachelementinSEMisexpandedusinglocallyde˝ned basisfunctions.ThislocalitybecomesaseriouschallengeindevelopingSEMforfractional operatorsastheirnon-localkernelrequiresagloballyde˝nitionofsolutionapproximation.This leadstoconstructionofextra history matrices,whichdonothappentobeneededininteger-order problems.Wedevelopanewhigh-order C 0 -continuousPetrospectralelementmethod foraone-dimensionalspace-fractionalHelmholtzequationoffractionalorder( 1 < 2 ),subject tohomogeneousboundaryconditions.Wecanusethestandardpolynomialmodalbasisfunctions intheweakformulationofproblembytransferringthefractionalportionofderivativeorderonto somepropernon-polynomialtestfunctions.Wecomputeallelementalmatrices,andthenformulate anewnon-localassemblingproceduretoconstructthegloballinearsystemfromtheelemental mass,sti˙ness,andhistorymatrices.Weshowthattheproposedschemescanaccuratelycapture boundaryandinteriorsingularitiesofsolutionbyminimalnumberofdomaindecompositions. 18 CHAPTER2 APETROV-GALERKINSPECTRALELEMENTMETHODFORFRACTIONAL ELLIPTICPROBLEMS 2.1Background Anumberoflocalnumericalmethods,prominently˝nitedi˙erencemethods(FDMs),have beendevelopedforsolvingfractionalpartialdi˙erentialequations(FPDEs)[33,48,65,70,93,103, 111,112,121,148,158,159,167,172,191,196].FixandRoop[55]developedthe˝rsttheoretical frameworkfortheleast-square˝niteelementmethod(FEM)approximationofafractional-order di˙erentialequation,whereoptimalerrorestimatesareprovenforpiecewiselinearelements. However,Roop[139]latershowedthatthemainhurdletoovercomeintheFEMisthenon-local natureofthefractionaloperator,whichleadstolargedensematrices;heshowedthateventhe constructionofsuchmatricespresentsdi˚culties.ErvinandRoop[52]presentedatheoretical frameworkforthevariationalsolutionofthesteadystatefractionaladvectiondispersionequation basedonFEMandprovedtheexistenceanduniquenessoftheresults.Jinetal.[80]proved theexistenceanduniquenessofaweaksolutiontothespace-fractionalparabolicequationusing FEM;theyshowedanenhancedregularityofthesolutionandderivedtheerrorestimatefor bothsemidiscreteandfullydiscretesolution.Well-posedness,regularityoftheweaksolution, stabilityofthediscretevariationalformulationanderrorestimateoftheFEMapproximationwere investigatedforfractionalellipticproblemsin[79].WangandYang[168]generalizedtheanalysis tothecaseoffractionalellipticproblemswithvariablecoe˚cient,analyzedtheregularityof thesolutioninH Ü olderspaces,andestablishedthewell-posednessofproposedPetrov-Galerkin formulation.WangandZhang[170]developedahigh-accuracypreservingspectralGalerkin methodfortheDirichletboundary-valueproblemofone-sidedvariable-coe˚cientconservative fractionaldi˙usionequations.Wangetal.[171]laterusedthediscontinuousPetrov-Galerkin frameworktodevelopaPetrov-GalerkinFEMforaclassofvariable-coe˚cientconservativeone- 19 dimensionalFPDEs,wheretheyalsoprovedtheerrorestimatesofthescheme.Moreover,Wanget al.[169]developedanindirectFEMfortheDirichletboundary-valueproblemsofCaputoFPDEs showingthereductioninthecomputationalworkfornumericalsolutionandmemoryrequirements. Therehasbeenrecentlymoreattentionande˙ortputondevelopingglobalandhigh-order approximations,whicharecapableofe˚cientlycapturingtheinherentnon-locale˙ects.ACheby- shevspectralelementmethod(SEM)forfractional-ordertransportwasadoptedbyHanert[67]and lateron,theideaofleast-squareFEMwasextendedtoSEMbyCarella[35].Morerecently,Deng andHesthevan[44]andXuandHesthaven[178]developedlocaldiscontinuousGalerkin(DG) methodsforsolvingspace-fractionaldi˙usionandconvection-di˙usionproblems. TwonewspectraltheoriesonfractionalandtemperedfractionalSturm-Liouvilleproblems(TF- SLPs)havebeendevelopedbyZayernourietal.in[183,186].Thisapproach˝rstfractionalizes andthentempersthewell-knowntheoryofSturm-Liouvilleeigen-problems.Theexpliciteigen- functionsofTFSLPsareanalyticallyobtainedintermsof temperedJacobipoly-fractonomials . Thesepoly-fractonomialshavebeensuccessfullyemployedindevelopingaseriesofhigh-order ande˚cientPetrov-Galerkinspectralanddiscontinuousspectralelementmethods[184,187,190]. In[188],ZayernouriandKarniadakisdevelopedaspectralandspectralelementmethodforFODEs withanexponentialaccuracy.TheyalsodevelopedahighlyaccuratediscontinuousSEMfortime- andspace-fractionaladvectionequationin[187].Dehghanetal.[42]consideredLegendreSEM inspaceandFDMintimeforsolvingtime-fractionalsub-di˙usionequation.Su[157]provideda parallelspectralelementmethodforthefractionalLorenzsystemandacomparisonofthemethod withFEMandFDM. TheSEMdiscretizationhasthebene˝tofdomaindecompositionintonon-overlappingelements, whichpotentiallyprovideageometrical˛exibility,especiallyforadaptivityaswellascomplex domains.Moreover,high-orderapproximationswithineachelementyieldafastrateofconvergence eveninthecasesofnon-smoothand/orrapidtransientsinthesolution.Therefore,atractable computationalcostofthemethodcanbeachievedbyasuccessfulcombinationof h-re˝nement , wherethesolutionisrough,and p-re˝nement ,wherethesolutionissmooth. 20 Inthischapter,weconsidertheone-dimensionalspace-fractionalHelmholtzequationoforder 2¹ 1 ; 2 ¼ ,subjecttohomogeneousboundaryconditions.Weformulateaweakform,inwhich thefractionalportion 2¹ 0 ; 1 ¼ istransferedontosomeproperfractionalordertestfunctions viaintegration-by-parts.Thissettingenablesustoemploythestandardpolynomialmodalbasis functions,usedinSEM[81].Subsequently,wedevelopanew C 0 -continuousPetrov-Galerkin SEM,followingtherecentspectraltheoryoffractionalSturm-Liouvilleproblem,wherethetest functionsareofJacobipoly-fractonomialsofsecondkind[186].Weinvestigatetwodistinct choicesofbasis/testfunctions:i) local basis/testfunctions,andii) local basiswith global test functions,whichenablestheconstructionofelementalmass/sti˙nessmatricesinthestandard domain 1 ; 1 ¼ .Weexplicitlycomputetheelementalsti˙nessmatricesusingtheorthogonality ofJacobipolynomials.Moreover,wee˚cientlyobtainthenon-local(history)sti˙nessmatrices, inwhichthenon-localityispresented analytically .Ononehand,weformulateanew non-local assemblingprocedure inordertoconstructthegloballinearsystemfromthelocal(elemental) mass/sti˙nessmatricesandhistorymatrices.Ontheotherhand,weformulateaprocedurefor non- localscattering toobtaintheelementalexpansioncoe˚cientsfromtheglobaldegreesoffreedom. Wedemonstratethee˚ciencyofthePetrov-Galerkinmethodsandshowthatthechoiceoflocal bases/testfunctionsleadstoabetteraccuracyandconditioning.Moreover,foruniformgrids,we computethehistorymatriceso˙-line.Thestoredhistorymatricescanberetrievedlaterinthe constructionofthegloballinearsystem.Weshowthegreatimprovementinthecomputational costbyperformingtheretrievalprocedurecomparedtoon-linecomputation.Wealsointroducea non-uniform kernel-based gridgenerationinadditionto geometricallyprogressive gridgeneration approaches.Furthermore,weinvestigatetheperformanceofthedevelopedschemesbyconsidering twocasesofsmoothandsingularsolutions,wherethesingularitycanoccuratboundarypoints ortheinteriordomain.Finally,westudythee˙ectofhistoryfadingviaasystematicanalysis, whereweconsiderthehistoryuptosomespeci˝celementandlettherestfade.Thisresultsinless computationalcost,whileweshowthattheaccuracyisstillpreserved.Themaincontributionsof thisworkarelistedinthefollowing: 21 ‹ Developmentofanewfastandaccurate C 0 -continuousPetrov-Galerkinspectralelement method,employinglocalbasis/testfunctions,wherethetestfunctionsareJacobipoly- fractonomials. ‹ Reducingthenumberofhistorymatrixcalculationfrom N el ¹ N el 1 º 2 to ¹ N el 1 º forauniformly partitioneddomain. ‹ Analyticalexpressionsofnon-locale˙ectsinuniformgridsleadingtofastcomputationof thehistorymatrices. ‹ Anewprocedurefortheassemblyofthegloballinearsystem. ‹ Performingo˙-linecomputationofhistorymatricesandon-lineretrievalofthestoredma- trices. ‹ Non-uniformernel-based"gridgenerationforresolvingsteepgradientsandsingularities. Theorganizationofthischapterisasfollows:section2.2providesproblemde˝nition,derivation oftheweakformandexpressionsforthelocalbasisandlocal/globaltestfunctions.Insection 2.3,wepresentaPetrov-Galerkinmethod,employingthelocalbasis/testfunctionsinadditionto formulatingthenon-localassemblingandnon-localscatteringprocedures,followedbyadiscussion onhowtocomputethehistorymatrices o˙-line .Wealsopresentthetwonon-uniformgrid generationapproaches.Insection2.4,wepresentaPetrov-Galerkinmethod,employingthelocal basiswithglobaltestfunctions.Insection2.5,wedemonstratethecomputationale˚ciencyof thetwoproposedschemesbyconsideringseveralnumericalexamplesofsmoothandsingular solutions.Finally,weperformtheo˙-linecomputationandretrievalprocedureofhistorymatrices andasystematichistoryfadinganalysis. 22 2.2De˝nitions 2.2.1ProblemDe˝nition WestudythefollowingfractionalHelmholtzequationoforder = 1 + , 2¹ 0 ; 1 ¼ : RL 0 D x u ¹ x º u ¹ x º = f ¹ x º ; 8 x 2 (2.1) u ¹ 0 º = u ¹ L º = 0 ; 8 x 2 @ ; (2.2) where = » 0 ; L ¼ .Wemultiplybothsidesof(2.1)bysomepropertestfunction v ¹ x º andtransfer thefractionalportion, ,ofderivativeontothetestfunctionbytakingthefractionalintegration- by-parts,followingtheproofofLemma2.4in[87].Therefore,weobtainthefollowingbilinear form: a ¹ u ; v º = l ¹ v º ; (2.3) inwhich a ¹ u ; v º = du dx ; RL x D L v u ; v ; (2.4) l ¹ v º = f ; v ; (2.5) where ¹ ; º denotestheusual L 2 innerproduct. 2.2.2LocalBasisFunctions Wepartitionthecomputationaldomaininto N el non-overlappingelements e = » x e 1 ; x e ¼ such that = [ Nel e = 1 e ,seeFig.2.1.Therefore,thebilinearform(2.4)canbewrittenas a ¹ u ; v ºˇ a ¹ u ; v º = N el Õ e = 1 du ¹ e º N dx ; RL x D L v e N el Õ e = 1 u ¹ e º N ; v e ; (2.6) where u ¹ e º N istheapproximationsolutionineachelement,whichisgivenby u ¹ e º N ¹ x º = P Õ p = 0 ^ u ¹ e º p p ¹ x º ; x 2 e ; (2.7) 23 Figure2.1:Domainpartitioning usingthebasisfunctions p ¹ x º 'sintheelement.Thus,theapproximatedsolutionoverthewhole domaincanbewrittenas u ˇ u ¹ x º = N el Õ e = 1 P Õ p = 0 ^ u ¹ e º p p ¹ x º : (2.8) Themodalbases p ¹ x º arede˝nedinthestandard(reference)domain[-1,1]as: p ¹ x ¹ ºº = 8 > > > > > > > >< > > > > > > > > : 1 2 ; p = 0 ; ¹ 1 2 º¹ 1 + 2 º P 1 ; 1 p 1 ¹ º ; p = 1 ; 2 ; ; P 1 ; 1 + 2 ; p = P : (2.9) Thechoiceofbasisfunctionsisthesameasinstandardspectralelementmethodsforinteger-order PDEs(seee.g.,[81]). 2.2.3TestFunctions:Localvs.Global Wechoosetwotypesoftestfunctions v :i) local testfunctions,andii) global testfunctions,given for " = 1 ; 2 ; ; N el asfollows: v local k ¹ x º = v " k ¹ x º = 8 > > > >< > > > > : ¹ 2 º P k + 1 ¹ x " º ; 8 x 2 " ; 0 ; other w ise ; ; k = 0 ; 1 ; ; P ; (2.10) 24 inwhich ¹ 2 º P k + 1 ¹ x " º representstheJacobi poly-fractonomial ofsecondkind,de˝nedinthe correspondingintervals " = » x " 1 ; x " ¼ ;and v g lobal k ¹ x º = v " k ¹ x º = 8 > > > >< > > > > : ¹ 2 º P k + 1 ¹ x 1 ˘ " º ; 8 x 2» 0 ; x " ¼ ; 0 ; other w ise ; k = 0 ; 1 ; ; P ; (2.11) where ¹ 2 º P k + 1 ¹ x 1 ˘ " º representstheJacobi poly-fractonomial ofsecondkind,de˝nedinthecorre- spondingintervals » 0 ; x " ¼ .ThestructureofJacobipoly-fractonomialaregivenin(1.25).Itshould benotedthatforeachelement " ,thecorrespondinglocaltestfunctionhasnonzerovalueonlyin theelementandvanisheselsewhere,unlikethecorrespondingglobaltestfunction,whichvanishes onlywhere x > x " . 2.3Petrov-GalerkinMethodwithLocalTestFunctions WedevelopthePetrov-Galerkinschemebysubstituting(2.7)and(2.10)into(2.6)toobtain: N el Õ e = 1 P Õ p = 0 ^ u ¹ e º p d p ¹ x º dx ; RL x D L v " k ¹ x º e N el Õ e = 1 P Õ p = 0 ^ u ¹ e º p p ¹ x º ; v " k ¹ x º e = f ; v " k ¹ x º ;" = 1 ; 2 ; ; N el ; k = 0 ; 1 ; ; P : (2.12) Sincethelocaltestfunctionvanishes 8 x 2 e , " ,wehave N el Õ e = 1 P Õ p = 0 ^ u ¹ e º p p ¹ x º ; v " k ¹ x º e = P Õ p = 0 ^ u ¹ " º p p ¹ x º ; v " k ¹ x º " ; f ; v " k ¹ x º = f ; v " k ¹ x º " : Moreover,forevery " ,theright-sidedfractionalderivative, RL x D L v " k ¹ x º = 1 ¹ 1 º d dx ¹ L x v " k ¹ s º ¹ s x º ds ; x 2 e ; istakenfrom x 2 e to x = L ,where e = 1 ; 2 ; ; N el throughthesummationovertheelements and s variesfrom x 2 e to L .Thelocaltestfunctionvanishes 8 x 2 e , " ,thusif e >" ( x > x " ,seeFig.2.2top),then RL x D L v " k ¹ x º = 1 ¹ 1 º d dx ¹ L x 0 ¹ s x º ds = 0 ; (2.13) 25 Figure2.2:Locationofthe(dummy)elementnumber, e ,withrespecttothecurrentelement, " .If e >" ,(top),then RL x D L v " k ¹ x º = 0 .If e = " ,(middle),then RL x D L v " k ¹ x º = RL x D x " h ¹ 2 º P k + 1 ¹ x º i . If e <" ,(bottom),then RL x D L v " k ¹ x º = H ¹ " º k ¹ x º . andif e <" ( x < x " 1 ,seeFig.2.2bottom),then RL x D L v " k ¹ x º = 1 ¹ 1 º d dx ¹ x " x " 1 ¹ 2 º P k + 1 ¹ s º ¹ s x º ds H ¹ " º k ¹ x º ; (2.14) andif e = " ,( x " 1 < x < x " ,seeFig.2.2middle),then RL x D L v " k ¹ x º = 1 ¹ 1 º d dx ¹ x " x ¹ 2 º P k + 1 ¹ s º ¹ s x º ds = RL x D x " h ¹ 2 º P k + 1 ¹ x º i : (2.15) Hence,for " = 1 ; 2 ; ; N el and k = 0 ; 1 ; ; P , RL x D L v " k ¹ x º = 8 > > > > > > > >< > > > > > > > > : 0 ; 8 x 2 e ; e >"; RL x D x " h ¹ 2 º P k + 1 ¹ x º i ; 8 x 2 e ; e = "; H ¹ " º k ¹ x º ; 8 x 2 e ; e <": (2.16) Therefore,thebilinearform(2.12)canbewrittenas " 1 Õ e = 1 P Õ p = 0 ^ u ¹ e º p d p ¹ x º dx ; H ¹ " º k ¹ x º e + P Õ p = 0 ^ u ¹ " º p d p ¹ x º dx ; RL x D x " h ¹ 2 º P k + 1 ¹ x º i " P Õ p = 0 ^ u ¹ " º p p ¹ x º ; ¹ 2 º P k + 1 ¹ x º " = f ; ¹ 2 º P k + 1 ¹ x º " ; (2.17) 26 andtheweakformisobtainedas " 1 Õ e = 1 P Õ p = 0 ^ u ¹ e º p ^ S ¹ e ;" º kp + P Õ p = 0 ^ u ¹ " º p h S ¹ " º kp M ¹ " º kp i = f ¹ " º k ; 8 > > >< > > > : " = 1 ; 2 ; ; N el ; k = 0 ; 1 ; ; P ; (2.18) inwhich ^ S ¹ e ;" º kp = d p dx ; H ¹ " º k ¹ x º e ; e = 1 ; 2 ; ;" 1 ; (2.19) S ¹ " º kp = d p dx ; RL x D x " h ¹ 2 º P k + 1 ¹ x º i " ; M ¹ " º kp = p ¹ x º ; ¹ 2 º P k + 1 ¹ x º " ; f ¹ " º k = f ; ¹ 2 º P k + 1 ¹ x º " ; arerespectivelythe history ,localsti˙ness,localmassmatrices,andlocalforcevector. 2.3.1Elemental(Local)Operations:theconstructionoflocalmatricesS ¹ " º andM ¹ " º ,and vectorf ¹ " º Here,weprovidetheanalyticallyobtainedexpressionsofthelocalsti˙nessmatrixaswellas theproperquadraturerulestoconstructthelocalmassmatrixandforcevectorforallelements " = 1 ; 2 ; ; N el . Elemental(Local)Sti˙nessMatrixS ¹ " º :giventhestructureofthebasisfunctionsandusing (1.26),we˝rstobtainthe˝rst( p = 0 )andlastcolumn( p = P )ofthethelocalsti˙nessmatrix S ¹ " º , andthen,therestofentriescorrespondingtotheinteriormodes.Hence, S ¹ " º k 0 = ¹ x " x " 1 d 0 dx RL x D x " h ¹ 2 º P k + 1 ¹ x º i dx ; (2.20) = Jac ¹ "; º ¹ 1 1 ¹ 1 2 º¹ d dx º » 1 + k + ¼ » 1 + k ¼ P k ¹ º¹ dx d º d ; = Jac ¹ "; º ¹ 1 + k + º 2 ¹ 1 + k º ¹ 1 1 P k ¹ º d ; = Jac ¹ "; º ¹ 1 + k + º ¹ 1 + k º k ; 0 ; ¹ bytheorthogonality º 27 inwhichtheJacobianconstant,associatedwiththeelement " andthefractionalorder ,is Jac ¹ "; º = ¹ 2 x " x " 1 º .Thus,the˝rstcolumnofthelocalsti˙nessmatrixisobtainedas S ¹ " º k 0 = 2 x " x " 1 º ¹ 1 + k + º ¹ 1 + k º k ; 0 ; k = 0 ; 1 ; ; P : (2.21) Similarly,wecanobtainthelastcolumnofthelocalsti˙nessmatrix S ¹ " º kP as S ¹ " º kP = ¹ 2 x " x " 1 º ¹ 1 + k + º ¹ 1 + k º k ; 0 = S ¹ " º k 0 ; k = 0 ; 1 ; ; P : (2.22) Inordertoobtaintherestofentriesof S ¹ " º kp ( k = 0 ; 1 ; ; P and p = 1 ; 2 ; ; P 1 ),wecarryout theintegration-by-partsandtransferanotherderivativeontothetestfunction,takingintoaccount thattheinteriormodesvanishattheboundarypoints x " and x " 1 .Therefore, S ¹ " º kp = ¹ x " x " 1 d p dx RL x D x " h ¹ 2 º P k + 1 ¹ x º i dx ; (2.23) = ¹ x " x " 1 p ¹ x º d dx RL x D x " h ¹ 2 º P k + 1 ¹ x º i dx ; = ¹ 1 1 p ¹ º d d d dx Jac ¹ "; º RL D 1 h ¹ 2 º P k + 1 ¹ º i dx d d ; = Jac ¹ "; º ¹ 1 + k + º 4 ¹ 1 + k º ¹ 1 1 ¹ 1 º¹ 1 + º P 1 ; 1 p 1 ¹ º d d h P k ¹ º i d ; = Jac ¹ "; º ¹ 1 + k + º 4 ¹ 1 + k º k + 1 2 ¹ 1 1 ¹ 1 º¹ 1 + º P 1 ; 1 p 1 ¹ º P 1 ; 1 k 1 ¹ º d : Hence,for " = 1 ; 2 ; ; N el , S ¹ " º kp = 2 x " x " 1 º ¹ 1 + k + º¹ k + 1 º 8 ¹ 1 + k º C 1 ; 1 k 1 k ; p ; k = 0 ; ; P ; p = 1 ; ; P 1 ; (2.24) where C 1 ; 1 k 1 representsthecorrespondingorthogonalityconstantofJacobipolynomialsoforder k 1 withparameters = = 1 .Wenotethattheentriesof S ¹ " º kp areobtainedanalytically,using theorthogonalityofJacobipolynomial.Also,theinteriormodesleadtoa diagonal matrixdueto k ; p .Fig.2.3showsthesparsityofthelocalsti˙nessmatrix. 28 Figure2.3:Sparsityoflocalsti˙nessmatrix Elemental(Local)MassMatrixM ¹ " º :giventhestructureofbasisfunctionsandde˝nition (2.19),we˝rstobtainthecorresponding˝rst( p = 0 )andlastcolumn( p = P )ofthelocalmass matrix M ¹ " º ,andthen,wecomputetherestofentriesassociatedwiththeinteriormodes,using properquadraturerules.Therefore, M ¹ " º k 0 = ¹ x " x " 1 4 º Q Õ q = 1 w 1 + ; 0 q P ; k ¹ z 1 + ; 0 q º ; M ¹ " º kP = ¹ x " x " 1 4 º Q Õ q = 1 w ; 1 q P ; k ¹ z ; 1 q º ; M ¹ " º kp = ¹ x " x " 1 8 º Q Õ q = 1 w 1 + ; 1 q P ; k ¹ z 1 + ; 1 q º ; where k = 0 ; 1 ; ; P and f w ; q ; z ; q g Q q = 1 aretheGauss-Lobatto-Jacobiweightsandpoints correspondingtotheparameters and . Elemental(Local)LoadVectorf ¹ " º :thelocalloadvectorisobtainedas: f ¹ " º k = ¹ x " x " 1 f ¹ x º ¹ 2 º P k + 1 ¹ x º dx = ¹ x " x " 1 2 º ¹ 1 1 ¹ 1 º f ¹ x ¹ ºº P ; k ¹ º d : Hence,for " = 1 ; 2 ; ; N el , f ¹ " º k = ¹ x " x " 1 2 º Q Õ q = 1 w ; 0 q f ¹ x " ¹ q ºº P ; k ¹ z ; 0 q º ; k = 0 ; 1 ; ; P ; 29 where f w ; 0 q ; z ; 0 q g Q q = 1 aretheGauss-Lobatto-Jacobiweightsandpointscorrespondingtothe parameters = and = 0 . 2.3.2Non-LocalOperation:theconstructionofhistorymatrix ^ S ¹ e ;" º Themostchallengingpartofconstructingthelinearsystemistocomputetheglobalhistory matrix ^ S ¹ e ;" º .Thehistorymatrixrelatesthecurrentelement " = 1 ; 2 ; ; N el toitspastelements e = 1 ; 2 ; ;" 1 by ^ S ¹ e ;" º kp = ¹ x e x e 1 d p dx H ¹ " º k ¹ x º dx ; k = 0 ; ; P ; p = 1 ; ; P 1 ; (2.25) where H ¹ " º k ¹ x º isgivenin(2.14)as H ¹ " º k ¹ x º = 1 ¹ 1 º d dx ¹ x " x " 1 ¹ 2 º P k + 1 ¹ s º ¹ s x º ds ; = ¹ 1 º ¹ x " x " 1 ¹ 2 º P k + 1 ¹ s º ¹ s x º 1 + ds ; inwhich, x 2 e = » x e 1 ; x e ¼ and s 2 " = » x " 1 ; x " ¼ .Byperformingthefollowinga˚ne mappings s = x " + x " 1 2 + x " x " 1 2 ; x = x e + x e 1 2 + x e x e 1 2 ˘; from e and " tothestandarddomain » 1 ; 1 ¼ ,thehistoryfunction H ¹ "; e º k ¹ ˘ º = H ¹ " º k ¹ x º isobtained as H ¹ "; e º k ¹ ˘ º (2.26) = ¹ x " x " 1 2 º ¹ 1 º ¹ 1 1 ¹ 2 º P k + 1 ¹ º d h ¹ x " + x " 1 x e + x e 1 º 2 + x " x " 1 2 x e x e 1 2 ˘ i 1 + : Ifthemeshisorthen x " + x " 1 2 = 2 " 1 2 x ; x e + x e 1 2 = 2 e 1 2 x ; x " x " 1 2 = x e x e 1 2 = x 2 ; (2.27) 30 andthus, H ¹ "; e º k ¹ ˘ º = x 2 ¹ 1 º ¹ 1 1 ¹ 2 º P k + 1 ¹ º h ¹ 2 " 1 2 e 1 º 2 x + x 2 ¹ ˘ º i 1 + d ; = ¹ 1 º ¹ 2 x º ¹ 1 1 ¹ 2 º P k + 1 ¹ º h 2 ¹ " e º + ˘ i 1 + d ; = ¹ 1 º ¹ 2 x º ¹ 1 1 ¹ 2 º P k + 1 ¹ º h 2 " + ˘ i 1 + d ; (2.28) where " = " e > 0 ,denotestheelementdi˙erencebetweenthecurrentelement " andthe e -th element.Next,weexpandthepoly-fractonomials ¹ 2 º P k + 1 ¹ º intermsoffractonomials ¹ 1 º + m as ¹ 2 º P k + 1 ¹ º = ¹ 1 º P ; k ¹ º = k Õ m = 0 C km ¹ 1 º + m ; (2.29) inwhich C km = k + m m k + k m 1 2 º m isalower-trianglematrix.Therefore,(2.28)canbewrittenas H ¹ "; e º k ¹ ˘ º = ¹ 1 º ¹ 2 x º k Õ m = 0 C km h ¹ "; e º m ¹ ˘ º ; (2.30) wherewecall h ¹ "; e º m ¹ ˘ º ¹ 1 1 ¹ 1 º + m » 2 " + ˘ ¼ 1 + d ; m = 0 ; 1 ; ; k ; (2.31) the (modal)memorymode .Also, h ¹ "; e º m ¹ ˘ º canbeobtainedanalyticallyas h ¹ "; e º m ¹ ˘ º = 2 ¹ " ˘ š 2 º 1 + m + h h m ; I ¹ ˘; " º + h m ; II ¹ ˘; " º + h m ; III ¹ ˘; " º i ; (2.32) inwhich h m ; I ¹ ˘; " º = Z I ¹ ˘; " º 2 F 1 1 ; 1 + m ; 2 + m + ; Z I ¹ ˘; " º ; (2.33) h m ; II ¹ ˘; " º = 1 2 " + ˘ 2 " + ˘ ¹ 2 m + º Z II ¹ ˘; " º 2 F 1 1 ; 1 + m ; 2 + m + ; Z II ¹ ˘; " º ; h m ; III ¹ ˘; " º = Z III ¹ ˘; " º 2 F 1 1 ; 1 + m ; 2 + m + ; Z III ¹ ˘; " º ; 31 andthegroupvariablesare Z I ¹ ˘; " º = 1 1 + 2 " ˘ , Z II ¹ ˘; " º = 2 1 2 " + ˘ ,and Z III ¹ ˘; " º = 1 2 " + ˘ .Therefore,by(2.30)andobtainingthederivativeofbasisfunctioninthestandarddomain, theentriesofthehistorymatrixcanbee˚cientlycomputed,usingaGaussquadrature.Hence: ^ S ¹ "; e º kp ^ S ¹ " º kp = ¹ 1 1 d p d ˘ H k ¹ ˘; " º d ˘; k ; p = 0 ; 1 ; ; P : (2.34) Remark2.3.1. Wenotethatwhenauniformmeshisemployed,thehistoryfunction H ¹ "; e º k ¹ ˘ º H k ¹ ˘; " º ,de˝nedinthestandarddomain,onlydependsonthedi˙erence " = " e . Thisissigni˝cantsinceoneonlyneedstoconstruct N el 1 historyfunction,andthus,history matrices ^ S ¹ e ;" º . 2.3.3AssemblingtheGlobalSystemwithLocalTestFunctions Wegeneralizethenotionofgloballinearsystemassemblybytakingintoaccountthepresenceof thehistorysti˙nessmatricesandrecallingthatthecorrespondinglocalmassmatrix M ¹ " º orthe localloadvector f ¹ " º donotcontributetoanyhistorycalculations.Weimposethe C 0 continuit y byemployingthearraysmap[e][p],de˝nedas map » e ¼» p ¼ = P ¹ e 1 º + p ; p = 1 ; 2 ; ; P ; e = 1 ; 2 ; ; N el ; (2.35) asforinstanceinMathematica,the˝rstentryofavectorislabelledby1ratherthan0asinC++. Then,thecorresponding ¹ P + 1 º¹ P + 1 º linearsystem,whichisassociatedwiththeelement " ,isobtainedas M ¹ " º = S ¹ " º M ¹ " º : (2.36) Weassemblethecorrespondinggloballinearmatrix M G andthegloballoadvector F G asfollows: 32 do " = 1 ; N el do k = 1 ; P + 1 F G h map » " ¼» k ¼ i = f ¹ " º » k ¼ do p = 1 ; P + 1 M G h map » " ¼» k ¼ ih map » " ¼» p ¼ i = M G h map » " ¼» k ¼ ih map » " ¼» p ¼ i + M ¹ " º » k ¼» p ¼ do e = 1 ;" 1 M G h map » " ¼» k ¼ ih map » e ¼» p ¼ i = M G h map » " ¼» k ¼ ih map » e ¼» p ¼ i + ^ S ¹ " º » k ¼» p ¼ End Thisglobaloperationleadstothefollowinglinearsystem: M G ^ u G = F G ; (2.37) inwhich ^ u G denotestheglobaldegreesoffreedom.ThehomogeneousDirichletboundarycondi- tionsrequirethe˝rstandlastentriesoftheglobaldegreeoffreedomtobezero,i.e. ^ u 1 0 = ^ u N el P = 0 . Thisisenforcedbyremovingthe˝rstandlastrowsaswellasthe˝rstandlastcolumnsoftheglobal matrix,inadditiontoremovingthe˝rstandlastentriesoftheloadvector.Wealsonotethatthe C 0 -continuityanddecompositionofbasisfunctionsintoboundaryandinteriormodesleadtothe standardscatteringprocessfromtheglobaltolocaldegreesoffreedom(seee.g.,[81]). 2.3.4O˙-LineComputationofHistoryMatricesandHistoryRetrieval Asmentionedinremark2.3.1(onuniformgridgeneration),thehistorymatricessolelydependon theelementdi˙erence, " = " e .Thus,foralllocalelements " ,where " = 1 ; 2 ; ; N el ,the historymatricescorrespondingtothepastelement e withsimilarelementdi˙erence,arethesame. SeeFig.2.4,wheresimilarly-coloredblocksrepresentthesamehistorymatrixandonecanseethat, forexample,allthehistorymatricesadjacenttothelocalsti˙nessmatriceshavethesameelement di˙erence, " = 1 ,andthusareinthesamecolor.Therefore,givennumberofelement N el ,we onlyneedtoconstructthetotalnumberof N el 1 historymatrices. 33 Figure2.4:Theassembledglobalmatrixcorrespondingtoauniformgridwith N el = 9 .Inthis globalmatrix, M ¹ " º = S ¹ " º M ¹ " º , " = 1 ; 2 ; ; N el ,representsthelocalmatrix,associatedwith theelement " .To˝llthelower-triangularblockmatrices,weconstructonly ¹ N el 1 º history matrices ^ S ¹ " º ,where " = 1 ; 2 ;::; N el 1 ,ratherthan N el ¹ N el 1 º 2 matrices. Foramaximumnumberofelements, N el j max ,andamaximumnumberofmodes, P j max ,we cancomputeo˙-lineandstorethetotal N el j max 1 historymatricesofsize ¹ P j max + 1 º¹ P j max + 1 º , whichwecanfetchlaterforanyspeci˝c N el N el j max and P P j max . 2.3.5Non-UniformKernel-BasedGrids Wepresentanon-uniformgridgenerationbasedonthepower-lawkernelinthede˝nitionof fractionalderivative.Therearedi˙erentsourcesofsingularityintheproposedproblemthatcan becausedmainlyduetotheforcefunction f ¹ x º .However,eveniftheforcetermissmooththe underlyingkernelofafractionalderivativeleadstoformationofsingularitiesattheboundaries. Herein,weproposeanewkernel-basedgridgenerationmethodthatconsidersasu˚cientlysmall boundarylayeratthevicinityofsingularpointandpartitionsthatparticularregionnon-uniformly.In 34 Figure2.5:Historycomputationandretrieval. thisapproach,wetreatthekerneloftheform 1 x ˙ asadensityfunctionandthen,weconstructthegrid suchthattheintegralofkernelfunctionovereachelement e 2» x e 1 ; x e ¼ (intheboundarylayer)is constant.Sincetheoperatorisaleftsidedfractionalderivative,werepresentthenon-uniformgrid Figure2.6:Kernel-basednon-uniformgridintheboundarylayer; L b and N b arethelengthofand thenumberofelementsintheboundarylayer,respectively. re˝nementattheleftboundary.Let L b bethelengthofboundarylayerand ¯ L b 0 1 x ˙ dx = L 1 ˙ b 1 ˙ = A . Then,theintegralovereachelementis 1 A ¹ x e x e 1 1 x ˙ dx = 1 L 1 ˙ b h ¹ x e 1 + x e º 1 ˙ x 1 ˙ e 1 i = C ; where x e = x e x e 1 and C isaconstant.Thus, x e = h x 1 ˙ e 1 + C L 1 ˙ b i 1 1 ˙ x e 1 : Startingfrom x 0 = 0 andcalculatingtherestofgridlocationssuccessively,weobtain x e = e 1 1 ˙ ; e = 1 ; 2 ; ; N b ; elementnumbers ; (2.38) 35 inwhich = L b C 1 1 ˙ and N b isthenumberofelementsintheboundarylayer.Theconstant C is obtainedbytheconstraint Í N b e = 1 x e = L b andhence, C = © « N b Õ e = 1 e 1 1 ˙ ¹ e 1 º 1 1 ˙ ª ® ¬ ˙ 1 : Weconsider ˙ = 1 andthuswhen = 1 ,werecovertheuniformgrid x e = L b N b e ,wherethe kernelis 1 , C = 1 N b , = L b N b .Wenotethatintheboundarylayer,wherethegridisnon-uniform, equations(2.27)-(2.31)nolongerhold.Thus,using(2.38),weobtain H ¹ "; e º k ¹ ˘ º = ¹ 1 º 2 " 1 ¹ " 1 º 1 ¹ 1 1 ¹ 2 º P k + 1 ¹ º d Z ; (2.39) inwhich, Z = 2 1 + h " 1 + ¹ " 1 º 1 e 1 + ¹ e 1 º 1 + " 1 ¹ " 1 º 1 e 1 ¹ e 1 º 1 ˘ i 1 + : Therefore,by(2.25),theentriesofthehistorymatrixfortheboundarylayerelements,where " = 1 ; 2 ; ; N b and e = 1 ; 2 ; ;" 1 ,canbenumericallyobtainedas ^ S ¹ "; e º kp = ¹ 1 1 d p d ˘ H ¹ "; e º k ¹ ˘ º d ˘; k ; p = 0 ; 1 ; ; P ; (2.40) ThesematricesarethesmallsquaresintheupperleftcornerofFig.2.7(interactionofboundarylayer elements e and ).Fortheinteriorelements, " = N b + 1 ; N b + 2 ; ; N el ,when N b + 1 e 1 , thegridisuniformandtherefore,weuse(2.34)toobtainthehistorymatrices.Thesematricesare shownasthebigsquaresinFig.2.7(interactionofinteriorelements e and ).However,when 1 e N b ,thegridisnon-uniformandweuse(2.40)toobtainthehistorymatrices.These matricesareshownasskinnyrectanglesinFig.2.7(interactionofinteriorelementswithboundary layerelements). Inuniformgridgeneration,thehistoryfunction(2.28)onlydependsonelementdi˙erence , whichleadstoafastande˚cientconstructionofhistorymatrices(seeRemark2.3.1).However,in 36 Figure2.7:Theassembledglobalmatrixcorrespondingto N el = 11 with N b = 4 non-uniform boundaryelementsand 7 uniforminteriorelements.Inthisglobalmatrix, M ¹ " º = S ¹ " º M ¹ " º , " = 1 ; 2 ; ; N el ,representsthelocalmatrix,associatedwiththeelement " .Thelower-triangle consistsofthreeparts:1)Thesmallsquare N b ¹ N b 1 º 2 historymatrices(interactionofboundary elements, " = 1 ; 2 ; ; N b ).2)Thebigsquarehistorymatrices(interactionofinteriorelements, " = N b + 1 ; ; N el ).3)Theskinnyrectangular ¹ N el N b º N b historymatrices(interactionof boundaryelementswithinteriorelements). non-uniformkernel-basedgridgeneration,thisisnotthecaseanymoreandconstructionofhistory matricesiscomputationallyexpensive.Improvingthehistoryconstructiononnon-uniformgrids requiresfurtherinvestigations,tobedoneinourfutureworks. 2.3.6Non-UniformGeometricallyProgressiveGrids Inadditiontothenon-uniformgridgenerationbasedonthekerneloffractionalderivative,we consideranon-uniformgridusinggeometricallyprogressiveseries[4,15].Inthiscase,thelength ofelementsareincreasedbyaconstantfactor r (seeFig.2.8).Byconsideringthelengthof˝rst elementtobe ,weconstructthegridas x 0 = 0 ; x 1 = ; x 2 = ¹ 1 + r º ; x 3 = ¹ 1 + r + r 2 º andso 37 Figure2.8:Non-uniformgeometricallyprogressivegrid. on.Hence, x e = e 1 Õ i = 0 r i = r e 1 r 1 ; e = 1 ; 2 ; ; N b : (2.41) Choosing r and N b ,theconstant isobtainedbytheconstraint x N b = L b ,whichgives = L b r 1 r N b 1 . Sincethegridisnon-uniform,equations(2.27)-(2.31)donotholdanymore.Thus,using(2.41), weobtain H ¹ "; e º k ¹ ˘ º = ¹ 1 º ¹ 2 º r " ¹ e 1 º ¹ 1 1 ¹ 2 º P k + 1 ¹ º d h r + 1 r 1 ¹ r " 1 º + ¹ r " ˘ º i 1 + ; (2.42) where " = " e > 0 ,denotesthe elementdi˙erence betweenthecurrentelement " andthe e -th element.Usingthesameexpansionasin(2.29),wecanwrite(2.42)as H ¹ "; e º k ¹ ˘ º = ¹ 1 º ¹ 2 º r " ¹ e 1 º k Õ m = 0 C km ~ h ¹ "; e º m ¹ ˘ º ; (2.43) wherethe(modal)memorymode ~ h ¹ "; e º m ¹ ˘ º = ¹ 1 1 ¹ 1 º + m » r + 1 r 1 ¹ r " 1 º + ¹ r " ˘ º¼ 1 + d ; m = 0 ; 1 ; ; k ; (2.44) canbeobtainedanalyticallyusinghypergeometricfunctions.Therefore,by(2.43),theentriesofthe historymatrixcanbee˚cientlycomputedusingtheGaussquadraturein(2.34).Theconstructionof theassembledgloballinearsystemisthesameaskernel-basedgridgenerationapproach.Wenote thatsimilartouniformgrid,inthenon-uniformgridgenerationusingthegeometricalprogression, thehistoryfunctionsdependontheelementdi˙erence " = " e ,leadingtoafastande˚cient constructionofhistorymatrices. 38 2.4Petrov-GalerkinMethodwithGlobalTestFunctions Inthissection,similartothecaseoflocaltestfunctions,wedevelopthePetrov-Galerkinscheme bysubstituting(2.7)into(2.6)andconsideringtheglobaltestfunction,givenin(2.11)toobtain: N el Õ e = 1 P Õ p = 0 ^ u ¹ e º p d p ¹ x º dx ; RL x D L v " k ¹ x º e N el Õ e = 1 P Õ p = 0 ^ u ¹ e º p p ¹ x º ; v " k ¹ x º e = N el Õ e = 1 f ; v " k ¹ x º e ;" = 1 ; 2 ; ; N el ; k = 0 ; 1 ; ; P : (2.45) Sincethetestfunctionvanishesonly 8 x 2 e , " and e >" ,(2.45)reducesto " Õ e = 1 P Õ p = 0 ^ u ¹ e º p d p dx ; RL x D x " v " k ¹ x º e " Õ e = 1 P Õ p = 0 ^ u ¹ e º p p ¹ x º ; v " k ¹ x º e = " Õ e = 1 f ; v " k ¹ x º e : Bysubstituting(2.11),weobtain " Õ e = 1 P Õ p = 0 ^ u ¹ e º p d p dx ; RL x D x " ¹ 2 º P k + 1 ¹ x 1 ˘ " º e " Õ e = 1 P Õ p = 0 ^ u ¹ e º p p ¹ x º ; ¹ 2 º P k + 1 ¹ x 1 ˘ " º e = ¹ x " 0 f ¹ x º ¹ 2 º P k + 1 ¹ x 1 ˘ " º dx ;" = 1 ; 2 ; ; N el ; k = 0 ; 1 ; ; P ; whichcanbewritteninthematrixformas " Õ e = 1 P Õ p = 0 ^ u ¹ e º p h ^ S ¹ "; e º kp ^ M ¹ "; e º kp i = f ¹ " º k ;" = 1 ; 2 ; ; N el ; k = 0 ; 1 ; ; P ; (2.46) where ^ S ¹ "; e º kp = d " p dx ; RL x D x " h ¹ 2 º P k + 1 ¹ x 1 ˘ " º i " ; (2.47) ^ M ¹ "; e º kp = " p ¹ x º ; ¹ 2 º P k + 1 ¹ x 1 ˘ " º " ; (2.48) f ¹ " º k = ¹ x " 0 f ¹ 2 º P k + 1 ¹ x 1 ˘ " º dx : (2.49) 39 Remark2.4.1. Thebene˝tofchoosingsuchglobaltestfunctionsisnowclearsincewecan analyticallyevaluate RL x D x " ¹ 2 º P k + 1 ¹ x 1 ˘ " º .However,wenotethatthischoiceoftestfunctions introducesaworkassociatedwiththeconstructionofthetorymass ^ M ¹ "; e º , 8 e = 1 ; 2 ; ;" 1 ,when , 0 . Remark2.4.2. Thechoiceofglobaltestfunctionsleadstoextracostofquadraturecarriedout overtheincreasing-in-lengthdomainsofintegrationin (2.49) .Dependingonthebehaviourofthe force-term f ¹ x º ,thisapproachmightrequireadaptive/multi-elementquadraturerulestoobtainthe correspondingentriesofthedesiredprecision. 2.4.1Elemental(Local)Operations:theconstructionoff ¹ " º Here,theconstructionoftheloadvectoristheonlyoperationthatcouldberegardedas operationsHence, f ¹ " º k = ¹ x " 0 f ¹ x º ¹ 2 º P k + 1 ¹ x 1 ˘ " º dx = ¹ x " 2 º ¹ 1 1 ¹ 1 º f ¹ x 1 ˘ " ¹ ºº P ; k ¹ º d ; andthus, f ¹ " º k = ¹ x " 2 º Q Õ q = 1 w ; 0 q f ¹ x 1 ˘ " ¹ q ºº P ; k ¹ z ; 0 q º ; where f w ; 0 q ; z ; 0 q g Q q = 1 aretheGauss-Lobatto-Jacobiweightsandpointscorrespondingtothe parameters = and = 0 . 2.4.2GlobalOperations:theconstructionof ^ S ¹ "; e º and ^ M ¹ "; e º Thecorrespondingsti˙nessandmassmatricesareglobalinnatureandweobtaintheirentriesusing properGaussquadraturerules. 40 2.4.3AssemblingtheGlobalSystemwithGlobalTestFunctions Weextendthenotionofgloballinearsystemassemblybytakingintoaccountthepresenceofthe historysti˙nessandmassmatrices.Wesimilarlyimposethe C 0 continuit y byemployingthe samearrays map » e ¼» p ¼ ,de˝nedin(2.35).Letusde˝nethe ¹ P + 1 º¹ P + 1 º matrix ^ M ¹ "; e º = ^ S ¹ "; e º ^ M ¹ "; e º ; (2.50) 8 "; e ˝xed.Then,weassemblethecorrespondinggloballinearmatrix M G andtheglobalload vector F G asfollows: do " = 1 ; N el do k = 1 ; P + 1 F G h map » " ¼» k ¼ i = f ¹ " º » k ¼ do p = 1 ; P + 1 do e = 1 ; " M G h map » " ¼» k ¼ ih map » e ¼» p ¼ i = M G h map » " ¼» k ¼ ih map » e ¼» p ¼ i + ^ M ¹ "; e º » k ¼» p ¼ End Thisleadstoalinearsystemsimilartothatin(2.37),showninFig.2.9,wherethehomogeneous Dirichletboundaryconditionsareenforcedinasimilarfashionasbefore.Wenotethatthescattering operationfromglobaltolocaldegreesoffreedomissimilartothestandardscatteringprocess. 2.5NumericalExamples WeconsidernumericalexamplesofthetwoPGschemeswehaveproposed.Weprovide examplesofsmoothandsingularsolutionswithsingularitiesatboundarypointsandintheinterior domain,whereweshowthee˚ciencyofdevelopedschemesincapturingthesingularities.Wealso performtheo˙-linecomputationofhistorymatricesandshowtheimprovementofcomputational cost.Moreover,weconstructnon-uniformkernel-basedandgeometricallyprogressivegridsand presentthesuccessofthetwoapproachesincapturingsingularsolutions.Furthermore,we 41 Figure2.9:Theassembledglobalmatrixcorrespondingto N el = 5 elementswhenglobaltest functionsareemployed.Inthisglobalmatrix, M ¹ " º = ^ S ¹ " º ^ M ¹ " º , " = 1 ; 2 ; ; N el ,represents thelocalmatrix,associatedwiththeelement " .To˝llthelower-triangularblockmatrices,we mustconstruct N el ¹ N el 1 º 2 historymatrices ^ S ¹ "; e º . investigatethenon-locale˙ectsfordi˙erentcasesofhistoryfading.Inthissection,weconsider thecomputationaldomain L = 1 . 2.5.1SmoothProblems Intheproposedschemes,thechoiceofbasisfunctionsarepolynomials,enablingtheschemeto accuratelyande˚cientlyapproximatethesmoothsolutionsoverthewholedomain.Weconsider twosmoothsolutionsoftheform u ext = x 7 x 6 and u ext = x 6 sin ¹ 2 ˇ x º .Thecorrespondingforce functionsareobtainedbysubstitutingtheexactsolutionsinto(2.1)(with = 0 ).ByemployingPG SEM,usinglocalbasis/testfunctionsandlocalbasiswithglobaltestfunctions(developedinSec. 2.3andSec.2.4,respectively),weobservethattheformerleadstoabetterapproximabilityand conditionnumber.Fig.2.10presentsthe L 2 -normerrorofthePGSEM,employinglocalbasis/test functions,whereweshowtheexponentialconvergenceoftheschemeinapproximatingthetwo 42 smoothsolutions.Theconditionnumberoftheresultingassembledglobalmatrix,usingthetwo developedschemesarealsopresentedinTable2.1.Weshowthatthechoiceoflocalbases/test functionsleadstoabetterconditioningfordi˙erentnumberofelementsandmodes. Figure2.10:PGSEMwithlocalbasis/testfunctions.Plottedistheerrorwithrespecttothe polynomialdegreeofeachelement(spectralorder). Table2.1:Conditionnumberoftheresultingassembledglobalmatrixforthetwochoicesoflocal bases/testfunctions(left)andlocalbaseswithglobaltestfunctions(right)fordi˙erentnumberof elementsandmodes. (LocalTestFunctions) PN el = 2 N el = 10 37.1386.13 513.21153.86 1035.39420.24 (GlobalTestFunction) PN el = 2 N el = 10 33.46 10 4 1.84 10 16 54.3 10 7 7.2 10 16 102.73 10 15 5.1 10 17 2.5.2HistoryRetrieval AsdiscussedinSec.2.3.4,alargenumberofhistorymatricescanbecomputedo˙-line,stored, andretrievedforlateruse.Theretrievalprocess,comparedtoon-lineconstructionofthehistory matrices,leadstohighercomputationale˚ciency.Inthissection,byconsidering 1000 elements, wecomputeandstore 999 historymatricesfordi˙erentnumberofmodes, P = 2 ; 3 ; and 4 (here = 1 2 ).Then,fordi˙erentnumberofelements,wecomputetheCPUtimerequiredforconstructing andsolvingthelinearsystem,obtainedbyretrievingthestoredhistorymatricesfromharddrive. 43 WealsocomputetheCPUtimerequiredforconstructingandsolvingthelinearsystem,obtained byon-linecomputationofthehistorymatrices.Table2.2showsthatinthecaseof p = 4 andfor N el = 10 , N el = 100 , N el = 500 ,and N el = 1000 ,theretrievalprocessisalmost4,5,and10times faster,respectively.Thus,thehigher p is,thefasterandmoree˚cienttheretrievalbecomes. Table2.2:CPUtimeofconstructingandsolvingthelinearsystembasedono˙-lineretrievaland on-linecalculationofhistorymatrices. CPUTime N el = 10 N el = 100 N el = 500 N el = 1000 P O˙-lineOn-lineO˙-lineOn-lineO˙-lineOn-lineO˙-lineOn-line retrievalcomputationretrievalcomputationretrievalcomputationretrievalcomputation 22.65207.254024.710583.5229141.3525429.3147370.6895790.3478 34.758018.907346.0826161.8042266.04411308.8327746.49594423.7671 48.814032.292284.8645499.9988485.77155599.40621392.870514709.4902 2.5.3SingularProblems ThedevelopedPGspectralelementmethod,comparedtosingle-domainspectralmethods,further leadstoaccuratesolutionseveninthepresenceofsingularitiesvia hp -re˝nementsatthevicinity ofsingularities,whilestillemployingsmoothpolynomialbases.Theerrorintheboundarylayer iscontrolledbyconsideringsu˚cientnumberofmodesintheboundarylayerelements.Theerror intheinteriordomainisthenimprovedbyperforming p -re˝nementinthoseelements.Inorder toinvestigatetheperformanceoftheschemeincapturingasingularity,weconsiderthreetypes ofsingularities,including:i)single-boundarysingularity,ii)full-boundarysingularity,andiii) interiorsingularity(whendiscontinuousforcefunctionsareapplied). I)Single-BoundarySingularity: weconsidertwosingularsolutionsoftheform u ext = ¹ 1 x º x 2 + and u ext = ¹ 1 x º x 5 + withleftboundarysingularity.Wepartitionthedomain intotwonon-overlappingelements,includingoneboundaryelementoflength L b atthevicinity ofsingularpointinadditiontoaninteriorelementfortherestofcomputationaldomain.The schematicofcorrespondingglobalsystemisshowninFig.2.11(left).Table2.3showsthe exponentialconvergenceof L 2 -normerrorintheinteriordomain.Theerrorintheboundarylayer 44 elementisthencontrolledbychoosingsu˚cientnumberofmodesintheboundaryelement.The resultsareobtainedforthetwocasesof L b = 10 2 L and L b = 10 4 L . Table2.3:Single-BoundarySingularity: L 2 -normerrorintheboundaryandinteriorelements usingPGSEMwithlocalbasis/testfunctions.Here, L b representsthesizeofleftboundaryelement, P b and P I denotethenumberofmodesintheboundaryandinteriorelementsrespectively. u ext = ¹ 1 x º x 2 + , = 1 š 2 BoundaryElementError P b L b = 10 1 LL b = 10 2 LL b = 10 4 L 6 1 : 29387 10 7 1 : 29634 10 10 1 : 19525 10 16 10 1 : 46601 10 8 1 : 4193 10 11 4 : 07955 10 18 InteriorElementError, P b = 10 P I L b = 10 1 LL b = 10 2 LL b = 10 4 L 6 5 : 49133 10 6 2 : 6893 10 5 3 : 38957 10 5 10 9 : 39045 10 8 1 : 08594 10 6 1 : 91087 10 6 14 8 : 27224 10 8 1 : 13249 10 7 3 : 02065 10 7 u ext = ¹ 1 x º x 5 + , = 1 š 2 BoundaryElementError P b L b = 10 1 LL b = 10 2 LL b = 10 4 L 6 3 : 94221 10 11 2 : 96862 10 17 4 : 8243 10 29 10 7 : 07024 10 13 2 : 54089 10 18 2 : 26939 10 29 InteriorElementError, P b = 10 P I L b = 10 1 LL b = 10 2 LL b = 10 4 L 6 1 : 73622 10 5 3 : 80264 10 5 4 : 13249 10 5 10 1 : 3122 10 9 8 : 76951 10 9 1 : 10139 10 8 14 4 : 39611 10 12 1 : 07775 10 10 1 : 66044 10 10 II)Full-BoundarySingularity: weconsiderthesolutionoftheform u ext = ¹ 1 x º 3 + 1 x 3 + 2 withsingularpointsattwoends,i.e. x = 0 and x = 1 .Herein,wepartitionthedomainintothree non-overlappingelementsincludingtwoboundaryelementsoflength L b inthevicinityofsingular points,andoneinteriorelementfortherestofdomain.Theschematicofcorrespondingglobal systemisshowninFig.2.11(right).Similartopreviousexample,thePGSEMcanaccurately capturethesingularitiesatbothends,whereincreasingthenumberofmodesintheinteriorelement resultsinexponentialconvergence.Table2.4showsthe L 2 -normerrorintheboundarylayersand interiorelementswithtwochoicesof P b = 6 ; 10 and L b = 10 2 L ; 10 4 L . 45 Table2.4:Full-BoundarySingularity: L 2 -normerrorintheboundaryelement(BE)andinterior element(IE)byPGSEMwithlocalbasis/testfunctions.Here, u ext = ¹ 1 x º 3 + 1 x 3 + 2 with 1 = 1 4 ; 2 = 2 3 , L b representsthesizeofleftandrightboundaryelements, P b and P I denotethe numberofmodesintheboundaryandinteriorelementsrespectively. L b = 10 2 LL b = 10 4 L P b = 6 P I LeftBEErrorIEErrorRightBEError 6 2 : 73893 10 7 6 : 52605 10 5 3 : 51075 10 6 10 2 : 46964 10 11 1 : 52215 10 7 2 : 2902 10 9 14 3 : 08719 10 12 9 : 30483 10 9 2 : 69541 10 10 P b = 10 P I LeftBEErrorIEErrorRightBEError 6 2 : 73892 10 7 6 : 52605 10 5 3 : 51075 10 6 102.48058 10 11 1 : 52215 10 7 2 : 29003 10 9 14 3 : 19684 10 12 9 : 30511 10 9 2 : 69506 10 10 P b = 6 P I LeftBEErrorIEErrorRightBEError 6 3 : 61679 10 10 5 : 85397 10 5 4 : 60538 10 8 10 1 : 2676 10 10 2 : 43295 10 7 1 : 62151 10 10 14 1 : 53993 10 13 2 : 09933 10 8 1 : 97677 10 11 P b = 10 P I LeftBEErrorIEErrorRightBEError 6 3 : 61679 10 10 5 : 85397 10 5 4 : 60538 10 8 10 1 : 2676 10 12 2 : 43295 10 7 1 : 62151 10 10 14 1 : 53993 10 13 2 : 09933 10 8 1 : 97677 10 11 Figure2.11:Schematicofglobalmatricescorrespondingtothecaseofsingularsolutions.(left): leftboundarysingularity,(right):leftandrightboundarysingularities. ^ S ¹ bI º , ^ S ¹ Ib º ,and ^ S ¹ bb º denotetheinteractionofboundary/interior,interior/boundaryandboundary/boundaryelements, respectively. III)InteriorSingularity (DiscontinuousForceFunction) : weconsiderthesolutionswith singularityinthemiddleofdomain.Theforcefunction,obtainedbysubstitutingthesolutioninto (2.1),isconsideredtobediscontinuousatthepointofsingularity.Fig.2.12showsthetwoexact solutionsoftheform u ext 1 = x 2 ¹ 1 x º 2 j x 1 2 j (top)and u ext 2 = sin ¹ 3 ˇ x º x ¹ 1 x ºj x 1 2 j (bottom) andtheircorrespondingforcefunctions.Wepartitionthedomainatthevicinityofsingularpoint usingtwonon-overlappinginteriorelements,inwhichthesolutionissmooth.ThePGschemewith localbasis/testfunctionsisshowntobeabletoaccuratelycapturethesingularityinthemiddleof thedomain.Inthecaseof u ext 1 ,weapproximatethesolutionintherangeofmachineprecisionwith 46 (a) (b) (c) (d) Figure2.12:InteriorSingularity.(left):exactsolutions,(right):thecorrespondingforcefunctions. P = 5 withineachelement.Wealsoshowtheexponentialrateofconvergenceinthecaseof u ext 2 byincreasingthenumberofmodes, P ,ineachelement.TheresultsareshowninFig.2.13. Figure2.13:InteriorSingularity:PGSEMwithlocalbasis/testfunctions.Plottedistheerrorwith respecttospectralorderineachelement. 2.5.4Non-UniformGrids Weconsiderasingularsolutionoftheform u ext = ¹ 1 x º x 1 + (here = 1 10 and = 0 )with singularityattheleftboundary.Inordertosolvetheproblem,weconsiderthreegridgeneration approacheswithsimilardegreesoffreedom,includingoneuniformandtwonon-uniformgrids 47 overthecomputationaldomain.Thenon-uniformgridsaregeneratedbasedonthepower-law kernelinthede˝nitionoffractionalderivativeandthegeometricprogressionseries(discussedin Sec.2.3.5andSec.2.3.6,respectively).Here,wechoose L b = L .Table2.5showsthe L 2 -norm errorconsideringtheuniformandnon-uniformgrids.Wekeepthetotaldegreesoffreedom˝xed, butweincreasethepolynomialorder P ineachsimulation.Thesuccessofthenon-uniformgrid inprovidingmoreaccurateresultsisobserved,wherefewernumberofelementsareused,while higherorderpolynomialareemployed.Werecallthatthesizeofboundarylayerhasbeensettoits maximumpossiblelength,i.e. L b = L .Clearly,onecanobtainevenmoreaccurateresultswhen L b issettomuchsmallerlength(e.g. 10 1 L , 10 3 L , etc. ). Table2.5: L 2 -normerror,usinguniformandnon-uniformgrids.Theexactsingularsolutionis u ext = ¹ 1 x º x 1 + with = 1 š 10 . UniformGridKernel-BasedNon-UniformGridGeometricallyProgressiveNon-UniformGrid N el = 50 , P = 25 : 83943 10 4 2 : 33461 10 5 3 : 93956 10 4 N el = 25 , P = 43 : 04739 10 5 1 : 77458 10 7 1 : 38755 10 6 N el = 10 , P = 101 : 39586 10 5 2 : 10813 10 9 1 : 45695 10 9 2.5.5ASystematicMemoryFadingAnalysis Inordertoinvestigatethee˙ectoftruncatingthehistorymatrices,weperformasystematicmemory fadinganalysis. In full memoryfading,wefadethememorybytruncatingthehistorymatrices,i.e.,weconsider thefullhistorymatricesuptosomespeci˝cnumberandthentruncatetherestofhistory.For instance,weconsideruptothe˝rst 4 historymatricesforeachelementandthuscompute ^ S 1 , ^ S 2 , ^ S 3 and ^ S 4 ,andtruncatetherest N el 1 4 matrices;seeFig.2.4forbettervisualization. In partial memoryfading,wefadethememorybypartiallycomputingthehistorymatrices. Similartothe full memoryfading,weconsiderthefullhistorymatricesuptosomespeci˝cnumber, however,fortherestofhistorymatriceswepartiallycomputetheentriesofmatrices.Inpartial memoryfading,weconsiderthreedi˙erentcasesasfollows. 48 (a):CaseI (b):CaseII (c):CaseIII Figure2.14: Memoryfading:(a)B-Binteraction,thecornerentries(b)B-BandB-Iinteraction,the boundaryentries(c)B-B,B-IandS-Iinteraction,boundaryanddiagonalentries ‹ CaseI: Boundary-Boundary(B-B)interaction.Inthiscase,weonlyconsidertheinteractions ofboundarymodeandboundarytestfunctions,i.e., p = 0 ; P and k = 0 ; P ,andthus,only computethecornerentries(SeeFig.2.14a). ‹ CaseII: Boundary-Boundary(B-B)andBoundary-Interior(B-I)interaction.Inadditionto thecornerentries,herewealsoconsidertheinteractionofboundarymode/testfunctionswith theinteriortest/modefunctions,i.e., 8 > > >< > > > : k = 0 ; p = 0 ; 1 ; ; P ; and k = P ; p = 0 ; 1 ; ; P p = 0 ; k = 0 ; 1 ; ; P ; and p = P ; k = 0 ; 1 ; ; P ; andthus,wecomputetheboundaryentries(SeeFig.2.14b). ‹ CaseIII: Boundary-Boundary(B-B),Boundary-Interior(B-I),Self-Interior(S-I)interac- tion.Inadditiontothelasttwocases,weconsidertheinteractionofeachmodewithits correspondingtestfunctionandthus,wecomputetheboundariesaswellasthediagonal entries(SeeFig.2.14c). Tables2.6and2.7showthe L 2 -normerrorforcasesoffullandpartialmemoryfading.Itis clearfromthecomputednormsthateveninthecaseoffadingmemory,wecanstillaccurately obtaintheapproximationsolution,howeverwithaproportionallossofaccuracydependingonthe lackofmodalinteraction. 49 Table2.6:Fullhistoryfading: L 2 -normerrorusingPGSEMwithlocalbasis/testfunctions,where u ext = x 7 x 6 , N el = 19 , P = 6 .The˝rstcolumninthetableshowsthenumberoffullyfaded historymatrices. Fullfading #fadedhistorymatrices = 1 š 10 = 1 š 2 = 9 š 10 0 9 : 26034 10 12 2 : 31391 10 11 4 : 24903 10 9 2 7 : 8905 10 11 1 : 26365 10 10 4 : 25456 10 9 5 1 : 42423 10 8 6 : 39474 10 8 1 : 95976 10 8 8 2 : 69431 10 7 2 : 47423 10 6 8 : 45001 10 7 11 2 : 09737 10 6 3 : 19995 10 5 1 : 37959 10 5 14 9 : 07427 10 6 2 : 44911 10 4 1 : 40684 10 4 17 2 : 94001 10 5 1 : 39043 10 3 1 : 6001 10 3 Table2.7:Partialhistoryfading: L 2 -normerrorusingPGSEMwithlocalbasis/testfunctions, where u ext = x 7 x 6 , N el = 19 , P = 6 .The˝rstcolumninthetablesshowsnumberofpartially fadedhistorymatrices. PartialfadingcaseI #fadedhistorymatrices = 1 š 10 = 1 š 2 = 9 š 10 0 9 : 26034 10 12 2 : 31391 10 11 4 : 24903 10 9 2 7 : 8905 10 11 1 : 26365 10 10 4 : 25456 10 9 5 1 : 42423 10 8 6 : 39474 10 8 1 : 95976 10 8 8 2 : 69431 10 7 2 : 47423 10 6 8 : 45001 10 7 11 2 : 09737 10 6 3 : 19995 10 5 1 : 37959 10 5 14 9 : 07427 10 6 2 : 44911 10 4 1 : 40684 10 4 17 2 : 94001 10 5 1 : 39043 10 3 1 : 6001 10 3 PartialfadingcaseII #fadedhistorymatrices = 1 š 10 = 1 š 2 = 9 š 10 0 9 : 26034 10 12 2 : 31391 10 11 4 : 24903 10 9 2 9 : 27241 10 12 2 : 34361 10 11 4 : 2491 10 9 5 3 : 37716 10 11 6 : 6476 10 10 4 : 44832 10 9 8 3 : 8092 10 10 1 : 99961 10 8 1 : 36941 10 8 11 1 : 47228 10 9 2 : 2715 10 7 1 : 47786 10 7 14 1 : 15821 10 8 1 : 64103 10 6 1 : 47098 10 6 17 5 : 06103 10 7 7 : 87929 10 6 1 : 85274 10 5 PartialfadingcaseIII #fadedhistorymatrices = 1 š 10 = 1 š 2 = 9 š 10 0 9 : 26034 10 12 2 : 31391 10 11 4 : 24903 10 9 2 9 : 26023 10 12 2 : 3113 10 11 4 : 24903 10 9 5 1 : 18462 10 11 7 : 7854 10 11 4 : 23683 10 9 8 1 : 60656 10 10 3 : 38055 10 9 3 : 65689 10 9 11 1 : 32413 10 9 4 : 35421 10 8 8 : 84638 10 9 14 7 : 10271 10 9 3 : 87096 10 7 1 : 7226 10 7 17 2 : 87023 10 8 3 : 71057 10 6 5 : 12104 10 6 50 CHAPTER3 DISTRIBUTED-ORDERFRACTIONALODES:PETROV-GALERKINANDSPECTRAL COLLOCATIONMETHOD 3.1Background Distributedorderfractionaloperatorso˙erarigoroustoolformathematicalmodelingofmulti- physicsphenomena.Inthiscase,thedi˙erentialorderisdistributedoverarangeofvaluesrather thanbeingjusta˝xedfractionasitisinstandard/fractionalODEs/PDEs.Thereisarapidly growinginterestintheuseoffractionalderivativesintheconstructionofmathematicalmodels, whichcontaindistributedordertermsoftheform ¹ ˙ 2 ˙ 1 ˚ ¹ ˙ º a D ˙ t u ¹ t º d ˙ = f ¹ t º ; t > a ; inthe˝eldofuncertaintyquanti˝cationastheinherentuncertaintyofexperimentaldatacanbe directlyincorporatedintothedi˙erentialoperators;see[6,12,47,114,154],forsomeworkon numericalmethods.Almostallofthenumericalschemesdevelopedforsuchmodelsare˝nite- di˙erencemethods.Whilethetreatmentoffractionaldi˙erentialequationswitha˝xedfractional ordercouldbememorydemandingduetothelocalityofthesemethodsandtheirlow-accuracy,the mainchallengeremainstheadditionale˙ectofthediscretizationofthedistributedordermodel, whichmayleadtoexceedingcomputationalcostofnumericalsimulations. Tothebestofourknowledge,the˝rstnumericalstudyofdistributedorderdi˙erentialequations (DODEs)wasperformedbyDiethelmandFordin[46],whereatwo-stagebasicframeworkwas developed.Inthe˝rststage,thedistributedorderdi˙erentiationtermwasapproximatedusinga quadraturerule,andinthesecondstage,asuitablemulti-termnumericalmethodwasemployed. Theylaterperformedthecorrespondingerroranalysisofthemethodin[49].Subsequently,mostof thenumericalstudieshavefollowedthesameapproachyettheyvaryinthediscretizationmethodin thesecondstage.Thedistributedordertime-fractionaldi˙usionequationwasnumericallystudied in[57]andthecorrespondingstabilityandconvergencestudyoftheschemewasprovidedin 51 [58].Addinganonlinearsource,[124]studiedthedistributedorderreactiondi˙usionequation followingthesamescheme.In[98],thesecondstageofthedistributedorderdi˙usionequationwas establishedusingareproducingkernelmethod.Thedistributedordertimefractionaldi˙usion-wave equationwasinvestigatedbydevelopingacompactdi˙erenceschemein[179].Othernumerical studiesinclude:animplicitnumericalmethodofatemporaldistributedorderandtwo-sidedspace- fractionaladvection-dispersionequationin[69],high-orderdi˙erenceschemesin[60],alternating directionimplicit(ADI)di˙erenceschemeswiththeextrapolationmethodforone-dimensional casein[62]andtwo-dimensionalproblemin[61],andanoperationalmatrixtechniquein[50]. TwonewspectraltheoriesonfractionalandtemperedfractionalSturm-Liouvilleproblems (TFSLPs)havebeenrecentlydevelopedbyZayernourietal.in[183,186].Thisapproach˝rst fractionalizesandthentempersthewell-knowntheoryofSturm-Liouvilleeigen-problems.The expliciteigenfunctionsofTFSLPsareanalyticallyobtainedintermsof temperedJacobipoly- fractonomials .Recently,in[185,187,188],Jacobipoly-fractonomialsweresuccessfullyemployed indevelopingaseriesofhigh-orderande˚cientPetrov-Galerkinspectralanddiscontinuousspectral elementmethodsofGalerkinandPetrov-GalerkinprojectiontypeforfractionalODEs.Totreat nonlinearproblemsthe collocation schemesarerelativelyeasytoimplement.Khaderin[83] presentedaChebyshevcollocationmethodforthediscretizationofthespace-fractionaldi˙usion equation.Morerecently,KhaderandHendy[84]developedaLegendrepseudospectralmethod forfractional-orderdelaydi˙erentialequations.Forfasttreatmentofnonlinearandmulti-term fractionalPDEssuchasthefractionalBurgers'equation,anewspectralmethod,called fractional spectralcollocationmethod ,wasdevelopedin[189].Thisnewclassofcollocationschemes introducesanewfamilyoffractionalLagrangeinterpolants,mimickingthestructureoftheJacobi poly-fractonomials.Forvariable-orderfractionalPDEs,afastandspectrallyaccuratecollocation methodwasdevelopedandimplementedin[190]. Inthischapter,we˝rstintroducethe distributedSobolevspaces andtheirassociatednorms. Weshowtheirequivalencetothede˝nedleft-sideandright-sidenormsaswell.Byemploying Riemann-Liouvillederivatives,wede˝nethedistributedorderdi˙erentialequationandthenobtain 52 itsvariationalform.WedevelopaPetrov-Galerkin(PG)spectralmethodfollowingtherecentthe- oryoffractionalSturm-Liouvilleeigen-problems(FSLP)in[186]andemploythecorresponding eigenfunctions,namelythe JacobiPoly-fractonomial sof˝rstkindasthebasesandthe JacobiPoly- fractonomial sofsecondkindastestfunctions.WedevelopaspectrallyaccurateGauss-Legendre quadratureruleintheconstructionofthelinearsystem,whereweinvestigatethestabilityand erroranalysisofthescheme.Inaddition,weconstructaspectrally-accuratefractionalspectral collocationscheme,whereweemployfractionalLagrangeinterpolantssatisfyingtheKronecker deltapropertyatthecollocationpoints,andthen,weobtainthecorrespondingfractionaldi˙erenti- ationmatrices.Wedemonstratethecomputationale˚ciencyofbothschemesconsideringseveral numericalexamplesanddistributionfunctions. Theorganizationofthischapterisasfollows:section3.2providespreliminaryde˝nitionsalong withusefullemmas.WerecallfractionalSobolevspaces,andthen,introducetheirgeneralizationto socalled distributedSobolevspace andassociatednorms,whichprovidesthenaturalsettingofour probleminthisstudy.Wefurthermoreobtainsomeequivalentnormstofacilitatethecorresponding analysisofourmethods.Insection3.3,wederiveanddiscretizethecorrespondingvariational formoftheproblemandsubsequentlyweprovethestabilityandconvergencerateofthescheme. Inaddition,wedevelopafractionalcollocationmethodinsection3.5andtesttheperformanceof thetwomethodsinsection3.6. 3.2Preliminaries Lemma3.2.1. Let ˙;> 0 .ThefractionalderivativeoftheJacobipoly-fractonomials,[186],of ˝rst( i = 1 )andsecondkind( i = 2 )aregivenby RL D ˙ n ¹ i º P n ¹ ˘ º o = ¹ n + º ¹ n + ˙ º ¹ i º P ˙ n ¹ ˘ º ; (3.1) andarealsoofJacobipoly-fractonomialtype,where RL D ˙ RL 1 D ˙ x when i = 1 , RL D ˙ RL x D ˙ 1 when i = 2 . Proof. Seesection(3.7.1). 53 Remark3.2.2. Lemma3.2.1showsthatthestructureofJacobipoly-fractonomialsispreserved undertheactionoffractionalderivatives.Moreover,wenotethatwhen ˙ = inLemma3.2.1,the fractionalderivativesofJacobipoly-fractonomialsareobtainedintermsofLegendrepolynomials, whichhasbeenreportedin[186]. 3.2.1FractionalSobolevSpaces By H s ¹ R º , s 0 ,wedenotethefractionalSobolevspaceon R ,de˝nedas H s ¹ R º = f v 2 L 2 ¹ R ºj¹ 1 + j ! j 2 º s 2 F¹ v º¹ ! º2 L 2 ¹ R ºg ; (3.2) whichisendowedwiththenorm kk s ; R = k¹ 1 + j ! j 2 º s 2 F¹º¹ ! ºk L 2 ¹ R º ; (3.3) where F¹ v º representstheFouriertransformof v .Subsequently,wedenoteby H s ¹ I º , s 0 the fractionalSobolevspaceonany˝niteclosedinterval I ,de˝nedas H s ¹ I º = f v 2 L 2 ¹ I ºj9 ~ v 2 H s ¹ R º s : t : ~ v j I = v g ; (3.4) withthenorm kk s ; I = inf ~ v 2 H s ¹ R º ; ~ v j I = ¹º k ~ v k s ; R : (3.5) Wenotethatthede˝nitionof H s ¹ I º andthecorrespondingnormreliesontheFouriertransformation ofthefunction.Otherusefulnormsassociatedwith H s ¹ I º ,e.g.,when I = » x L ; x R ¼ ,havebeenalso introducedin[99], kk l ; s ; I = kk 2 L 2 ¹ I º + k RL x L D x ¹ºk 2 L 2 ¹ I º 1 2 ; (3.6) kk r ; s ; I = kk 2 L 2 ¹ I º + k RL x D x R ¹ºk 2 L 2 ¹ I º 1 2 ; (3.7) suchthattheleft-side kk l ; s ; I ,theright-sided kk r ; s ; I ,and kk s ; I areshowntobeequivalent. 54 Next,let ˚ 2 L 1 ¹» min ; max ¼º , 0 min < max ,benonnegative.By ˚ H¹ R º ,wedenotethe distributed fractionalSobolevspaceon R ,de˝nedas ˚ H¹ R º = f v 2 L 2 ¹ R ºj ¹ max min h ˚ ¹ º¹ 1 + j ! j 2 º i 1 2 F¹ v º¹ ! º d 2 L 2 ¹ R ºg ; (3.8) whichisendowedwiththenorm kk ˚; R = ¹ max min ˚ ¹ º ¹ 1 + j ! j 2 º 2 F¹º¹ ! º 2 L 2 ¹ R º d ! 1 2 : (3.9) 3.2.2DistributedFractionalSobolevSpaces Wedenoteby ˚ H¹ I º the distributed fractionalSobolevspaceonthe˝niteclosedinterval I ,de˝ned as ˚ H¹ I º = f v 2 L 2 ¹ I ºj9 ~ v 2 ˚ H¹ R º s : t : ~ v j I = v g ; (3.10) withthenorm kk ˚; I = inf ~ v 2 ˚ H¹ R º ; ~ v j I = ¹º k ~ v k ˚; R : (3.11) Moreover,weintroducethefollowingusefulnorms,associatedwith ˚ H¹ I º : kk l ;˚; I = kk 2 L 2 ¹ I º + ¹ max min ˚ ¹ º RL x L D x ¹º 2 L 2 ¹ I º d ! 1 2 ; (3.12) and kk r ;˚; I = kk 2 L 2 ¹ I º + ¹ max min ˚ ¹ º RL x L D x ¹º 2 L 2 ¹ I º d ! 1 2 : (3.13) Wenotethatwhen ˚> 0 iscontinuousin I , ˚ H¹ R º isequivalentto H max ¹ R º .However, ingeneral,thechoiceof ˚ canarbitrarilycon˝nethedomainofintegrationinpractice.Inother words, min and max areonlythetheoreticallowerandupperterminalsinthede˝nitionof distributedorderfractionalderivative.Forinstance,inadistributedsub-di˙usionproblem,the temporalderivativeisassociatedwith min = 0 and max = 1 ,andinasuper-di˙usionproblem, thetheoreticalupperterminal max = 2 .Inthisstudyweparticularlyaimtolet ˚ bede˝nedinany possiblesubsetoftheinterval » min ; max ¼ .Hence,ineachrealizationofaphysicalprocess(e.g. 55 Figure3.1: SchematicofdistributedfractionalSobolevspace ˚ H¹ R º :(left) ˚ = ¹ max º hence ˚ H¹ R º = H max ¹ R º ;(middle) ˚ de˝nedonacompactsupportin » min ; max ¼ ,hence, ˚ H¹ R º˙ H max ¹ R º ;(right) ˚ = ¹ min º ,where ˚ H¹ R º = H min ¹ R º . sub-orsuper-di˙usion) ˚ canbeobtainedfromdata,wherethetheoreticalsettingoftheproblem remainsinvariantyetrequiringthesolutiontohavelessregularity(since ˚ H¹ R º˙ H max ¹ R º in general,seeFig.3.1). Inthefollowingtheorem,weprovetheequivalence(shownbythenotation ˘ )oftheaforemen- tionednorms. Theorem3.2.3. Let ˚ 2 L 1 ¹» min ; max ¼º benon-negative.Then,thenorms kk ˚; I , kk l ;˚; I ,and kk r ;˚; I areequivalent. Proof. Seesection(3.7.2). Lemma3.2.4. [99]:Forall 0 < 1 ,if u 2 H 1 ¹» a ; b ¼º suchthat u ¹ a º = 0 ,and w 2 H š 2 ¹» a ; b ¼º , then ¹ a D s u ; w º = ¹ a D š 2 s u ; s D š 2 b w º ; (3.14) where ¹ ; º representsthestandardinnerproductin = » a ; b ¼ . Lemma3.2.5. Let 1 š 2 << 1 , a and b bearbitrary˝niteorin˝niterealnumbers.Assume u 2 H 2 ¹ a ; b º suchthat u ¹ a º = 0 ,also x D b v isintegrablein ¹ a ; b º suchthat v ¹ b º = 0 .Then ¹ a D 2 x u ; v º = ¹ a D x u ; x D b v º : (3.15) 56 Proof. Seesection(3.7.3). Remark3.2.6. Unlikeotherexistingproofs(e.g.,seeProposition1in[197]),ourproofrequires v ¹ x º toonlyvanishattherightboundary(notethat v ¹ a º canbenon-zero),moreover,weonly requirethe -thderivative(ratherthanthe˝rstderivative)of v ¹ x º tobeintegrablein ¹ a ; b º . 3.3Distributed-OrderDi˙erentialEquations:ProblemDe˝nition Following[11],let 7! ˚ ¹ º beacontinuousmappingin » min ; max ¼ .Then,wede˝nethe distributedorderfractionalderivativeas D D ˚ u ¹ t º = ¹ max min ˚ ¹ º a D t u ¹ t º d ; t > a ; (3.16) where a D t denotesRiemann-Liouvillefractionalderivativeoforder .Next,weaimtosolvethe followingdi˙erentialequationofdistributedorder: D D ˚ u ¹ t º = f ¹ t ; u º ; 8 t 2¹ 0 ; T ¼ ; (3.17) u ¹ 0 º = 0 ; ¹ max º2¹ 0 ; 1 ¼ ; (3.18) u ¹ 0 º = du dt j t = 0 = 0 ; ¹ max º2¹ 1 ; 2 ¼ : (3.19) Inthesequel,wepresentdi˙erentapproachestodiscretizetheaforementioneddi˙erentialoperator. Dueto(1.22),theCaputoandRiemann-Liouvillefractionalderivativesoforder 2¹ 0 ; 1 º coincide witheachotherwhen u ¹ a º = 0 .Therefore,inthisstudy,weemploythede˝nitionofthedistributed fractionalderivativesofRiemann-Liouvillesenseandremovethepre-superscript RL forsimplicity. 3.3.1VariationalFormulation Inordertoobtainthevariationalformwemultiply(4.6)byapropertestfunction v (de˝nedlater) andintegrateoverthecomputationaldomain: ¹ D D ˚ u ¹ t º v ¹ t º d = ¹ f ¹ t ; u º v ¹ t º d : (3.20) 57 Usingthede˝nitionofdistributedorderfractionalderivativesde˝nedin(5.7)weget ¹ max min ˚ ¹ º ¹ 0 D t u ¹ t º v ¹ t º d d = ¹ f ¹ t ; u º v ¹ t º d ; (3.21) where 8 2¹ min ; max º ; ¹ 0 D t u v d = 0 D t u ; v denotesthewell-known L 2 -innerproduct.Giventheinitialconditions(4.7)and/or(4.8)andby Lemmas(7.3.3),wede˝nethebilinearformassociatedwith 2¹ min ; max º as a ¹ º ¹ u ; v º = 0 D t u ; v = ¹ 0 D š 2 t u ; t D š 2 T v º : (3.22) Wechoose v suchthat v ¹ T º = 0 and t D š 2 T v isintegrablein 8 2¹ min ; max º .Moreover,let U bethesolutionspace,de˝nedas U = f u 2 L 2 ¹ º : s ¹ max min ˚ ¹ º 0 D š 2 t u 2 L 2 ¹ º d < 1 (3.23) s : t : u ¹ 0 º = 0 if max 2¹ 0 ; 1 ¼ u ¹ 0 º = du dt j t = 0 = 0 if max 2¹ 1 ; 2 ¼ g andlet V bethetestfunctionspacegivenby V = f v 2 L 2 ¹ º : s ¹ max min ˚ ¹ º t D š 2 T v 2 L 2 ¹ º d < 1 s : t : v ¹ T º = 0 g : (3.24) Theproblemthusreadsas:˝nd u 2 U suchthat a ¹ u ; v º = l ¹ v º ; 8 v 2 V where a ¹ u ; v º : = ¹ max min ˚ ¹ º a ¹ º ¹ u ; v º d (3.25) denotesthe distributed bilinearformand l ¹ v º : = ¹ f ; v º . 58 3.4Petrov-GalerkinMethod:ModalExpansion InthePetrov-Galerkin(PG)method,wefollowtherecenttheoryoffractionalSturm-Liouville eigen-problems(FSLP)in[186]andemploythecorrespondingeigenfunctions,knownasthe Jacobi Poly-fractonomial s(of˝rstkind)giveninthestandarddomain 1 ; 1 ¼ by ¹ 1 º P n ¹ ˘ º = ¹ 1 + ˘ º P ; n 1 ¹ ˘ º ;˘ 2 1 ; 1 ¼ ; (3.26) as non-polynomial basisfunctionsconsistingofafractionaltermmultipliedbytheJacobipolyno- mial P ; n 1 ¹ ˘ º ,henceweemploy(3.26)inconstructionofabasistoformulateaprojectiontype scheme,namely modal expansion.Werepresentthesolutionintermsoftheelementsofthebasis space U N givenasfollows U N = span ¹ 1 º P n ¹ ˘ º ;˘ 2 1 ; 1 ¼ ; n = 1 ; 2 ; ; N ; (3.27) viathepoly-fractonomial modal expansionas u N ¹ ˘ º = N Õ n = 1 c n ¹ 1 º P n ¹ ˘ º ; (3.28) inwhich istobe˝xedasafractionalparameter apriori dependingontherangeofdistribution orderinterval,i.e. 2¹ 0 ; 1 º if max 2¹ 0 ; 1 ¼ and 2¹ 1 ; 2 º if max 2¹ 1 ; 2 ¼ .Itcanalsobetunned tocapturepossiblesingularitiesintheexactsolutionifsomeknowledgeaboutthatisavailable. Moreover,inthePGscheme,weemployanotherspaceoftestfunctions V N ,howeverofthe samedimension,givenby V N = span ¹ 2 º P k ¹ ˘ º ;˘ 2 1 ; 1 ¼ ; k = 1 ; 2 ; ; N ; (3.29) inwhich ¹ 2 º P k ¹ ˘ º = ¹ 1 ˘ º P ; k 1 ¹ ˘ º denotestheJacobipoly-fractonomialofsecondkind,which istheexpliciteigenfunctionoffractionalSturm-Liouvilleproblemofsecondkindin[186]. Itshouldbenotedthatsince ˚ ¹ º 0 and ¯ max min ˚ ¹ º d = 1 itisnotdi˚culttoseethat U N ˆ U and V N ˆ V when ischosenproperly.Therefore,thebilinearform(7.28)reducestothe discrete bilinearform 59 a h ¹ u N ; v N º = ¹ 0 D š 2 t u N ; t D š 2 T v N º (3.30) andthustheproblemreadsas:˝nd u N 2 U N suchthat a h ¹ u N ; v N º = l h ¹ v N º ; 8 v N 2 V N ; (3.31) where a h ¹ u N ; v N º : = ¯ max min ˚ ¹ º a h ¹ u N ; v N º d denotesthe discrete distributedbilinearformand l h ¹ v N º : = ¹ f ; v N º representstheloadvector. Bysubstitutingtheexpansion(3.28),choosing v N = ¹ 2 º P k ¹ ˘ º2 V N , k = 1 ; 2 ; ; N andusing (1.21),thediscretedistributedbilinearformin(4.23)canbewrittenas ¹ max min ˚ ¹ º N Õ n = 1 c n ¹ 2 T º 1 D š 2 ˘ » ¹ 1 º P n ¹ ˘ º¼ ; ˘ D š 2 1 » ¹ 2 º P k ¹ ˘ º¼ d : (3.32) FromLemma3.2.1,wehave 1 D š 2 ˘ » ¹ 1 º P n ¹ ˘ º¼ = ¹ n + º ¹ n + º ¹ 1 º P n ¹ ˘ º (3.33) RL ˘ D š 2 1 » ¹ 2 º P k ¹ ˘ º¼ ; = ¹ k + º ¹ k + º ¹ 2 º P k ¹ ˘ º ; (3.34) where = š 2 .Thus,bychangingtheorderofsummation,theintegral(3.32)takestheform N Õ n = 1 c n ¹ max min ˚ ¹ º¹ 2 T º ¹ n + º ¹ n + º ¹ k + º ¹ k + º ¹ 1 º P n ¹ ˘ º ; ¹ 2 º P k ¹ ˘ º d (3.35) = N Õ n = 1 c n ¹ max min ˚ ¹ º¹ 2 T º ¹ n + º ¹ n + º ¹ k + º ¹ k + º ¹ 1 1 ¹ 1 º P n ¹ ˘ º ¹ 2 º P k ¹ ˘ º d ˘ d ; wherebychangingtheorderofintegrationsweget a h ¹ u N ; v N º (3.36) = N Õ n = 1 c n ¹ 1 1 " ¹ max min ˚ ¹ º¹ 2 T º ¹ n + º ¹ n + º ¹ k + º ¹ k + º ¹ 1 º P n ¹ ˘ º ¹ 2 º P k ¹ ˘ º d # d ˘: 60 Theorem3.4.1 (Spectrally/ExponentiallyAccurateQuadratureRulein -Dimension) . PartA: 8 ˘ = ˘ 0 2 1 ; 1 ¼ ˝xed,and 8 n 2 ” [f 0 g ,theJacobipolynomial P ; n ¹ ˘ 0 º isapolynomialof order n in . PartB:Let ˚ 2 H r ¹» min ; max ¼º , r > 0 .Then 8 max š 2 ¹ max min ˚ ¹ º¹ 2 T º ¹ n + º ¹ n + º ¹ k + º ¹ k + º ¹ 1 º P n ¹ ˘ º ¹ 2 º P k ¹ ˘ º d Q Õ q = 1 e w q ˚ ¹ q º¹ 2 T º q ¹ n + º ¹ n + q º ¹ k + º ¹ k + q º ¹ 1 º P q n ¹ ˘ º ¹ 2 º P q k ¹ ˘ º C Q r k ˚ k H r ¹» min ; max ¼º ; where C > 0 , ˚ N ¹ º = Í N n = 0 e ˆ n P n ¹ º denotesthepolynomialexpansionof ˚ ¹ º ,and f q ; e w q g Q q = 1 representsthesetofGauss-Legendrequadraturepointsandweights. PartC:If ˚ ¹ º issmooth,thequadraturerulein -dimensionbecomesexponentiallyaccuratein Q . Proof. Seesection(3.7.4). Bytheorem(4.3.2)andperformingana˚nemappingfrom » 2 min ; max ¼ tothestandard domain st 2 1 ; 1 ¼ ,theinnerintegralin(3.36)canbeevaluatedwithspectralaccuracyby employingaGauss-Legendrequadraturerule.Thenbychangingtheorderofsummationand integral a h ¹ u N ; v N º = l h ¹ v N º canbewrittenas: N Õ n = 1 Q Õ j = 1 c n J w j ˚ j ¹ 2 T º j ¹ n + º ¹ n + j º ¹ k + º ¹ k + j º ¹ 1 1 ¹ 1 º P j n ¹ ˘ º ¹ 2 º P j k ¹ ˘ º d ˘ (3.37) = f ; ¹ 2 º P k ¹ ˘ º ; k = 1 ; 2 ;:::; N ; where J = d d st = ¹ max min º 2 isconstantand j = ¹ st j º , ˚ j = ˚ ¹ ¹ st j ºº , j = j š 2 and st j and w j arethequadraturepointsandweightsrespectively.Thelinearsystemisthen constructedas S ® c = ® F ; (3.38) 61 inwhichtheentriesofthesti˙nessmatrix S andforcevector ® F aregivenby S kn = Q 1 Õ j = 0 J w j ˚ j ¹ 2 T º j C kn ¹ 1 1 ¹ 1 º P j n ¹ ˘ º ¹ 2 º P j k ¹ ˘ º d ˘ (3.39) and F k = ¹ 1 1 f ¹ ˘ º ¹ 2 º P k ¹ ˘ º d ˘ (3.40) respectively,where C kn ¹ n + º ¹ n + j º ¹ k + º ¹ k + j º : Remark3.4.2. Foreach˝xed j andgiventhestructureof ¹ 1 º P j n ¹ ˘ º and ¹ 2 º P j k ¹ ˘ º ,theabove integrationstaketheform ¹ 1 1 ¹ 1 º P j n ¹ ˘ º ¹ 2 º P j k ¹ ˘ º d ˘ = ¹ 1 1 ¹ 1 ˘ º j ¹ 1 + ˘ º j P j ; j k 1 ¹ ˘ º P j ; j n 1 ¹ ˘ º d ˘; ¹ 1 1 f ¹ ˘ º ¹ 2 º P k ¹ ˘ º d ˘ = ¹ 1 1 ¹ 1 ˘ º f ¹ ˘ º P ; k 1 ¹ ˘ º d ˘; andtherefore,thefullsti˙nessmatrix S andvector ® F canbeconstructedaccuratelyusingaproper Gauss-Lobatto-Jacobirulecorrespondingtotheweightfunction ¹ 1 ˘ º j ¹ 1 + ˘ º j and ¹ 1 ˘ º respectively. 3.4.1DiscreteStabilityAnalysis Inthissection,weinvestigatethestabilityofthenumericalscheme,developedbasedonthe aforementionedchoiceofsolutionandtestfunctionspaceconsideringthebilinearformin(4.23). Theorem3.4.3. Thescheme (4.23) isstableandthefollowinginequalityholds inf u N 2 U N sup v N 2 V N a h ¹ u N ; v N º k u N k U N k v N k V N : Proof. Recallingfrom(4.15) a h ¹ u N ; v N º = ¹ max min ˚ ¹ º a h ¹ u N ; v N º d ; 62 where,bylemma(3.2.1), a h ¹ u N ; v N º = a h ¹ N Õ n = 1 a n ¹ 1 º P n ¹ ˘ º ; N Õ k = 1 a k ¹ 2 º P k ¹ ˘ ºº ; 8 2¹ min ; max º ; = N Õ n = 1 N Õ k = 1 a n a k C ; n ; k ¹ 1 1 ¹ 1 ˘ º ~ ¹ 1 + ˘ º ~ ¹ 1 + ˘ º 2~ P ~ ; ~ k 1 ¹ ˘ º P ~ ; ~ n 1 ¹ ˘ º d ˘; inwhich, ~ = š 2 , C ; n ; k = ¹ n + º ¹ n + ~ º ¹ k + º ¹ k + ~ º and ¹ 1 + ˘ º ~ isreplacedby ¹ 1 + ˘ º ~ ¹ 1 + ˘ º 2~ .Welet ~ > 1 š 2 ,hencethefunction ¹ 1 + ˘ º 2~ isnonnegative,nondecreasing,continuousandintegrable intheintegrationdomain.Therefore, a h ¹ u N ; v N º N Õ n = 1 N Õ k = 1 a n a k C ; n ; k C 0 ¹ 1 1 ¹ 1 ˘ º ~ ¹ 1 + ˘ º ~ P ~ ; ~ k 1 ¹ ˘ º P ~ ; ~ n 1 ¹ ˘ º d ˘: Moreover,theJacobipolynomial P ~ ; ~ n 1 ¹ ˘ º canbeexpandedas: P ~ ; ~ n 1 ¹ ˘ º = n 1 Õ j = 0 n 1 + j j n 1 + ~ n 1 j 1 º j n + 1 ¹ 1 2 º j ¹ 1 + ˘ º j : Bymultiplyinganddividingeachtermwithinthesummationby n 1 ~ n 1 j weget P ~ ; ~ n 1 ¹ ˘ º = n 1 Õ j = 0 A ~ n ; j n 1 + j j n 1 ~ n 1 j 1 º j n + 1 ¹ 1 2 º j ¹ 1 + ˘ º j ; where A ~ n ; j = n 1 + ~ n 1 j š n 1 ~ n 1 j isnondecreasing,positiveandbounded 8 n ; j ; ~ .Therefore,there exists C 1 = C 1 ¹ n º > 0 suchthat a h ¹ u N ; v N º N Õ n = 1 N Õ k = 1 a n a k C ; n ; k C 0 C 1 ¹ n º ¹ 1 1 ¹ 1 ˘ º ~ ¹ 1 + ˘ º ~ P ~ ; ~ k 1 ¹ ˘ º P ~ ; ~ n 1 ¹ ˘ º d ˘ C 0 N Õ k = 1 a 2 k C ; k ; k C 1 ¹ n º ~ ; ~ k 1 ; inwhich ~ ; ~ k 1 = 2 2 k 1 ¹ k + º ¹ k º ¹ k 1 º ! ¹ k º .Hence, a h ¹ u N ; v N º C 0 ¹ max min ˚ ¹ º N Õ k = 1 a 2 k C ; k ; k C 1 ¹ n º ~ ; ~ k 1 d : (3.41) 63 Moreover,wehave k v N k 2 V N = ¹ max min ˚ ¹ º 1 D š 2 x v N 2 L 2 ¹ º d ; wherebyconsidering v n = Í N k = 1 a k ¹ 2 º P k ¹ ˘ º ,wecanwrite 8 2¹ min ; max º , 1 D š 2 x v N 2 L 2 ¹ º = ¹ 1 1 ¹ N Õ k = 1 a k ¹ k + º ¹ k + ~ º ¹ 1 ˘ º ~ P ~ ; ~ k 1 ¹ ˘ ºº 2 d ˘; ¹ 1 1 N Õ k = 1 a 2 k ¹ k + º ¹ k + ~ º 2 ¹ 1 ˘ º 2~ ¹ P ~ ; ~ k 1 ¹ ˘ ºº 2 d ˘; (ByJensenInequality) : Bymultiplyingtheintegrandby ¹ 1 + ˘ º ~ ¹ 1 + ˘ º ~ andchangingtheorderofsummationand integration,weobtain 1 D š 2 x v N 2 L 2 ¹ º N Õ k = 1 a 2 k ¹ k + º ¹ k + ~ º 2 ¹ 1 1 ¹ 1 ˘ º ~ ¹ 1 + ˘ º ~ ¹ 1 ˘ 2 º ~ P ~ ; ~ k 1 ¹ ˘ º P ~ ; ~ k 1 ¹ ˘ º d ˘; N Õ k = 1 a 2 k ¹ k + º ¹ k + ~ º 2 C 2 ¹ 1 1 ¹ 1 ˘ º ~ ¹ 1 + ˘ º ~ P ~ ; ~ k 1 ¹ ˘ º P ~ ; ~ k 1 ¹ ˘ º d ˘; since ~ > 1 š 2 andconsequently ¹ 1 ˘ 2 º ~ isanonnegativeandintegrableinthedomainof integration.BytheorthogonalityofJacobipolynomials,weget 1 D š 2 x v N 2 L 2 ¹ º C 2 N Õ k = 1 a 2 k ¹ k + º ¹ k + ~ º 2 ~ ; ~ k 1 ; andthus k v N k V N v u t C 2 ¹ max min ˚ ¹ º N Õ k = 1 a 2 k ¹ k + º ¹ k + ~ º 2 ~ ; ~ k 1 d : (3.42) Similarlyfor k u N k 2 U N : k u N k 2 U N = ¹ max min ˚ ¹ º 1 D š 2 x u N 2 L 2 ¹ º d ; where 8 2¹ min ; max º : 1 D š 2 x u N 2 L 2 ¹ º = ¹ 1 1 ¹ N Õ n = 1 a n ¹ n + º ¹ n + ~ º ¹ 1 + ˘ º ~ P ~ ; ~ n 1 ¹ ˘ ºº 2 d ˘; ¹ 1 1 N Õ n = 1 a 2 n ¹ n + º ¹ n + ~ º 2 ¹ 1 + ˘ º 2~ ¹ P ~ ; ~ n 1 ¹ ˘ ºº 2 d ˘; ByJensenInequality : 64 Followingsimilarsteps,andbymultiplyingtheintegrandby ¹ 1 ˘ º ~ ¹ 1 ˘ º ~ andchangingthe orderofsummationandintegration,weobtain 1 D š 2 x u N 2 L 2 ¹ º ; N Õ n = 1 a 2 n ¹ n + º ¹ n + ~ º 2 ¹ 1 1 ¹ 1 ˘ º ~ ¹ 1 + ˘ º ~ ¹ 1 ˘ 2 º ~ P ~ ; ~ n 1 ¹ ˘ º P ~ ; ~ n 1 ¹ ˘ º d ˘; N Õ n = 1 a 2 n ¹ n + º ¹ n + ~ º 2 C 3 ¹ 1 1 ¹ 1 ˘ º ~ ¹ 1 + ˘ º ~ P ~ ; ~ n 1 ¹ ˘ º P ~ ; ~ n 1 ¹ ˘ º d ˘; since ¹ 1 ˘ 2 º ~ isanonnegativeandintegrableinthedomainofintegral.Next,bytheorthogonality ofJacobipolynomials, 1 D š 2 x u N 2 L 2 ¹ º C 3 N Õ n = 1 a 2 n ¹ n + º ¹ n + ~ º 2 ~ ; ~ n 1 : Therefore, k u N k U N v u t C 3 ¹ max min ˚ ¹ º N Õ n = 1 a 2 n ¹ n + º ¹ n + ~ º 2 ~ ; ~ n 1 d ; (3.43) where ~ ; ~ n 1 = ~ ; ~ n 1 . Therefore,using(3.41),(3.42)and(3.43), inf u N 2 U N sup v N 2 V N a h ¹ u N ; v N º k u N k U N k v N k V N C 0 C 1 min p C 2 C 3 = : 3.4.2ProjectionErrorAnalysis Inthissection,weinvestigatetheerrorduetotheprojectionofthetruesolutionontothede˝ned setofbasisfunctions. Theorem3.4.4. Let d r u dt r 2 U ,thatis, ¯ max min ˚ ¹ º 0 D r + š 2 t u 2 L 2 ¹ º d < 1 and u N denotesthe projectionoftheexactsolution u .Then, k u u N k 2 U C N 2 r ¹ max min ˚ ¹ º 0 D r + š 2 t u 2 L 2 ¹ º d : (3.44) 65 Proof. Byperformingana˚nemappingfrom t 2» 0 ; T ¼ tothestandarddomain ˘ 2 1 ; 1 ¼ ,we expandtheexactsolution u intermsofthefollowingin˝niteseriesofJacobipoly-fractonomials u ¹ ˘ º = 1 Õ n = 1 c n ¹ 1 º P n ¹ ˘ º : (3.45) Then,wenotethatbyusing(3.2.1)and(1.21), 0 D r + š 2 t u ¹ ˘ ¹ t ºº = ¹ 2 T º r + š 2 1 I š 2 ˘ d r d ˘ r 1 D ˘ u ¹ ˘ º ; = ¹ 2 T º r + š 2 1 Õ n = 1 c n ¹ n + º ¹ n º 1 I š 2 ˘ d r d ˘ r » P n 1 ¹ ˘ º¼ ; where, d r d ˘ r » P n 1 ¹ ˘ º¼ = 8 > >< > > : ¹ n 1 + r º ! 2 r ¹ n 1 º ! P r ; r n 1 r ¹ ˘ º r < n ; 0 r n : Thus,bymultiplyingwithaproperweightfunction, w ¹ ˘ º = ¹ 1 + ˘ º r š 2 + š 2 ¹ 1 ˘ º r š 2 ,the right-hand-sideof(3.44)takestheform ¹ max min ˚ ¹ º¹ 2 T º 2 r + š 2 ¹ 1 + ˘ º r š 2 + š 2 ¹ 1 ˘ º r š 2 (3.46) 1 Õ n = r + 1 c n ¹ n + º ¹ n º ¹ n 1 + r º ! 2 r ¹ n 1 º ! 1 I š 2 ˘ P r ; r n 1 r ¹ ˘ º 2 L 2 ¹ º d : ByexpandingtheJacobipolynomialas P r ; r n 1 r ¹ ˘ º = 1 º n 1 r n 1 r Õ j = 0 n 1 + r + j j n 1 n 1 r j ¹ 1 2 º j ¹ 1 + ˘ º j ; andchangingtheorderofsummationandtheintegration,weobtainthefractionalintegralas 1 I š 2 ˘ P r ; r n 1 r ¹ ˘ º ; (3.47) = 1 º n 1 r ¹ 1 + ˘ º š 2 n 1 r Õ j = 0 n 1 + r + j j n 1 n 1 r j ¹ 1 2 º j ¹ 1 + j º ¹ 1 + j + š 2 º ¹ 1 + ˘ º j ; = 1 º n 1 r ¹ 1 + ˘ º š 2 n 1 r Õ q = 0 ~ c q ¹ º P r ; r q ¹ ˘ º ; 66 where,thecoe˚cient, ~ c q ¹ º ,canbeobtainedusingtheorthogonalityofJacobipolynomials.Hence, bytaking C n = c n 1 º n 1 ¹ n + º ¹ n º ,(3.46)takestheform ¹ max min ˚ ¹ º¹ 2 T º 2 r + ¹ 1 + ˘ º r š 2 ¹ 1 ˘ º r š 2 1 Õ n = r + 1 C n ¹ n 1 + r º ! 2 r ¹ n 1 º ! 1 º r n 1 r Õ q = 0 ~ c q ¹ º P r ; r q ¹ ˘ º 2 L 2 ¹ º d ; (3.48) = ¹ max min ˚ ¹ º¹ 2 T º 2 r + 2 6 6 6 6 4 ¹ 1 1 ¹ 1 + ˘ º r ¹ 1 ˘ º r © « 1 Õ n = r + 1 C n ¹ n 1 + r º ! 2 r ¹ n 1 º ! n 1 r Õ q = 0 ~ c q ¹ º P r ; r q ¹ ˘ º ª ® ¬ 2 d ˘ 3 7 7 7 7 5 d : Then,wechangetheorderoftwosummationsinordertousetheorthogonalityofJacobipolynomials andobtain ¹ max min ˚ ¹ º¹ 2 T º 2 r + 2 6 6 6 6 4 ¹ 1 1 ¹ 1 + ˘ º r ¹ 1 ˘ º r © « 1 Õ q = 0 1 Õ n = r + 1 + q C n ¹ n 1 + r º ! 2 r ¹ n 1 º ! ~ c q ¹ º P r ; r q ¹ ˘ º ª ® ¬ 2 d ˘ 3 7 7 7 7 5 d ; (3.49) = ¹ max min ˚ ¹ º¹ 2 T º 2 r + 2 6 6 6 6 4 ¹ 1 1 ¹ 1 + ˘ º r ¹ 1 ˘ º r 1 Õ q = 0 © « 1 Õ n = r + 1 + q C n ¹ n 1 + r º ! 2 r ¹ n 1 º ! ª ® ¬ 2 ~ c 2 q ¹ º¹ P r ; r q ¹ ˘ ºº 2 d ˘ 3 7 7 7 7 5 d ; = 1 Õ q = 0 © « 1 Õ n = r + 1 + q C n ¹ n 1 + r º ! 2 r ¹ n 1 º ! ª ® ¬ 2 ¹ 1 1 ¹ 1 + ˘ º r ¹ 1 ˘ º r ¹ P r ; r q ¹ ˘ ºº 2 d ˘ ¹ max min ˚ ¹ º¹ 2 T º 2 r + ~ c 2 q ¹ º d ; = 1 Õ q = 0 © « 1 Õ n = r + 1 + q C n ¹ n 1 + r º ! 2 r ¹ n 1 º ! ª ® ¬ 2 2 2 r + 1 2 q + 2 r + 1 ¹¹ q + 1 º ! º 2 q ! ¹ q + 2 r º ! ¹ max min ˚ ¹ º¹ 2 T º 2 r + ~ c 2 q ¹ º d ; = 2 2 r + 1 ¹ 2 T º 2 r ¹ 2 r º 2 1 Õ q = 0 © « 1 Õ n = r + 1 + q C n ¹ n 1 + r º ! ¹ n 1 º ! ª ® ¬ 2 ¹ q + 1 º 2 q ! ¹ 2 q + 2 r + 1 º¹ q + 2 r º ! ¹ max min ˚ ¹ º¹ 2 T º ~ c 2 q ¹ º d : Moreover,usingtheapproximationofthesolutiongivenin(3.28)andbymultiplyingwiththe properweightfunctions,theleft-hand-sideof(3.44)takestheform ¹ 1 + ˘ º + š 2 ¹ u u N º 2 U ; (3.50) = ¹ max min ˚ ¹ º¹ 2 T º ¹ 1 + ˘ º + š 2 1 Õ n = N + 1 c n ¹ n + º ¹ n º 1 I š 2 ˘ P n 1 ¹ ˘ º 2 L 2 ¹ º d ; 67 inwhich, 1 D š 2 ˘ = 1 D š 2 + ˘ = 1 I š 2 ˘ 1 D ˘ andthefractionalderivativeistakenusing (3.2.1).ByexpandingtheLegendrepolynomialas P n 1 ¹ ˘ º = 1 º n 1 n 1 Õ j = 0 n 1 + j j n 1 n 1 j ¹ 1 2 º j ¹ 1 + ˘ º j ; andfollowingsimilarstepsasin(3.47),weobtainthefractionalintegralas 1 I š 2 ˘ P n 1 ¹ ˘ º = 1 º n 1 ¹ 1 + ˘ º š 2 n 1 Õ q = 0 ~ a q ¹ º P q ¹ ˘ º ; (3.51) wherethecoe˚cient, ~ a q ¹ º ,canbeobtainedusingtheorthogonalityofLegendrepolynomials. Hence,(3.50)takestheform ¹ 1 + ˘ º + š 2 ¹ u u N º 2 U = ¹ max min ˚ ¹ º¹ 2 T º ¹ 1 1 © « 1 Õ n = N + 1 C n n 1 Õ q = 0 ~ a q ¹ º P q ¹ ˘ º ª ® ¬ 2 d ˘ d ; (3.52) inwhich, C n = c n 1 º n 1 ¹ n + º ¹ n º .Wechangetheorderoftwosummationstousetheorthogonality ofLegendrepolynomialsandobtain ¹ max min ˚ ¹ º¹ 2 T º ¹ 1 1 © « N Õ q = 0 1 Õ n = q + 1 C n ~ a q ¹ º P q ¹ ˘ º + 1 Õ q = N + 1 1 Õ n = q + 1 C n ~ a q ¹ º P q ¹ ˘ º ª ® ¬ 2 d ˘ d ; = ¹ max min ˚ ¹ º¹ 2 T º N Õ q = 0 © « 1 Õ n = q + 1 C n ª ® ¬ 2 ~ a 2 q ¹ º ¹ 1 1 ¹ P q ¹ ˘ ºº 2 d ˘ + 1 Õ q = N + 1 © « 1 Õ n = q + 1 C n ª ® ¬ 2 ~ a 2 q ¹ º ¹ 1 1 ¹ P q ¹ ˘ ºº 2 d ˘ ! d ; 1 Õ q = N + 1 © « 1 Õ n = q + 1 C n ª ® ¬ 2 2 2 q + 1 ¹ max min ˚ ¹ º¹ 2 T º ~ a 2 q ¹ º d ; N ! ¹ N r º ! 2 1 Õ q = N + 1 © « 1 Õ n = q + 1 C n ¹ n 1 + r º ! ¹ n 1 º ! ª ® ¬ 2 2 2 q + 1 ¹ q + 1 º 2 q ! ¹ 2 q + 2 r + 1 º¹ q + 2 r º ! ¹ max min ˚ ¹ º¹ 2 T º ~ a 2 q ¹ º d ; N ! ¹ N r º ! 2 ¹ max min ˚ ¹ º¹ 2 T º 2 r + ¹ 1 + ˘ º r š 2 + š 2 ¹ 1 ˘ º r š 2 1 D r + š 2 ˘ u 2 L 2 ¹ º d ; N ! ¹ N r º ! 2 ¹ max min ˚ ¹ º 0 D r + š 2 t u 2 L 2 ¹ º d : (3.53) 68 Therefore, k u u N k 2 U ¹ 1 + ˘ º + š 2 ¹ u u N º 2 U C N 2 r ¹ max min ˚ ¹ º 0 D r + š 2 t u 2 L 2 ¹ º d : Remark3.4.5. Sincetheinf-supconditionholds(seeTheorem (3.4.3) ),bytheBanach-Ne c as- Babu s katheorem[51],theerrorinthenumericalschemeislessthatorequaltoaconstanttimes theprojectionerror.Choosingtheprojection u N inTheorem (3.4.4) ,weinferthespectralaccuracy ofthescheme. 3.5FractionalCollocationMethod:NodalExpansion Next,werepresentthesolutionviathefollowingpoly-fractonomial nodal expansionas u N ¹ ˘ º = N Õ j = 1 u N ¹ ˘ j º h j ¹ ˘ º ; (3.54) where h j ¹ ˘ º represent fractionalLagrangeinterpolants FLIs,whicharealloffractionalorder ¹ N + 1 º andconstructedusingtheaforementionedinterpolationspoints 1 = ˘ 1 <˘ 2 < < ˘ N = 1 as: h j ¹ ˘ º = ˘ x 1 x j x 1 N Ö k = 1 k , j ˘ x k x j x k ; j = 2 ; 3 ; ; N : (3.55) BecauseofthehomogeneousDirichletboundarycondition(s)in(4.7)and(4.8), u N 1 º = 0 ,and thusweonlyconstruct h j ¹ ˘ º for j = 2 ; 3 ; ; N .WenotethatFLIssatisfytheKroneckerdelta property,i.e., h j ¹ ˘ k º = jk ,atinterpolationpoints,howevertheyvaryasapoly-fractonomial between ˘ k 's. 3.5.1Di˙erentiationMatricesD andD 1 + , 2¹ 0 ; 1 º Bybreakingthedomainofintegrationin ,(5.7)takestheform ¹ 1 min ˚ ¹ º 0 D t u ¹ t º d + ¹ max 1 ˚ ¹ º 0 D t u ¹ t º d = f ¹ t ; u º ; 8 t 2¹ 0 ; T ¼ : (3.56) 69 Following[189],weobtainthecorrespondingfractionaldi˙erentiationmatrices D and D 1 + , 2¹ 0 ; 1 º bysubstituting(3.55)in(3.54)andtakingthe -thorderfractionalderivative.These matricesaregivenas: D ij = 1 ¹ ˘ j + 1 º N Õ n = 1 j n n 1 Õ q = d e b nq ¹ ˘ i + 1 º q + ; (3.57) and D 1 + ij = 1 ¹ ˘ j + 1 º h N Õ n = 1 j n n 1 Õ q = d e b nq ¹ q + º¹ ˘ i + 1 º q + 1 i ; (3.58) inwhich d e denotestheceilingof and b nq = 1 º n + q 1 ¹ 1 2 º q © « n 1 + q q ª ® ® ¬ © « n 1 + n 1 q ª ® ® ¬ ¹ q + + 1 º ¹ q + + 1 º : (3.59) Thecoe˚cients, j n ,arethecoe˚cientsinexpansionofthepolynomial p j ¹ ˘ º = Î N k = 1 k , j ˘ ˘ k ˘ j ˘ k in termsofJacobipolynomialsas N Ö k = 1 k , j ˘ ˘ k x j x k = N Õ n = 1 j n P ; n 1 ¹ ˘ º : (3.60) DuetotheorthogonalityoftheJacobipoly-fractonomials P ; n 1 ¹ ˘ º withrespecttotheweight function w ¹ ˘ º = ¹ 1 ˘ º ¹ 1 + ˘ º ,thesecoe˚cientscanbecomputede˚cientlyonly once by employingaproperGuass-Lobatto-Jacobiquadraturerule. Therefore,bysubstitutingthenodalexpansion(3.55)into(3.56),performingana˚nemapping from » min ; max ¼ tothestandarddomain 1 ; 1 ¼ ,andemployingaproperquadraturerulein -domain,(3.56)canbewrittenas N Õ j = 2 2 6 6 6 6 4 Q Õ q w q ˚ ¹ q º¹ 2 T º q D q ij + D 1 + q ij 3 7 7 7 7 5 u N ¹ ˘ j º = f ¹ ˘ i º ; (3.61) N Õ j = 2 D i ; j u N ¹ ˘ j º = f ¹ ˘ i º ; i = 2 ; 3 ; ; N : 70 Remark3.5.1. Multi-termproblemscanbegeneralizedtothedistributedordercounterparts throughthede˝nitionofdistributionfunction ˚ ¹ º .Forinstance,iftheoperatorconsistsof multiplefractionalorders 0 < 1 < 2 < < P 2 ,thecorrespondingmulti-termproblem p = P Õ p = 1 0 D p t u ¹ t º = f ¹ t º canberepresentedasadistributedorderproblemoftheform (5.7) ,inwhich ˚ ¹ º = Í p = P p = 1 ¹ p º .Wenotethatinthiscase,thedistributedfractionalSobolevspace, ˚ H¹ R º ,coincideswith thefractionalSobolevspace, H P ¹ R º .Thechoiceofcollocation/interpolationpointsisthekeyto constructwell-conditionedlinearsystemswithoptimalapproximability.Inthepresentwork,we leave inexpansion (3.54) asafreeinterpolationparametertocapturepossiblesingularitiesand employthezerosofLegendrepolynomialsastheinterpolationcollocation/interpolationpoints. 3.6NumericalSimulations Inordertoexaminetheconvergenceoftheschemeswith modal and nodal expansions,we considerproblemswithsmoothandnon-smoothsolutions. 3.6.1SmoothSolutions Let 2» 0 ; 2 ¼ andconsiderthefollowingtwocases: ‹ CaseI: u ext = t 5 , ˚ ¹ º = ¹ 6 ºš 5! , f ¹ t º = ¹ t 5 t 3 º log ¹ t º ‹ CaseII: u ext = t 3 , ˚ ¹ º = ¹ 4 º sinh ¹ º , f ¹ t º = 6 t ¹ t 2 cosh ¹ 2 sinh ¹ 2 º log ¹ t ºº ¹ log ¹ t º 2 1 º . Bytakingthesimulationtime T = 2 andfordi˙erentchoicesof ,weprovidetheconvergence studyin L 1 -norm, L 2 -norm, H 1 -normand ˚ H -normusingthePGschemeandin L 1 -norm usingthecollocationscheme.Itisobservedthatthechoiceof hasanimportante˙ectonthe convergencebehaviourofthescheme.Forinstance,sincetheexactsolutionisapolynomial,as ! 1 ,werecovertheexponentialconvergenceincapturingtheexactsolution. 71 Table3.1:Case-I;PGschemeconvergencestudyin L 1 -norm, L 2 -norm, H 1 -normand ˚ H -norm, where T = 2 . 1 = 1 + 10 4 N L 1 -Error L 2 -Error H 1 -Error ˚ H -Error 2 9 : 497843 : 3806320 : 6046 : 52507 4 0 : 1634860 : 08233680 : 8027570 : 187176 6 9 : 71043 10 8 6 : 7433 10 8 8 : 37613 10 7 1 : 70551 10 7 8 2 : 9053 10 9 2 : 32457 10 9 3 : 53574 10 8 6 : 59486 10 9 10 2 : 27748 10 10 2 : 01002 10 10 3 : 67074 10 9 6 : 38469 10 10 1 = 1 : 1 N L 1 -Error L 2 -Error H 1 -Error ˚ H -Error 2 9 : 67763 : 289823 : 30046 : 38693 4 0 : 1604340 : 06613040 : 8728090 : 157957 6 0 : 00009479420 : 00005897840 : 001074580 : 00015822 8 3 : 10668 10 6 2 : 19939 10 6 0 : 00005077376 : 59429 10 6 10 2 : 48519 10 7 1 : 9822 10 7 5 : 5753 10 6 6 : 61409 10 7 1 = 1 : 5 N L 1 -Error L 2 -Error H 1 -Error ˚ H -Error 2 9 : 84763 : 1068135 : 84575 : 96161 4 0 : 1025340 : 02649740 : 9491830 : 0718496 6 0 : 0005849950 : 000151060 : 01172350 : 000524729 8 0 : 00002726557 : 37649 10 6 0 : 0009891580 : 0000306404 10 2 : 75271 10 6 7 : 75346 10 7 0 : 0001588233 : 72512 10 6 Tables3.1and3.2showtheconvergencebehaviourofthesimulationresultsbasedonthe PGschemeforthetwocase-IandIIrespectively.Tables3.3showstheconvergencebehaviour ofthesimulationresultsbasedonthecollocationschemeforthetwocasesIandII.Table3.4 showstheconvergencebehaviourofthesimulationresultsbasedonthecollocationschemefor thecasewheretheexactsolutionisthesameascase-Ibutthedistributionfunctionis ˚ ¹ º = Í 4 p = 1 ¹ p º withthefractionalorders f 1 š 10 ; 1 š 2 ; 13 š 10 ; 19 š 10 g andtheforcingfunctionis f ¹ t º = Í 4 p = 1 120 ¹ 6 p º t 5 p . 3.6.2Non-SmoothSolutions SincetheexactsolutionisnotalwaysknownandincontrasttothestandardfractionalODEswhere theforcingtermgivessomeregularityinformationabouttheexactsolution,indistributedorder problemssuchapredictionisratherdi˚culttomake.Hence,thefractionalparameter canplay theroleofa ˝ne-tuningknob givingthepossibilityofsearchingforthebest/optimalcase,wherethe highestratecanbeachievedwithminimaldegreesoffreedom.Here,welet 2» 0 ; 1 ¼ andconsider 72 Table3.2:Case-II;PGschemeconvergencestudyin L 1 -norm, L 2 -norm, H 1 -normand ˚ H - norm,where T = 2 . 1 = 1 + 10 4 N L 1 -Error L 2 -Error H 1 -Error ˚ H -Error 2 0 : 3791340 : 3252531 : 443921 : 86897 4 6 : 80222 10 7 6 : 33141 10 7 4 : 61395 10 6 5 : 29606 10 6 6 5 : 22608 10 8 4 : 52071 10 8 4 : 80236 10 7 5 : 08899 10 7 8 1 : 27547 10 8 9 : 98313 10 9 1 : 0532 10 7 1 : 049 10 7 10 7 : 31142 10 9 7 : 21402 10 9 3 : 44882 10 8 3 : 39574 10 8 1 = 1 : 1 N L 1 -Error L 2 -Error H 1 -Error ˚ H -Error 2 0 : 3696820 : 2638291 : 453841 : 62458 4 0 : 0006465570 : 0005699950 : 005486080 : 00499413 6 0 : 00004583340 : 00004389260 : 0006360230 : 000511403 8 7 : 74333 10 6 7 : 36329 10 6 0 : 0001471770 : 000107932 10 2 : 02013 10 6 1 : 84714 10 6 0 : 0000482120 : 0000327428 1 = 1 : 5 N L 1 -Error L 2 -Error H 1 -Error ˚ H -Error 2 0 : 2885080 : 1148711 : 254710 : 848595 4 0 : 004039160 : 001639790 : 05116670 : 0190804 6 0 : 0004060950 : 0001698170 : 01069090 : 00268063 8 0 : 00007893520 : 00003369390 : 003586980 : 000671243 10 0 : 00002192759 : 49574 10 6 0 : 001537710 : 000228446 Table3.3:Case-IandII;collocationschemeconvergencestudyin L 1 -norm,where T = 2 . = 1 10 10 = 7 š 10 = 1 š 10 NCase-ICase-IICase-ICase-IICase-ICase-II 2 2 : 59 10 + 1 5 : 743 : 0 10 + 1 8 : 844 : 3 10 + 1 19 : 91 4 6 : 81 10 1 5 : 30 10 12 1 : 10 10 + 1 2 : 58 10 1 2 : 51 10 + 1 1 : 01 10 1 6 3 : 87 10 13 2 : 15 10 13 1 : 43 10 3 1 : 52 10 3 3 : 48 10 3 8 : 03 10 3 8 1 : 10 10 14 2 : 68 10 14 3 : 10 10 5 3 : 34 10 4 8 : 38 10 5 1 : 83 10 3 10 8 : 75 10 15 7 : 01 10 15 2 : 12 10 6 1 : 12 10 4 1 : 0 10 5 6 : 25 10 4 Table3.4:Multi-termcase;collocationschemeconvergencestudyin L 1 -norm,where T = 2 . N = 1 10 10 = 7 š 10 = 1 š 10 6 2 : 99596 10 4 1 : 93088 10 3 7 : 10859 10 2 10 4 : 40056 10 7 7 : 90916 10 6 1 : 95735 10 4 14 9 : 35031 10 9 3 : 39228 10 7 7 : 99603 10 6 18 9 : 15918 10 10 3 : 53369 10 8 8 : 27226 10 7 thefollowingtwocasesofsingularsolution,wherebytheproperchoiceof wecaneasilycapture thesingularityofthesolution. ‹ CaseIII: u ext = t 0 , ˚ ¹ º = ¹ 1 + 0 ºš¹ 0 º ! , 0 = 1 š 10 ; 9 š 10 , ‹ CaseIV: u ext = t 0 sin ¹ t º , ˚ ¹ º , 0 = 75 š 100 ; 25 š 100 . Incase-III,weareabletoobtaintheexactsolutiononlywithonetermbychoosing = 0 .In 73 case-IV,wetake = 0 andexpand sin ¹ t º usingTaylorseries.Table3.5showsthe L 2 -norm convergenceofthePGschemefortwodi˙erentdistributionfunctions. Table3.5:Case-IV;PGschemeconvergencestudyin L 2 -norm,where T = 2 . = 75 š 100 = 25 š 100 N ˚ ¹ º = 1 ˚ ¹ º = Normal ˚ ¹ º = 1 ˚ ¹ º = Normal 2 1 : 56682 10 1 1 : 62765 10 1 1 : 5773 10 1 1 : 548 10 1 4 3 : 13043 10 3 3 : 3898 10 3 3 : 4228 10 3 3 : 28626 10 3 6 2 : 55359 10 5 2 : 81522 10 5 2 : 8956 10 5 2 : 76729 10 5 8 1 : 13562 10 7 1 : 2512 10 7 4 : 24126 10 7 1 : 40114 10 7 10 2 : 60471 10 9 7 : 84647 10 10 3 : 9524 10 7 5 : 49882 10 8 3.6.3ConditionNumber Theconditionnumberoftheconstructedlinearsystemisobtainedfordi˙erentdistributionfunc- tions, ˚ ¹ º .Tables3.6and3.7show,respectively,theconditionnumberoftheconstructedlinear systemforcase-IandIIbasedonPGandcollocationschemefortheaforementioneddistribution functions. Table3.6:Case-IandII;PGschemeconditionnumberoftheconstructedlinearsystem,where T = 2 . = 2 10 8 = 2 10 1 = 1 + 1 š 2 = 1 + 1 š 10 NCase-ICase-IICase-ICase-IICase-ICase-IICase-ICase-II 6 29706 : 6824863 : 5014319 : 4652168 : 87661 : 7014570 : 108151 : 9289357 : 19267 10 240000 : 5533494 : 890197 : 38811817 : 91339 : 2855130 : 92542 : 7541119 : 35597 14 882010 : 62118283279501 : 7835395 : 61941 : 9838190 : 30947 : 33577013 : 0337 18 2 : 2811229 10 6 301479633307 : 0679324 : 02505 : 5107247 : 62750 : 71342816 : 7944 Table3.7:Case-IandII;collocationschemeconditionnumberoftheconstructedlinearsystem, where T = 2 . = 1 10 8 = 1 10 1 = 1 š 2 = 1 š 10 NCase-ICase-IICase-ICase-IICase-ICase-IICase-ICase-II 6 67 : 5606345 : 04560 : 3467302 : 7443 : 6649191 : 05836 : 0056340 : 539 10 386 : 3392781 : 51325 : 0372309 : 25214 : 9351515 : 11202 : 8263554 : 95 14 1330 : 1110646 : 11076 : 148518 : 56685 : 1665435 : 91713 : 00216539 : 5 18 3388 : 9528619 : 52665 : 3222290 : 41661 : 1613964 : 93397 : 250911 : 1 Moreover,threecasesforthedistributionfunctionareconsidered:1)thedistributionismore biasedtowardtheleftofdomain,2)thedistributionissymmetricand3)thedistributionismore biasedtowardtherightofdomain,namely leftbiased , symmetric and rightbiased respectively. Thedistributionfunctionsarewell-knownnormal,exponential,log-normal,Cauchy,Laplace,Beta 74 andMaxwelldistributions,however,theyaretruncatedandnormalized,seeFig.4.5.Forthese distributions,theconditionnumberoftheconstructedlinearsystembasedonthetwomethodsis computedandprovidedinTables3.8,3.9and3.10. (a) (b) (c) Figure3.2: Distributionfunctions:(a) Leftbiased (b) Symmetric (c) Rightbiased Table3.8:Leftbiaseddistributionfunction;PG(top)andcollocation(bottom)schemecondition numberoftheconstructedlinearsystem,where T = 2 . NLogNormalExponentialNormalLaplaceCauchyBeta = 2 1 š 10 6 62101 : 513022785410 : 421714 : 73136170458 : 4 10 1 : 28119 10 6 2 : 62266 10 6 2 : 15167 10 6 1865272946301 : 51681 10 6 14 9 : 84911 10 6 1 : 61563 10 7 1 : 98724 10 7 6681781 : 04066 10 6 1 : 22216 10 6 18 4 : 48721 10 7 5 : 34428 10 7 1 : 0748 10 8 1 : 62018 10 6 2 : 41399 10 6 5 : 86944 10 6 = 1 + 1 š 10 6 200 : 626505 : 679300 : 64371 : 5348100 : 504233 : 849 10 654 : 2591397 : 741309 : 1291 : 8644140 : 467816 : 254 14 1322 : 631969 : 153437 : 7264 : 409398 : 41931780 : 96 18 2145 : 6121787037 : 9970 : 754191 : 87673134 : 64 NLogNormalExponentialNormalLaplaceCauchyBeta = 1 1 š 10 6 20 : 10019 : 4298913 : 357451 : 910339 : 506416 : 8765 10 48 : 336425 : 622926 : 2852237 : 89169 : 11438 : 3283 14 91 : 086655 : 571246 : 5097714 : 563503 : 22965 : 4666 18 143 : 171126 : 1973 : 33881672 : 611185 : 1597 : 5852 = 1 š 10 6 15 : 73195 : 6732311 : 427733 : 434126 : 392413 : 3573 10 40 : 235918 : 095822 : 8619150 : 375102 : 28530 : 7949 14 71 : 005437 : 166435 : 1324435 : 867309 : 08952 : 9303 18 110 : 72580 : 950648 : 00511032 : 88742 : 02581 : 7202 75 Table3.9:Symmetricdistributionfunction;PG(left)andcollocation(right)schemecondition numberoftheconstructedlinearsystem,where T = 2 . NUniformNormalCauchyLaplaceBeta = 2 1 š 10 6 3104 : 633116261 : 69328 : 536969 : 2213404 : 7 10 17244 : 21922020580110 : 251304 : 3157767 14 52095 : 4601 : 28827 10 6 308549179737817268 18 117338 : 894 : 86517 10 6 8033624416282 : 77821 10 6 = 1 + 1 š 10 6 9 : 545167744 : 085727 : 685817 : 987237 : 4224 10 11 : 21126985 : 382230 : 009221 : 36164 : 2331 14 12 : 360897127 : 84430 : 934320 : 199787 : 2794 18 15 : 900925172 : 88834 : 212622 : 1982107 : 403 NUniformNormalCauchyLaplaceBeta = 1 1 š 10 6 219 : 73381 : 9543118 : 433140 : 92295 : 0983 10 1592 : 05284 : 171598 : 247798 : 174375 : 673 14 5769 : 24620 : 0561788 : 852549 : 17905 : 972 18 14944 : 91097 : 374122 : 76115 : 651737 : 18 = 1 š 10 6 183 : 98455 : 890872 : 087678 : 425262 : 069 10 1854 : 46198 : 226360 : 943506 : 994248 : 186 14 8678 : 37439 : 3161113 : 831595 : 84589 : 611 18 26873 : 1786 : 6552600 : 563954 : 721171 : 64 Table3.10:Rightbiaseddistributionfunction;PG(left)andcollocation(right)schemecondition numberoftheconstructedlinearsystem,where T = 2 . NNormalLaplaceCauchyBeta = 2 1 š 10 6 2985 : 082274 : 392368 : 822105 : 03 10 23439 : 313948 : 915471 : 913474 : 7 14 92925 : 745201 : 452626 : 945325 : 7 18 259993107082129454110813 = 1 + 1 š 10 6 6 : 998016 : 609336 : 543086 : 18762 10 10 : 80499 : 8152310 : 086110 : 1067 14 15 : 156314 : 295814 : 026514 : 6423 18 19 : 441518 : 400318 : 844518 : 8791 NNormalLaplaceCauchyBeta = 1 1 š 10 6 329 : 01328 : 433333 : 499360 : 053 10 2022 : 092330 : 682275 : 432501 : 41 14 6299 : 738170 : 647736 : 878579 : 48 18 14429 : 720615 : 919067 : 621250 = 1 š 10 6 278 : 525333 : 384322 : 187378 : 704 10 1539 : 252647 : 092246 : 712727 : 03 14 4701 : 3810365 : 38041 : 129884 : 8 18 10622 : 628764 : 720895 : 425800 : 8 3.7ProofofLemmasandTheorems 3.7.1ProofofLemma (3.2.1) Proof. Following[5]andfor ˙> 0 , > 1 , > 1 ,and 8 x 2 1 ; 1 ¼ wehave ¹ 1 + x º + ˙ P ˙; + ˙ n ¹ x º P ˙; + ˙ n 1 º = ¹ + ˙ + 1 º ¹ + 1 º ¹ ˙ º P ; n 1 º ¹ x 1 ¹ 1 + s º P ; n ¹ s º ¹ x s º 1 ˙ ds ; (3.62) and ¹ 1 x º + ˙ P + ˙; ˙ n ¹ x º P + ˙; ˙ n ¹ + 1 º = ¹ + ˙ + 1 º ¹ + 1 º ¹ ˙ º P ; n ¹ + 1 º ¹ 1 x ¹ 1 s º P ; n ¹ s º ¹ s x º 1 ˙ ds : (3.63) Bythede˝nitionoftheleft-sidedRiemann-Liouvilleintegral RL 1 I ˙ x andevaluatingthespecial end-values P ˙; + ˙ n 1 º and P ; n 1 º ,wecanre-write(3.62)as RL 1 I ˙ x n ¹ 1 + x º P ; n ¹ x º o = ¹ n + + 1 º ¹ n + + ˙ + 1 º ¹ 1 + x º + ˙ P ˙; + ˙ n ¹ x º ; where,bytakingthefractionalderivative RL 1 D ˙ x onthebothsides,weobtain RL 1 D ˙ x n ¹ 1 + x º + ˙ P ˙; + ˙ n ¹ x º o = ¹ n + + ˙ + 1 º ¹ n + + 1 º ¹ 1 + x º P ; n ¹ x º : (3.64) 76 Hence,taking + ˙ = , ˙ = in(3.64),andshiftingfrom n to n 1 ,weobtain RL 1 D ˙ x n ¹ 1 º P n ¹ x º o = ¹ n + º ¹ n + ˙ º ¹ 1 + x º ˙ P ˙ ; ˙ n 1 ¹ x º ; (3.65) = ¹ n + º ¹ n + ˙ º ¹ 1 + x º P ; n 1 ¹ x º ; = ¹ n + º ¹ n + ˙ º ¹ 1 º P n ¹ x º ; where = ˙ .Moreover,bythede˝nitionoftheright-sidedRiemann-Liouvilleintegral RL x I ˙ 1 andevaluatingthespecialend-values P ˙; + ˙ n ¹ + 1 º and P ; n ¹ + 1 º ,wecanre-write(3.63)as RL x I ˙ 1 n ¹ 1 x º P ; n ¹ x º o = ¹ n + + 1 º ¹ n + + ˙ + 1 º ¹ 1 x º + ˙ P + ˙; ˙ n ¹ x º : Inasimilarfashion,bytakingthefractionalderivative RL x D ˙ 1 onthebothsides,weobtain RL x D ˙ 1 n ¹ 1 x º + ˙ P + ˙; ˙ n ¹ x º o = ¹ n + + ˙ + 1 º ¹ n + + 1 º ¹ 1 x º P ; n ¹ x º : (3.66) Next,bytaking + ˙ = , ˙ = in(3.66),andagainshiftingfrom n to n 1 wehave RL x D ˙ 1 n ¹ 2 º P n ¹ x º o = ¹ n + º ¹ n + ˙ º ¹ 1 x º ˙ P ˙;˙ n 1 ¹ x º : (3.67) = ¹ n + º ¹ n + ˙ º ¹ 1 x º P ; n 1 ¹ x º ; = ¹ n + º ¹ n + ˙ º ¹ 2 º P n ¹ x º ; andthatcompletestheproof. 3.7.2ProofofTheorem (3.2.3) Proof. Let ˚ beboundedin ¹ min ; max º .Then, C 1 ˚ min A kk 2 l ;˚; I C 2 ˚ max A ; (3.68) C 3 ˚ min B kk 2 r ;˚; I C 4 ˚ max B ; 77 where A = kk 2 L 2 ¹ I º + ¹ max min RL x L D x ¹º 2 L 2 ¹ I º d ; B = kk 2 L 2 ¹ I º + ¹ max min RL x D x R ¹º 2 L 2 ¹ I º d ; and C 1 , C 2 , C 3 ,and C 4 arepositiveconstants.From[99],weknowthat 8 = s ˝xed, kk l ; s ; I ˘ kk r ; s ; I thatis RL x L D s x ¹º 2 L 2 ¹ I º ˘ RL x D s x R ¹º 2 L 2 ¹ I º ,hencelet ® s = f s 1 ; s 2 ; ; s Q g ,andsimilarly 8 s = s q theaforementionedequivalenceholds.Therefore,anylinearcombinationof Q Õ q = 1 w q RL x L D s x ¹º 2 L 2 ¹ I º ˘ Q Õ q = 1 w q RL x D s x R ¹º 2 L 2 ¹ I º : Taking Q !1 andassuming f w q ; s q g tobeRiemannintegralweightsandpointsin » min ; max ¼ , ¹ max min RL x L D s x ¹º 2 L 2 ¹ I º ˘ ¹ max min RL x D s x R ¹º 2 L 2 ¹ I º : Byadding kk 2 L 2 ¹ I º tothebothsidesoftheaboveequivalence,weobtain A ˘ B ;andby(3.68), kk l ;˚; I ˘kk r ;˚; I . Inaddition,from[99],weknowthat 8 = s ˝xed, kk 2 s ; I ˘kk 2 l ; s ; I .Let ® s = f s 1 ; s 2 ; ; s Q g thus 8 s = s q 2 ® s ; kk 2 s q ; I ˘kk 2 l ; s q ; I .Therefore,foranylinearcombinationof Í Q q = 1 w q kk 2 s q ; I ˘ Í Q q = 1 w q kk 2 l ; s q ; I .Taking Q !1 ,weobtain: ¹ max min kk 2 ; I d ˘ ¹ max min kk 2 l ;; I d ; wheretherighthandsideoftheequivalenceis ¹ max min kk 2 L 2 ¹ I º + RL x L D x ¹º 2 L 2 ¹ I º d = ¹ max min ºkk 2 L 2 ¹ I º + ¹ max min RL x L D x ¹º 2 L 2 ¹ I º d ; ˘kk 2 l ;˚; I ˘kk 2 r ;˚; I : Therefore, ¹ max min kk 2 ; I d ˘kk 2 l ;˚; I ˘kk 2 r ;˚; I : (3.69) Wecanalsoshowthat ˚ min ¹ max min ¹ 1 + j ! j 2 º 2 F¹º¹ ! º 2 L 2 ¹ R º d kk 2 ˚; R ˚ max ¹ max min ¹ 1 + j ! j 2 º 2 F¹º¹ ! º 2 L 2 ¹ R º d : 78 Becauseofthenon-negativityofthenorms,wehave ˚ min inf ~ v 2 ˚ H¹ R º ; ~ v j I = ¹º ¯ max min ¹ 1 + j ! j 2 º 2 F¹ ~ v º¹ ! º 2 L 2 ¹ R º d inf ~ v 2 ˚ H¹ R º ; ~ v j I = ¹ ~ v º kk 2 ˚; R (3.70) ˚ max inf ~ v 2 ˚ H¹ R º ; ~ v j I = ¹º ¯ max min ¹ 1 + j ! j 2 º 2 F¹ ~ v º¹ ! º 2 L 2 ¹ R º d : Ingeneral, ˚ H¹ R ºˆ H max ¹ R º , 8 2» min ; max ¼ .Therefore,wehave: inf ~ v 2 ˚ H¹ R º ; ~ v j I = ¹º ¹ max min ¹ 1 + j ! j 2 º 2 F¹ ~ v º¹ ! º 2 L 2 ¹ R º d ; = ¹ max min inf ~ v 2 ˚ H¹ R º ; ~ v j I = ¹º ¹ 1 + j ! j 2 º 2 F¹ ~ v º¹ ! º 2 L 2 ¹ R º d ; C ¹ max min inf ~ v 2 H ¹ R º ; ~ v j I = ¹º ¹ 1 + j ! j 2 º 2 F¹ ~ v º¹ ! º 2 L 2 ¹ R º d ; = C ¹ max min k k 2 ; I d : However,forsomechoicesof ˚ = ¹ min º andthus ˚ H¹ R º˙ H max ¹ R º , 8 2» min ; max ¼ . Therefore, ¹ max min inf ~ v 2 ˚ H¹ R º ; ~ v j I = ¹º ¹ 1 + j ! j 2 º 2 F¹ ~ v º¹ ! º 2 L 2 ¹ R º d ; e C ¹ max min inf ~ v 2 H ¹ R º ; ~ v j I = ¹º ¹ 1 + j ! j 2 º 2 F¹ ~ v º¹ ! º 2 L 2 ¹ R º d ; = e C ¹ max min k k 2 ; I d ; whichby(3.70)and(3.5),weget ¹ max min k k 2 ; I d ˘ k k 2 ˚; I : (3.71) Comparing(3.69)and(3.71),wehave k k 2 ˚; I ˘ k k 2 l ;˚; I ˘ k k 2 r ;˚; I : 79 Remark3.7.1. Wenotethatif ˚ = ¹ s º ,werecoverthestandard RL x L D s x ¹ u º = f ,wherethe equivalencebetweenthecorresponding kk l ; s ; I , kk r ; s ; I ,and kk s ; I hasbeenalreadyestablished. Moreover,wenotethatforthecase ˚ 2 L 1 ¹» min ; max º containing˝nitelymanysingularitiesat 1 ; 2 ; ; m ,thewholeinterval » min ; max ¼ andtheintegrationcanbewrittenas ¹ 1 min ˚ ¹ º RL x L D s x ¹ u º d + ¹ 2 1 ˚ ¹ º RL x L D s x ¹ u º d + + ¹ max m ˚ ¹ º RL x L D s x ¹ u º d ; whereallthepreviousstepsintheproofcanapplyineachinterval. 3.7.3ProofofLemma (7.3.4) Proof. Since u ¹ a º = 0 ,by(1.18) a D 2 x u = a D x ¹ a D x u º .Taking e u ¹ x º = a D x u ,wehave ¹ a D 2 x u ; v º = ¹ a D x e u ; v º ; = 1 ¹ 1 º ¹ b a h d dx ¹ x a e u ¹ s º ds ¹ x s º i v ¹ x º dx ; = n v ¹ x º ¹ 1 º ¹ x a e u ¹ s º ds ¹ x s º o x = b x = a 1 ¹ 1 º ¹ b a h ¹ x a e u ¹ s º ds ¹ x s º i d v ¹ x º dx dx ; b yv ¹ b º = 0 ; = v ¹ a º lim x ! a a I 1 x e u 1 ¹ 1 º ¹ b a ¹ b s d v ¹ x º dx dx ¹ x s º e u ¹ s º ds ; (3.72) whichmakesensewhentheinteriorterm ¯ b s d v ¹ x º dx dx ¹ x s º isintegrablein ¹ a ; b º .Takingintoaccount that v ¹ a º isbounded,wecanshowthattheboundaryterm v ¹ a º lim x ! a a I 1 x e u alsovanishesas lim x ! a a I 1 x e u = lim x ! a 1 ¹ 1 º ¹ x a e u ¹ s º ds ¹ x s º ; (3.73) lim x ! a 1 ¹ 1 º ¹ x a ds ¹ x s º k e u k L 1 ; = lim x ! a 1 ¹ 1 º ¹ x a º 1 1 k e u k L 1 = 0 : 80 Moreover,itiseasytocheckthat d ds ¹ b s v ¹ x º dx ¹ x s º = d ds n v ¹ x º¹ x s º 1 1 x = b x = s 1 1 ¹ b s d v ¹ x º dx ¹ x s º 1 dx o ; = d ds n 0 1 1 ¹ b s d v ¹ x º dx ¹ x s º 1 dx o ; = ¹ b s d v ¹ x º dx dx ¹ x s º : (3.74) Now,bysubstituting(3.74)into(3.72),weobtain ¹ a D 2 x u ; v º = ¹ b a e u ¹ s º n 1 ¹ 1 º ¹ d ds º ¹ b s v ¹ x º dx ¹ x s º o ds ; = ¹ e u ; x D b v º ; when x D b v iswell-de˝nedandisintegrableintheinterval » a ; b ¼ . 3.7.4ProofofTheorem (4.3.2) Proof. PartA : TheJacobipolynomials, P ; n ¹ ˘ º ,canbeconstructedviathethree-termrecursionrelation.By letting = ,thecorrespondingtreetermrecursionreducesto P ; n + 1 ¹ ˘ º = ¹ 2 n + 1 º ¹ n + 1 º ˘ P ; n ¹ ˘ º ¹ n 2 º n ¹ n + 1 º ˘ P ; n 1 ¹ ˘ º ; (3.75) andtherefore,theJacobipolynomialsevaluatedat ˘ = ˘ 0 2 1 ; 1 ¼ areobtainedinthefollowing standardform P ; 0 ¹ ˘ 0 º = ’ 0 ¹ º = 1 ; : zerothorderin (3.76) P ; 1 ¹ ˘ 0 º = ’ 1 ¹ º = + ˘ 0 ; : linearin P ; 2 ¹ ˘ 0 º = ’ 2 ¹ º = 1 2 2 + 3 2 ˘ 0 + 3 ˘ 2 0 1 2 ; : quadraticin P ; 3 ¹ ˘ 0 º = ’ 3 ¹ º = 1 6 3 + ˘ 0 2 + 15 ˘ 2 0 4 6 + 5 ˘ 3 0 3 ˘ 0 2 : : cubicin Now,let n = k ,thus, P ; k ¹ ˘ 0 º and P ; k 1 ¹ ˘ 0 º arerespectivelypolynomialsoforder k and k 1 in .Using(3.75)for n = k + 1 ,weget P ; k + 1 ¹ ˘ 0 º = ¹ 2 k + 1 º ¹ k + 1 º ˘ 0 P ; k ¹ ˘ 0 º ¹ k 2 º k ¹ k + 1 º ˘ 0 P ; k 1 ¹ ˘ 0 º ; 81 whichisapolynomialsoforder k + 1 in duetothesecondterm.Hence,bymathematicalinduction, P ; n ¹ ˘ 0 º = ’ n ¹ º isapolynomialoforder n in 8 ˘ 0 2 1 ; 1 ¼ .Similarlywiththesameargument, wecanshowthat P ; n ¹ ˘ 0 º = ’ n º isalsoapolynomialoforder n in 8 ˘ 0 2 1 ; 1 ¼ . PartB : Theinnerintegralofthediscretedistributedbilinearform(3.36)canbewrittenas ¹ max min ˚ ¹ º¹ 2 T º ¹ n + º ¹ n + º ¹ k + º ¹ k + º ¹ 1 º P n ¹ ˘ º ¹ 2 º P k ¹ ˘ º d = (3.77) ¹ n + º ¹ k + º ¹ max min ˚ ¹ º¹ 2 T º ¹ 1 + ˘ º ¹ n + º ¹ 1 ˘ º ¹ k + º P ; n 1 ¹ ˘ º P ; k 1 ¹ ˘ º d ; inwhich = š 2 .Bytheorem(4.3.2)partA, P 2 ; 2 n 1 ¹ ˘ º and P 2 ; 2 k 1 ¹ ˘ º arepolynomials in oforder n 1 and k 1 ,respectively, 8 ˘ 2 1 ; 1 ¼ ,and ˝xed.Thus, P 2 ; 2 n 1 ¹ ˘ º = n 1 Õ r = 0 ˙ r P r ¹ º ; (3.78) P 2 ; 2 k 1 ¹ ˘ º = k 1 Õ l = 0 e ˙ l P l ¹ º : (3.79) Byplugging(3.78)and(3.79)into(3.77),weobtain ¹ max min ˚ ¹ º¹ 2 T º ¹ 1 + ˘ º ¹ n + º ¹ 1 ˘ º ¹ k + º P ; n 1 ¹ ˘ º P ; k 1 ¹ ˘ º d = ¹ max min ˚ ¹ ºW ˘; kn ¹ º n 1 Õ r = 0 ˙ r P r ¹ º k 1 Õ l = 0 e ˙ l P l ¹ º d ; (3.80) inwhich W ˘; kn ¹ º = ¹ 2 T º ¹ 1 ˘ º ¹ n + º ¹ 1 + ˘ º ¹ k + º is smooth inanycompactsupportin » min ; max ¼ anditspolynomialexpansion W ˘; kn N ¹ º = W N ¹ º = Í N q = 0 ˆ q P q ¹ º convergesexponentiallyi.e., W ˘; kn ¹ ºW N ¹ º c 1 exp c 2 N c 3 º ; (3.81) inwhich k : k denotesthe L 2 -normin » min ; max ¼ .Ifthedistributionfunction ˚ 2 H r ¹» min ; max ¼º , r > 0 ,wehavethefollowingprojectionerror: k ˚ ¹ º ˚ N ¹ º k c 4 N r k ˚ k H r ¹» min ; max ¼º ; (3.82) 82 where ˚ N ¹ º = Í N n = 0 e ˆ n P n ¹ º .Consequently,theintegrandin(3.80)canbewell-approximated via ˚ ¹ ºW ˘; kn ¹ º n 1 Õ r = 0 ˙ r P r ¹ º k 1 Õ l = 0 e ˙ l P l ¹ ºˇ ˚ N ¹ ºW N ¹ º n 1 Õ r = 0 ˙ r P r ¹ º k 1 Õ l = 0 e ˙ l P l ¹ º : (3.83) Next,let I = ¹ max min ˚ ¹ ºW ˘; kn ¹ º n 1 Õ r = 0 ˙ r P r ¹ º k 1 Õ l = 0 e ˙ l P l ¹ º d ; (3.84) I N = ¹ max min ˚ N ¹ ºW N ¹ º n 1 Õ r = 0 ˙ r P r ¹ º k 1 Õ l = 0 e ˙ l P l ¹ º d ; where I N canbe accurately calculatedvia I N = Q Õ q = 1 e w q ˚ N ¹ q ºW N ¹ q º n 1 Õ r = 0 ˙ r P r ¹ q º k 1 Õ l = 0 e ˙ l P l ¹ q º ; (3.85) employingaGauss-Legendrequadraturerule,provided Q = 2 N .ThusbyCauchy-schwarzin- equality, j II N j p min max ˚ W ˘; kn n 1 Õ r = 0 ˙ r P r k 1 Õ l = 0 e ˙ l P l ˚ N W N n 1 Õ r = 0 ˙ r P r k 1 Õ l = 0 e ˙ l P l ; (3.86) 83 inwhich ˚ W ˘; kn n 1 Õ r = 0 ˙ r P r k 1 Õ l = 0 e ˙ l P l ! ˚ N W N n 1 Õ r = 0 ˙ r P r k 1 Õ l = 0 e ˙ l P l ! ; (3.87) n 1 Õ r = 0 ˙ r P r k 1 Õ l = 0 e ˙ l P l ˚ W ˘; kn ˚ N W N ; (byH Ü olderinequality) ; c 5 ˚ W ˘; kn ˚ N W N ; c 5 ¹ ˚ ˚ N º + ˚ N ¹W ˘; kn W N º + W N ˚ N W N ; c 5 ¹ ˚ ˚ N º¹W ˘; kn W N º + ˚ N ¹W ˘; kn W N º + ¹ ˚ ˚ N ºW N + ˚ N W N ˚ N W N ; c 5 ¹ ˚ ˚ N º¹W ˘; kn W N º + ˚ N ¹W ˘; kn W N º + k ¹ ˚ ˚ N ºW N k ; (bytriangleinequality) ; c 5 k ¹ ˚ ˚ N º k ¹W ˘; kn W N º + k ˚ N k ¹W ˘; kn W N º + k ¹ ˚ ˚ N º kk W N k ; c 5 c 4 N r k ˚ k H r ¹» min ; max ¼º : c 1 exp c 2 N c 3 º + k ˚ N k c 1 exp c 2 N c 3 º + k W N k c 4 N r k ˚ k H r ¹» min ; max ¼º ; (by(3.81)and(3.82)) ; c 6 N r k ˚ k H r ¹» min ; max ¼º : Hence,by(3.86)and(3.87)wecanshow j II N j C N r k ˚ k H r ¹» min ; max ¼º ; (3.88) andtherefore,by(3.78),(3.79),(3.84)and(3.85),weobtain ¹ max min ˚ ¹ º¹ 2 T º ¹ n + º ¹ n + º ¹ k + º ¹ k + º ¹ 1 º P n ¹ ˘ º ¹ 2 º P k ¹ ˘ º d Q Õ q = 1 e w q ˚ N ¹ q º¹ 2 T º q ¹ n + º ¹ n + q º ¹ k + º ¹ k + q º ¹ 1 º P q n ¹ ˘ º ¹ 2 º P q k ¹ ˘ º C Q r k ˚ k H r ¹» min ; max ¼º : PartC : If ˚ ¹ º issmooth,thentheapproximation ˚ N ¹ º ,in(3.82),convergeswithanexponential accuracyandsodoesthenormin(3.87).Thus, j II N j C 1 exp C 2 N C 3 º ; (3.89) 84 andtherefore,thequadraturerulebecomesexponentiallyaccuratein Q . 85 CHAPTER4 DISTRIBUTED-ORDERFRACTIONALPDES:FRACTIONALPSEUDO-SPECTRAL METHODS 4.1Background Incomparisontosingleorderfractionaloperator,distributivecharacterofDODEsincreases thecomputationalcostsduetointegrationinthedomainofderivativeorder,requiringe˚cientand accuratenumericalschemes.DiethelmandFordin[46],asoneofthe˝rstnumericalstudiesof DODEs,developedatwo-stagebasicframe-work,whereinthe˝rststage,thedistributedorder di˙erentiationtermwasapproximatedusingaquadraturerule,andinthesecondstage,asuitable multi-termnumericalmethodwasemployed.Mostofthesubsequentnumericalstudieshave followedthesameapproachyettheyvaryinthediscretizationmethodinthesecondstage,see e.g.,distributed-ordertime-fractionaldi˙usionequation[57],distributed-orderreactiondi˙usion equation[124],distributed-orderdi˙usionequationusingareproducingkernelmethod[98],and distributed-ordertimefractionaldi˙usion-waveequationbasedonacompactdi˙erencescheme [179].Othernumericalstudiesinclude:multi-termanddistributedorderproblemsinhalf-line [106],animplicitnumericalmethodofatemporaldistributedorderandtwo-sidedspace-fractional advection-dispersionequationin[69],high-orderdi˙erenceschemesin[60],alternatingdirection implicit(ADI)di˙erenceschemeswiththeextrapolationmethodforone-dimensionalcasein[62] andtwo-dimensionalproblemin[61],andanoperationalmatrixtechniquein[50]. Morerecently,Kharazmietal.[87]havedevelopedaPetrov-Galerkinspectralmethodfor DODEs,followingtherecenttheoryofFSLPin[186]andbyemployingJacobiPoly-fractonomials asthebasesandtestfunctions,wheretheyintroducedthedistributedSobolevspacesandtheir associatedequivalentnorms,andinvestigatedthestabilityanderroranalysisofthescheme.More importantly,theyhavedevelopedaspectrallyaccurateGauss-Legendrequadratureruleinthe constructionoflinearsystem.Weextendtheirdevelopedschemebyproposingapseudo-spectral 86 methodinordertoe˚cientlytreatthenonlinearDODEs. Inthischapter,weproposeaPetrov-Galerkin(PG)pseudo-spectralmethodforDODEs.We constructfractionalLagrangeinterpolantsof˝rstandsecondkindasbasisandtestfunctions, respectively.Weobtainthecorresponding weakdistributeddi˙erentiationmatrices fordistributed- orderoperatorswithone-andtwo-sidedfractionalderivatives.Wefurtherstudythee˙ectof distributionfunctionandinterpolationpointsontheconditionnumberoftheresultinglinearsystem andalsodesigndistributedpre-conditioners,basedonthedistributionfunction.Weshowthebetter conditioningoftheresultinglinearsystembycomparingtheproposedmethodwiththefractional spectralcollocationmethod,developedin[87],whichemployssimilarexpansionsbutinastrong senseofproblem.Moreover,thefractionalLagrangeinterpolantsarecomprisedofafractionalorder termmultipliedbystandardLagrangeinterpolants,whereweshowthattheinterpolationparameter inthefractionalpartcanbetunnedproperlytoaccuratelycaptureanysingularityofthesolution. Inaddition,thesefractionalinterpolantsenjoythepropertyofKroneckerdeltaattheinterpolation points,whichmakesitpossibletoe˚cientlytreatthealgebraicanddi˙erentialnonlinearitiessuch asinnonlinearreactiondi˙usionandBurgersequations.Wedemonstratethee˚ciencyofthe proposedschemesbyconsideringseveralnumericalexamplesincludingnonlinearity,inwhichwe showthespectralconvergenceoftheapproximatesolution. Theorganizationofthischapterisasfollows:Insection(4.2),weprovidepreliminaryde˝nitions alongwithusefullemmas,andthende˝netheinitialandboundaryvalueproblemsandobtainthe weakformulation.Insection(4.3),weimplementthepseudo-spectralmethodandderivetheweak distributeddi˙erentiationmatrices.Wefurtherstudytheconditionnumberofthelinearsystem anddesignthepre-conditioner.Weprovidenumericalexamplesinsection(4.4). 4.2De˝nitions 4.2.1DistributedFractionalSobolevSpaces We˝rstrecallthefractionalSobolevspace.By H s ¹ R º , s 0 ,wedenotetheFractionalSobolev spaceon R ,de˝nedas H s ¹ R º = f v 2 L 2 ¹ R ºj¹ 1 + j ! j 2 º s 2 F¹ v º¹ ! º2 L 2 ¹ R ºg ,whichisendowed 87 withthenorm kk s ; R = k¹ 1 + j ! j 2 º s 2 F¹º¹ ! ºk L 2 ¹ R º ,where F¹ v º representstheFouriertransform of v .Subsequently,wedenoteby H s ¹ I º , s 0 theFractionalSobolevspaceonthe˝nite closedinterval I ,de˝nedas H s ¹ I º = f v 2 L 2 ¹ R ºj9 ~ v 2 H s ¹ R º s : t : ~ v j I = v g withthenorm kk s ; I = inf ~ v 2 H s ¹ R º ; ~ v j I = ¹º k ~ v k s ; R .Wenotethatthede˝nitionof H s ¹ I º andthecorresponding normreliesontheFouriertransformationofthefunction.Otherleft-sided kk l ; s ; I andright- sided kk r ; s ; I usefulnorms,associatedwith H s ¹ I º ,e.g.,when I = » x L ; x R ¼ ,arealsogivenas kk l ; s ; I = kk 2 L 2 ¹ I º + k RL x L D x ¹ºk 2 L 2 ¹ I º 1 2 ,and kk r ; s ; I = kk 2 L 2 ¹ I º + k RL x D x R ¹ºk 2 L 2 ¹ I º 1 2 ,which areshownin[52,53]thatareequivalentto kk s ; I . De˝nition4.2.1. DistributedFractionalSobolevSpace [87]:Let ˚ 2 L 1 ¹» min ; max ¼º , 0 min < max benonnegative.Then,the distributed fractionalSobolevspaceon R anditsassociated normare ˚ H¹ R º = f v 2 L 2 ¹ R ºj ¹ max min h ˚ ¹ º¹ 1 + j ! j 2 º i 1 2 F¹ v º¹ ! º d 2 L 2 ¹ R ºg ; (4.1) kk ˚; R = ¹ max min ˚ ¹ º ¹ 1 + j ! j 2 º 2 F¹º¹ ! º 2 L 2 ¹ R º d ! 1 2 : (4.2) Subsequently,the distributed fractionalSobolevspaceonthe˝niteclosedinterval I ,i.e. ˚ H¹ I º ,is de˝nedas ˚ H¹ I º = f v 2 L 2 ¹ R ºj9 ~ v 2 ˚ H¹ R º s : t : ~ v j I = v g ; (4.3) withthenorm kk ˚; I = inf ~ v 2 ˚ H¹ R º ; ~ v j I = ¹º k ~ v k ˚; R .Moreover,thefollowingusefulnorms,associated with ˚ H¹ I º areintroducedas: kk l ;˚; I = kk 2 L 2 ¹ I º + ¹ max min ˚ ¹ º RL x L D x ¹º 2 L 2 ¹ I º d ! 1 2 (4.4) kk r ;˚; I = kk 2 L 2 ¹ I º + ¹ max min ˚ ¹ º RL x D x R ¹º 2 L 2 ¹ I º d ! 1 2 ; (4.5) whichareprovenin[87]tobeequivalenttothenorm kk ˚; I . Remark4.2.2. Thelowerandupperlimits f min ; max g areonlythetheoreticalterminals.How- ever,foranyrealizationofphysicalprocess,therangeofderivativeordersarenotexactlyequal 88 tothethesetheoreticalterminals.Moreover,thechoiceof ˚ canarbitrarilycon˝nethedomainof integrationinpractice,seeforexampleFig.4.1inthefollowingsections,andif ˚> 0 iscontinuous in ¹ min ; max º ,then ˚ H isequivalentto H max ,forsomesmall . Lemma4.2.3. [99]:Forall 0 < 1 ,if u 2 H 1 ¹» a ; b ¼º suchthat u ¹ a º = 0 ,and w 2 H š 2 ¹» a ; b ¼º , then ¹ a D s u ; w º = ¹ a D š 2 s u ; s D š 2 b w º ,where ¹ ; º representsthestandardinnerproduct in = » a ; b ¼ . Lemma4.2.4. [87]:Let 1 š 2 << 1 , a and b bearbitrary˝niteorin˝niterealnumbers. Assume u 2 H 2 ¹ a ; b º suchthat u ¹ a º = 0 , x D b v isintegrablein ¹ a ; b º ,and v ¹ b º = 0 .Then ¹ a D 2 x u ; v º = ¹ a D x u ; x D b v º . 4.2.2ProblemDe˝nition:Initial/BoundaryValueProblem Weconsiderthefollowingdistributed-orderfractionaldi˙erentialequation,subjecttotheproper initialconditions D D ˚ u ¹ t º = f ¹ t ; u º ; 8 t 2¹ 0 ; T ¼ ; (4.6) u ¹ 0 º = 0 ; max 2¹ 0 ; 1 ¼ ; (4.7) u ¹ 0 º = du dt j t = 0 = 0 ; max 2¹ 1 ; 2 ¼ ; (4.8) inwhichthedistributed-orderfractionalderivativeisde˝nedin[87],as D D ˚ u ¹ t º = ¹ max min ˚ ¹ º a D t u ¹ t º d ; t > a ; (4.9) where 7! ˚ ¹ º isacontinuousmappingin » min ; max ¼ and a D t denotesRiemann-Liouville fractionalderivativeoforder .Dueto(1.22),theCaputoandRiemann-Liouvillefractional derivativesoforder 2¹ 0 ; 1 º coincidewitheachotherwhen u ¹ 0 º = 0 .Therefore,inthisstudy, weemploythede˝nitionofthedistributedfractionalderivativesofRiemann-Liouvillesenseand removethepre-superscript RL forsimplicity. 89 Borrowingthesameideaasinde˝nitionofdistributed-orderfractionalderivative,wede˝ne thefollowingdistributed-orderfractionalderivativeinspace. D D ˚ u ¹ x º = ¹ max min ˚ ¹ º a D x u ¹ x º d ; a < x < b : (4.10) Byperformingana˚nemappingfromanycon˝neddomain » a ; b ¼ tothestandarddomain 1 ; 1 ¼ , wede˝netheboundaryvalueproblem,subjecttohomogeneousDirichletboundaryconditionsas D D ˚ u ¹ x º = f ¹ x ; u º ; 8 x 2 1 ; 1 º ; (4.11) u 1 º = u ¹ 1 º = 0 ; 1 < min < max 2 : (4.12) 4.2.3WeakFormulation InordertoobtaintheweakformoftheIVP(4.6)andBVP(4.11),wemultiplytheequationswith propertestfunctions,andthenintegrateoverthecorrespondingcomputationaldomain.Here,we showtheweakformderivationonlyfortheIVP.However,theformulationissimilarfortheBVP aswell.Thus, ¹ D D ˚ u ¹ t º v ¹ t º d = ¹ f ¹ t ; u º v ¹ t º d : (4.13) Usingthede˝nitionofdistributedorderfractionalderivativesde˝nedin(5.7),weget ¹ max min ˚ ¹ º 0 D t u ; v d = ¹ f ¹ t ; u º v ¹ t º d ; (4.14) where 0 D t u ; v = ¯ 0 D t u v d ; 8 2¹ min ; max º ,denotesthewell-known L 2 -innerprod- uct.Giventheinitialconditions(4.7)and/or(4.8)andbyLemmas(7.3.3)and(7.3.4),wede˝nethe bilinearformassociatedwith 2¹ min ; max º as a ¹ º ¹ u ; v º = 0 D t u ; v = ¹ 0 D š 2 t u ; t D š 2 T v º , inwhichwechoose v suchthat v ¹ T º = 0 and t D š 2 T v isintegrablein 8 2¹ min ; max º . Moreover,let U and V bethesolutionandtestspaces,respectivelyde˝nedas U = n u 2 ˚ H¹ º : u ¹ 0 º = 0 if max 2¹ 0 ; 1 ¼ ; u ¹ 0 º = du dt j t = 0 = 0 if max 2¹ 1 ; 2 ¼ o ; V = n v 2 ˚ H¹ º : v ¹ T º = 0 o ; 90 inwhich ˚ : ¹ min ; max º! R .Theproblemthusreadsas:˝nd u 2 U suchthat a ¹ u ; v º = l ¹ f º ; 8 v 2 V where a ¹ u ; v º : = ¹ max min ˚ ¹ º a ¹ º ¹ u ; v º d (4.15) denotesthe distributed bilinearformand l ¹ f º : = ¹ f ; v º . Wenotethatbyde˝ningthefollowingsolutionandtestspaces,weobtainthesame distributed bilinearformfortheBVPaswell,wherethesolutionandtestspacesare U = n u 2 ˚ H¹ º : u 1 º = u ¹ 1 º = 0 o ; V = n v 2 ˚ H¹ º : v ¹ 1 º = 0 o : (4.16) 4.3FractionalPseudo-SpectralMethod 4.3.1InitialValueProblem Weconsidertheweakform a ¹ u ; v º = l ¹ f º ; 8 v 2 V ,associatedwiththeIVP,andperformana˚ne mappingfrom » 0 ; T ¼ tothestandarddomain 1 ; 1 ¼ .Following[186],weemploythe fractional Lagrangeinterpolant s,giveninthestandarddomain 1 ; 1 ¼ by ¹ 1 º h j ¹ ˘ º = ˘ x 1 x j x 1 N + 1 Ö k = 1 k , j ˘ x k x j x k ; j = 1 ; 3 ; ; N + 1 ; (4.17) inwhich, 2¹ 0 ; 1 º ,the interpolationparameter ,isusedasatunableparameterforcapturing possiblesingularitiesinthesolution.Weconstruct N + 1 fractionalLagrangeinterpolantsoforder N + byde˝ning N + 1 basisinterpolatingpoints" ¹ 1 º W = f x j j x 1 = 1 ; x j 2 1 ; 1 º ; j = 2 ; 3 ; ; N + 1 g ,ofwhich,weassignthe˝rstdegreeoffreedom(the˝rstpoint, x 1 = 1 )to thefractionalpart.Becauseofhomogeneousinitialcondition,weonlyneedtoconstructthe interpolants, ¹ 1 º h j ¹ ˘ º ; j = 2 ; 3 ; ; N + 1 ,thus,weseekthesolution u N 2 U N = span n ¹ 1 º h j ¹ ˘ º ;˘ 2 1 ; 1 ¼ ; j = 2 ; 3 ; ; N + 1 o ; (4.18) oftheform u N ¹ ˘ º = N + 1 Õ j = 2 u j ¹ 1 º h j ¹ ˘ º ; (4.19) 91 where, u j = u N ¹ ˘ j º .Wenotethatintheconstructionoffractionalinterpolants(4.17),weinclude theleftboundary x 1 = 1 andstartfrom k = 1 .Therefore,wehavearegularityoforder 1 + at x 1 = 1 ,whichensuressatisfyingtheinitialconditions,where max 2¹ 1 ; 2 º . Moreover,weconstructanother N + 1 fractionalLagrangeinterpolantsoforder N + byde˝ning anewsetof N + 1 tinterpolatingpoints" ¹ 2 º W = f ~ x i j ~ x N + 1 = 1 ; ~ x i 2 1 ; 1 º ; i = 1 ; 2 ; ; N g , ofwhich,weassignthelastdegreeoffreedom(thelastpoint, x N + 1 = 1 )tothefractionalpart. Therefore, ¹ 2 º h i ¹ ˘ º = ~ x N + 1 ˘ ~ x N + 1 ~ x i N + 1 Ö k = 1 k , i ˘ ~ x k ~ x 1 ~ x k ; i = 1 ; 2 ; ; N : (4.20) Hence,wede˝nethespaceoftestfunctions V N ,ofsamedimensionasthesolutionspace,by V N = span n ¹ 2 º h i ¹ ˘ º ;˘ 2 1 ; 1 ¼ ; i = 1 ; 2 ; ; N o ; (4.21) whereincluding x N + 1 = 1 ensuresvanishingtestfunctionsattherightboundary. 4.3.2BoundaryValueProblem Inthiscaseduetotheboundaryconditions,weaddoneextrainterpolationpointandconsider N + 2 points.Therefore,weconsiderthefollowingtwosetsofinterpolationpoints ¹ 1 º W = f x j j x 1 = 1 ; x N + 2 = 1 ; x j 2 1 ; 1 º ; j = 2 ; 3 ; ; N + 1 g and ¹ 2 º W = f ~ x i j ~ x 1 = 1 ; ~ x N + 2 = 1 ; ~ x i 2 1 ; 1 º ; i = 2 ; ; N + 1 g ,anduse(4.17)and(4.20)toconstructthecorrespondinginterpolants, where k = 1 ; 2 ; ; N + 2 . Remark4.3.1. Suchaconstructionhastwobene˝ts:(i)duetothehomogeneousDirichletboundary conditions,weonlyneedtoconstructtheintrerpolantsfor j = 2 ; 3 ; ; N + 1 (andnotfor j = 1 and j = N + 2 ).Therefore,thediscretesolutionspaceandtheexpansion (4.18) remaininvariant. Moreover,thesizeofdiscretetestspacedoesnotchangeneither,howeveritselementsshiftfrom i = 1 ; 2 ; ; N forIVPto i = 2 ; 3 ; ; N + 1 forBVP;(ii)themathematicalframeworkofthe proposedscheme(derivationofthedi˙erentiationmatrix)alsoremainsinvariant.Therefore,one canusethesamederivationforthetwocasesIVPandBVPtoconstructthelinearsystem,aswill 92 bediscussedlater,byproperlychangingthelimitsofiterator i onnumberoftestfunctions.Wealso shouldnotethatinclusionof 1 and 1 ,inadditiontoletting k totakevales 1 and N + 2 ,imposes extraregularitytothesolutionattheboundaries,makingits˝rstderivativetovanishaswellasthe solutionitself. ForeithercaseofIVPorBVP,since ˚ ¹ º 0 and ¯ max min ˚ ¹ º d is˝nite,itisstraightforward tocheckthat U N ˆ U and V N ˆ V when ischosenproperly.Therefore,thebilinearform a ¹ º ¹ u ; v º reducestothe discrete bilinearform a h ¹ u N ; v N º = ¹ a D š 2 x u N ; x D š 2 b v N º ; 8 x 2» a ; b ¼ (4.22) andthus,theproblemreadsas:˝nd u N 2 U N suchthat a h ¹ u N ; v N º = l h ¹ f º ; 8 v N 2 V N ; (4.23) where a h ¹ u N ; v N º : = ¯ max min ˚ ¹ º a h ¹ u N ; v N º d denotesthe discrete distributedbilinearformand l h ¹ f º : = ¹ f ; v N º representstheloadvector. 4.3.3WeakDistributedDi˙erentiationMatrix Here,wederivethecorrespondingdi˙erentiationmatrixforIVP.Asmentionedearlier,thesame procedurecanbereadilyusedforBVPbychangingthelimitsofiterator i ,onceproperinterpolants areconstructed(Seesubsections4.3.1and4.3.2).Bysubstituting(4.19),choosing v N = ¹ 2 º h i ¹ ˘ º2 V N , i = 1 ; 2 ; ; N andusing(1.21),thediscretedistributedbilinearform(4.23)canbewrittenas a h ¹ u N ; v N º = ¹ max min ˚ ¹ º¹ 2 T º N + 1 Õ j = 2 u j 1 D š 2 ˘ » ¹ 1 º h j ¹ ˘ º¼ ; ˘ D š 2 1 » ¹ 2 º h i ¹ ˘ º¼ d ; (4.24) = N + 1 Õ j = 2 u j ¹ max min ˚ ¹ º¹ 2 T º 1 D š 2 ˘ » ¹ 1 º h j ¹ ˘ º¼ ; ˘ D š 2 1 » ¹ 2 º h i ¹ ˘ º¼ d = N + 1 Õ j = 2 D i ; j u j ; where D i ; j = ¹ max min ˚ ¹ º¹ 2 T º D i ; j d (4.25) 93 isthe weakdistributeddi˙erentiationmatrix ,andtheintegrand D i ; j = 1 D š 2 ˘ » ¹ 1 º h j ¹ ˘ º¼ ; ˘ D š 2 1 » ¹ 2 º h i ¹ ˘ º¼ isderivedasfollows.Giventhede˝nitionoffractionalLagrangeinterpolantsin(4.17)and(4.20), wehave D i ; j = a j 1 D š 2 ˘ »¹ 1 + ˘ º G j ¹ ˘ º¼ ; ~ a i ˘ D š 2 1 »¹ 1 ˘ º ~ G i ¹ ˘ º¼ ; i = 1 ; 2 ; ; N ; j = 2 ; 3 ; ; N + 1 ; (4.26) where, a j = 1 ¹ x j + 1 º , ~ a i = 1 ¹ 1 ~ x i º ,and G j ¹ ˘ º = N + 1 Ö k = 1 k , j ˘ x k x j x k ; ~ G i ¹ ˘ º = N + 1 Ö k = 1 k , i ˘ ~ x k ~ x 1 ~ x k : (4.27) Thefunctions G j ¹ ˘ º and ~ G i ¹ ˘ º arepolynomialsoforder N andthus,canbeexpandedintermsof Jacobipolynomialsas G j ¹ ˘ º = N + 1 Õ n = 1 ¹ 1 º j n p ; n 1 ¹ ˘ º ; j = 2 ; 3 ; ; N + 1 ; (4.28) ~ G i ¹ ˘ º = N + 1 Õ m = 1 ¹ 2 º i m p ; m 1 ¹ ˘ º ; i = 1 ; 2 ; ; N ; (4.29) forwhichthecoe˚cients ¹ 1 º j n and ¹ 2 º i m areobtainedanalytically,usingtheorthogonalityof Jacobipolynomials,seesection(4.5.1).Therefore, D i ; j = a j N + 1 Õ n = 1 ¹ 1 º j n 1 D š 2 ˘ »¹ 1 + ˘ º p ; n 1 ¹ ˘ º¼ ; ~ a i N + 1 Õ m = 1 ¹ 2 º i m ˘ D š 2 1 »¹ 1 ˘ º p ; m 1 ¹ ˘ º¼ ; (4.30) = a j N + 1 Õ n = 1 ¹ 1 º j n 1 D š 2 ˘ » ¹ 1 º P n ¹ ˘ º¼ ; ~ a i N + 1 Õ m = 1 ¹ 2 º i m ˘ D š 2 1 » ¹ 2 º P m ¹ ˘ º¼ ; where ¹ 1 º P n ¹ ˘ º and ¹ 2 º P m ¹ ˘ º areJacobipoly-fractonomialsof˝rstandsecondkind,respectively, ofwhichthefractionalderivativesare 1 D š 2 ˘ » ¹ 1 º P n ¹ ˘ º¼ = ¹ n + º ¹ n + š 2 º ¹ 1 º P š 2 n ¹ ˘ º ; (4.31) RL ˘ D š 2 1 » ¹ 2 º P m ¹ ˘ º¼ = ¹ m + º ¹ m + š 2 º ¹ 2 º P š 2 m ¹ ˘ º ; (4.32) 94 using(3.1).Let L D š 2 ; ˘; j ¹ ˘ º = a j N + 1 Õ n = 1 ¹ 1 º j n ¹ n + º ¹ n + š 2 º p + š 2 ; š 2 n 1 ¹ ˘ º ; (4.33) R D š 2 ; ˘; i ¹ ˘ º = ~ a i N + 1 Õ m = 1 ¹ 2 º i m ¹ m + º ¹ m + š 2 º p š 2 ; + š 2 m 1 ¹ ˘ º : (4.34) Hence, D i ; j = 1 D š 2 ˘ » ¹ 1 º h j ¹ ˘ º¼ ; ˘ D š 2 1 » ¹ 2 º h i ¹ ˘ º¼ ; (4.35) = ¹ 1 + ˘ º š 2 L D š 2 ; ˘; j ¹ ˘ º ; ¹ 1 ˘ º š 2 R D š 2 ; ˘; i ¹ ˘ º : Theexpressionin(4.35)isalsousedlaterinthesectionofnumericalresultstoobtainthecorre- spondingsystemofODEsinsolvingBurgersequation. Theorem4.3.2. Spectrally/ExponentiallyAccurateQuadratureRulein -Dimension[87]: PartA: 8 ˘ = ˘ 0 2 1 ; 1 ¼ ˝xed,and 8 n 2 ” [f 0 g ,theJacobipolynomial P ; n ¹ ˘ 0 º isa polynomialoforder n in . PartB:Let ˚ 2 H r ¹» min ; max ¼º , r > 0 .Then 8 max š 2 ¹ max min ˚ ¹ º¹ 2 T º ¹ n + º ¹ n + º ¹ k + º ¹ k + º ¹ 1 º P n ¹ ˘ º ¹ 2 º P k ¹ ˘ º d Q Õ q = 1 e w q ˚ ¹ q º¹ 2 T º q ¹ n + º ¹ n + q º ¹ k + º ¹ k + q º ¹ 1 º P q n ¹ ˘ º ¹ 2 º P q k ¹ ˘ º C Q r k ˚ k H r ¹» min ; max ¼º ; where C > 0 , ˚ N ¹ º = Í N n = 0 e ˆ n P n ¹ º denotesthepolynomialexpansionof ˚ ¹ º ,and f q ; e w q g Q q = 1 representsthesetofGauss-Legendrequadraturepointsandweights. PartC:If ˚ ¹ º issmooth,thequadraturerulein -dimensionbecomesexponentiallyaccuratein Q . Theconstructionofweakdistributeddi˙erentiationmatrix D i ; j requirestwostagesofquadra- turerule.The˝rststageisinthedomainof ,followingTheorem4.3.2,whereweobtainthe quadraturepointsandcorrespondingweights f q ; w q g Q q = 0 byperformingana˚nemappingfrom 95 2» min ; max ¼ tothestandarddomain st 2 1 ; 1 ¼ withJacobian J = ¹ max min º 2 .Therefore, D i ; j = Q Õ q = 0 J w q ˚ ¹ q º¹ 2 T º q D q i ; j (4.36) Inthesecondstage,foreachquadraturepoint q ,weobtainthepropersetofquadraturepointsin ˘ ,correspondingtotheweightfunctions ¹ 1 + ˘ º q š 2 and ¹ 1 ˘ º q š 2 andthus,wecarryout theintegral(4.35)tocomputetheentry D q i ; j . 4.3.4ConstructionofLinearSystem Thebilinearform a h ¹ u N ; v N º = l h ¹ v N º canberecastas: N + 1 Õ j = 2 D i ; j u j = f i ; i = 1 ; 2 ; ; N ; IVP i = 2 ; 3 ; ; N + 1 ; BVP (4.37) where D i ; j aregivenin(4.36)and f i = f ; ¹ 2 º h i ¹ ˘ º = ~ a i ¹ 1 1 ¹ 1 ˘ º f ¹ ˘ º ~ G¹ ˘ º d ˘; (4.38) arecomputedbyemployingaproperquadraturerulebasedontheregularityofforcefunction f . Thus,thelinearsystemisconstructedas D ® u = ® F ; (4.39) where ® u = f u 2 ; u 3 ; ; u N + 1 g .Inthesectionofnumericalresults,weusetheobtaineddi˙eren- tiationmatrixtosolvenon-linearequationsaswell,whereweprovidethecorrespondingsystem ofODEsforeachproblem.Wefollowthestepsbelowtoconstructthelinersystem,andthen, performtimeintegrationintimedependentproblems(Thefollowinginterpolationmatrix M and 96 nonlinear-inducedmatrix N arede˝nedlaterinsection4.4). ¹ i º Choosethepropersetsofinterpolationpoints : ¹ 1 º W ; ¹ 2 º W ¹ ii º Formthebasisandtestfunctions : ¹ 1 º h j ¹ ˘ º ; ¹ 2 º h i ¹ ˘ º ¹ iii º Obtainthequadraturepointsin domainintegal : q ; w q ¹ i v º Obtainthequadraturepointsinspaceintegral ¹ v º Constructthecorrespondinginterpolation,di˙erentiation,andnonlniear-inducedmatrices : M ; D ; N ¹ v i º 2 nd -orderAdamsBashforthtimeintegration 4.3.5ConditionNumberofLinearSystem Weconsiderdi˙erentchoicesofinterpolationpointsandstudytheconditionnumberoftheresulting linearsystem.Also,basedonthedistributionfunction ˚ ¹ º ,wedevelopseveralpre-conditioning matrices,correspondingtothechoiceofinterpolationpoints,andinvestigatetheire˚ciency. 4.3.5.1InterpolationPoints Fractionaloperatorsingeneralleadtofullandasymmetriclinearsystems,ofwhichthecondition numbersigni˝cantlygrowswiththenumberofmodes,thus,requiringdevelopinge˙ectiveand easytoconstructpre-conditioners.Inaddition,thedistributionfunction ˚ ¹ º andthechoiceof interpolationpointsareothermajorfactorsa˙ectingtheconditionnumberoftheresultinglinear system.Inthedistributedorderfractionaloperators,foreachrealizationofaphysicalprocess, thedistributionfunction ˚ ¹ º canbeobtainedfromobservabledataandarbitrarilycon˝nethe theoreticallowerandupperterminals min and max inpractice.Forexample,inasuper-di˙usive process,thedistributioncanbemorebiasedtowardtheupperbound max = 2 .Therefore,the choiceof ˚ ¹ º hasadirecte˙ectonthepropertiesofresultinglinearsystem.Toillustrate,we consider ˚ ¹ º tobeleft-andright-biasednormaldistributionfunctionwithvariousmeanand variance,showninFig.4.1(left).Weconstructthecorrespondingdi˙erentiationmatricesfor interpolationparameters = f 0 : 1 ; 0 : 9 g ,whereweusechoice ¹ ii º ofinterpolationpoints.The 97 Figure4.1: Initial/Boundaryvalueproblem:conditionnumberoftheresultinglinearsystemforleft- andright-biasednormaldistributions(left),where = 0 : 1 (middle), = 0 : 9 (right),andchoice ¹ ii º of interpolationpointsisused. conditionnumberofresultinglinearsystemisshowninFig.4.1(middle)and(right).Weobserve thatfortheright-biaseddistributions,whichhavethemeanvaluesclosertotheupperterminal, thechoiceof doesnothaveasigni˝cante˙ectontheconditionnumberofthelinearsystem. However,thisisnotthecaseforleft-sideddistributions,whereweobservethatthechoiceof = 0 : 1 ,closertomeanvalueofdistribution,resultsinbetterconditionnumber.Moreover,the resultsinFig.4.1suggeststhatdistributionswithlargervariance,whichincludeawiderrangeof derivativeorders,leadtohigherconditionnumber;andthissituationbecomesevenmoreadverse whenthedistributionisalsoleft-biased. TheconstructedfractionalLagrangeinterpolantsareallofsameorderanddonotformasetof hierarchicalbasis.Thus,foreachsetofinterpolationpoints,weneedtoreconstructtheinterpolants aswellasthedi˙erentiationmatrix.Therefore,thecorrespondinglinearsystemhighlydepends onthechoiceofinterpolationpoints.Here,weexaminefourmethodsthatyielddi˙erentsetsof interpolation/collocationpoints.Ineachcase,weformtwosetsof N pointsintheopenrange ¹ 1 ; 1 º andappend 1 toobtain ¹ 1 º W asthe˝rstsetofinterpolationspointstoconstructtheinterpolants, ¹ 1 º h j ¹ ˘ º ; j = 2 ; 3 ; ; N + 1 ,and 1 toobtain ¹ 2 º W asthesecondsetofinterpolationspointsto constructtheinterpolants, ¹ 2 º h i ¹ ˘ º ; 1 = 1 ; 2 ; ; N .Werefertotheaforementionedpointsas follows: Equidistantpoints: thischoiceisinspiredbythewell-knownFouriercollocationpoints.We formthetwosimilarsetof N pointsbyconsidering ˘ q = 1 + 2 q N + 1 ; q = 1 ; 2 ; ; N ,and then,obtain ¹ 1 º W and ¹ 2 º W . (i) 98 RootsofJacobipolynomials: weformthetwodi˙erentsetof N pointsbyconsidering thezerosof p ; N ¹ ˘ º and p ; N ¹ ˘ º ,whichareessentiallythe Gauss-Jacobi points.Then, weobtain ¹ 1 º W and ¹ 2 º W byappending 1 and 1 tothe˝rstandsecondset,respectively. Indeed,theobtainedinterpolationpointsarethe Gauss-Radau pointsasthezerosofJacobi polyfractonomials ¹ 1 º P N + 1 ¹ ˘ º and ¹ 2 º P N + 1 ¹ ˘ º . (ii) RootsofChebyshevpolynomials: weformthetwosimilarsetof N pointsbyconsidering ˘ q = cos 2 q + 1 N ˇ 2 ; q = 0 ; 1 ; ; N 1 astherootsofChebyshevpolynomial T N ¹ ˘ º ,which arealsoGausspoints.Hence,weobtain ¹ 1 º W and ¹ 2 º W byadding 1 and 1 . (iii) RootsofderivativeofChebyshevpolynomials: weformthetwosimilarsetof N pointsby considering ˘ q = cos ˇ q Ntot + 1 ; q = 1 ; 2 ; ; N astheroots d d ˘ T N + 1 ¹ ˘ º andthen,obtain thetwosets. (iv) Inordertoexaminethee˚ciencyofeachchoiceofinterpolationpoints,weconsiderthelinear distributed-orderinitialvalueproblem(4.6),wherewelet u ext = t 5 , ˚ ¹ º = ¹ 6 º ¹ 6 º , 2¹ 0 ; 2 º , and f ¹ t º = t 3 t 2 1 log ¹ t º .Wesolvetheproblem,usingthedevelopedscheme,inwhichweconsider theaforementionedfourchoicesofinterpolationpointsandthen,computeerrorandcondition numberoftheresultinglinearsystemfor = f 1 10 4 ; 0 : 1 ; 0 : 5 ; 0 : 9 g .Duetotheuniquenessof polynomials,weobservethatthechoiceofdi˙erentpointsdoesnota˙ecttherateofconvergence, however,hasagreate˙ectontheconditionnumberoftheconstructedlinearsystem.WeseeinFig. 4.2thatforthegiven ˚ ¹ º ,rootsofJacobipolynomials p ; N ¹ ˘ º and p ; N ¹ ˘ º (choice ¹ ii º )leadsto thelowestconditionnumber,andforrelativelysmallandlargevaluesof ,improvesthecondition numberbyalmosttwoordersofmagnitude.Inthenextsection,wedesignasetofpre-conditioners, whichcanfurtherimprovetheconditionnumberforthischoiceofinterpolationpoints. 4.3.5.2Pre-Conditioning Weintroduceadistributedpre-conditionermatrix M 1 ,whoseentriesaregivenby: M 1 lr = ¹ ˙ max ˙ min ' ¹ ˙ º ¹ 2 º P ˙ r ¹ ˘ l º d ˙; l ; r = 1 ; 2 ; ; N ; (4.40) 99 Figure4.2: Initial/Boundaryvalueproblem:conditionnumberoftheresultinglinearsystemfordi˙erent choicesofinterpolationpointsfordi˙erentinterpolationparameter,(lefttoright) = 1 10 4 , 0 : 1 , 0 : 5 , and 0 : 9 ,where ˚ ¹ º = ¹ 6 º ¹ 6 º and 2¹ 0 ; 2 º . whereweconsider f ˙ min ;˙ max g = f min ; max g and ˘ l N l = 1 tobethezerosof P ˙ 0 ; ˙ 0 N .Di˙erent choicesof ˙ 0 andalso ' ¹ ˙ º resultindi˙erentmatrices M .Weconsiderthefollowingchoicesto constructthepre-conditioningmatrix: (i) ' = ¹ ˙ ˙ 0 º ˙ 0 = (ii) ' = ˚˙ 0 = where istheinterpolationparameterand isthekroneckerdeltafunction. Inordertoinvestigatethee˚ciencyofintroducedpre-conditioner,weapplythemtothelinear system,obtainedintheprevioussectionforchoice ¹ ii º ofinterpolationpoints.Thus,welet ˚ ¹ º = ¹ 6 º ¹ 6 º , 2¹ 0 ; 2 º .WeshowinFig.4.3thatthedesignedpre-conditioner(ii)improvesthe conditionnumberofthelinearsystemmoree˚cientlyandatleast.Furthermore,weconsiderthe sameproblembutwitharight-biaseddistributionfunction,andshowinFig.4.4thatthedesigned pre-conditioner(i)ismoree˙ective.Wealsonotethatinbothcasesof ˚ ¹ º ,eitherchoicesof pre-conditionershavealmostsimilarperformance. 100 Figure4.3: Conditionnumberoftheoriginallinearsystemwith ˚ ¹ º = ¹ 6 º ¹ 6 º (left),andthepre- conditionedonefor = 1 š 10 (middle),and = 9 š 10 (right). Figure4.4: Conditionnumberofthelinearsystemforright-biasednormaldistribution.Theoriginallinear system(blacklines)andthepre-conditionedone(redandbluelines)for = 1 š 10 (left), = 5 š 10 (middle), and = 9 š 10 (right). 4.3.6WeakDistributedDi˙erentiationMatrix:Two-SidedDistributed-OrderBVPs Weextendourformulationtothetwo-sidedspacedistributed-orderderivative(inRiemann- Liouvillesense)inBVPsandconsiderthefollowingproblem D D ˚ u ¹ x º = ¹ max min ˚ ¹ º l 1 D x u ¹ x º + r x D 1 u ¹ x º d = f ¹ x ; u º ; (4.41) u 1 º = u ¹ 1 º = 0 ; 1 < min < max 2 8 x 2 1 ; 1 º ; forwhichthediscretebilinearformcanbewrittenas: a h ¹ u N ; v N º = l ¹ 1 D š 2 x u N ; x D š 2 1 v N º + r ¹ x D š 2 1 u N ; 1 D š 2 x v N º .Wefollowthederivationinimplementingthepseudo-spectralscheme, whereweneedtoconstructtwosetsoffractionalinterpolants.Thechallengeinsuchproblemis toconsidertheinterpolationpointssuchthatwecantakethetwosidedderivativeoftheresulting interpolantwithnoextracostandthus,respecttheformulationofweakdistributeddi˙erentiation matrix.Bypickingtheinterpolationparameter tobezero,thetwosetsofinterpolationpoints coincide.Therefore,thefractionalinterpolantsbecomethestandardLagrangeinterpolants,which 101 canbeexpandedintermsofLegendrepolynomial,andwecanreadilytaketheirright-andleft-sided derivatives.Hence, ¹ 1 º h j ¹ ˘ º = 0 = h j ¹ ˘ º = N + 2 Ö k = 1 k , j ˘ x k x j x k = N + 1 Õ n = 1 j n P n ¹ ˘ º ; j = 2 ; 3 ; ; N + 1 ; ¹ 2 º h i ¹ ˘ º = 0 = h i ¹ ˘ º = N + 2 Ö k = 1 k , i ˘ x k x i x k = N + 1 Õ m = 1 i m P m ¹ ˘ º ; i = 2 ; 3 ; ; N + 1 ; where P n ¹ ˘ º and P m ¹ ˘ º aretheLegendrepolynomial.WefollowsimilarderivationasinSec.4.3.3 andlet D i ; j = N + 1 Õ n = 1 j n 1 D š 2 ˘ » P n ¹ ˘ º¼ ; N + 1 Õ m = 1 i m ˘ D š 2 1 » P m ¹ ˘ º¼ = ¹ 1 + ˘ º š 2 L D š 2 ˘; j ¹ ˘ º ; ¹ 1 ˘ º š 2 R D š 2 ˘; i ¹ ˘ º ; whichcanbetakenbyaproperquadraturerulewiththecorrespondingweights ¹ 1 + ˘ º š 2 and ¹ 1 ˘ º š 2 ,where L D š 2 ˘; j ¹ ˘ º = N + 1 Õ n = 1 j n ¹ n + 1 º ¹ n š 2 + 1 º P š 2 ; š 2 n ¹ ˘ º¹ 1 + ˘ º š 2 ; R D š 2 ˘; i ¹ ˘ º = N + 1 Õ m = 1 i m ¹ m + 1 º ¹ m š 2 + 1 º P š 2 ; š 2 m ¹ ˘ º¹ 1 ˘ º š 2 : Thus,thediscretebilinearcanbewrittenas a h ¹ u N ; v N º = l D i ; j + r D j ; i andthe two-sided weakdistributeddi˙erentiationmatrix takestheform D i ; j = ¹ max min ˚ ¹ º l D i ; j + r D j ; i d : (4.42) Wenotethatif l = r ,thentheobtaineddi˙erentiationmatrixissymmetric. 4.4NumericalSimulations Inordertoexaminethee˚ciencyofproposednumericalschemes,weconsiderseveralnumerical examplesasfollows. 102 4.4.1Distributed-OrderIVP WerecalltheIVP(4.6)here D D ˚ u ¹ t º = f ¹ t º ; 8 t 2¹ 0 ; T ¼ ,subjecttozeroinitialconditions,which wesolvefortwosmoothandnon-smoothtestcases: ‹ CaseI: u ext = t 5 , ˚ ¹ º = ¹ 6 º ¹ 6 º , f ¹ t º = t 3 t 2 1 log ¹ t º , ‹ CaseII: u ext = t 5 + , ˚ ¹ º = ¹ 6 + º ¹ 6 + º , f ¹ t º = t 3 + t 2 1 log ¹ t º , where 2¹ 0 ; 2 º .followingtheprocedure,weconstructthelinearsystem(4.39).Bytakingthe simulationtime T = 2 andfordi˙erentchoicesof ,weinvestigatetheconvergenceofthesolution in L 2 -norm.Itisobservedthatthechoiceof hasanimportante˙ectontheaccuracyofscheme. Buttheexactsolutionisnotalwaysknownindistributed-orderproblems.Hence,thefractional parameter canplaytheroleofa ˝ne-tuningknob givingthepossibilityofsearchingforthe best/optimalcase,wherethehighestratecanbeachievedwithminimaldegreesoffreedom.As showninFig.4.5(a)forthecaseofsmoothsolution,theerrorreachesmachineprecisionmuch fasterbytunningtheparameter tobecloseto 1 ,wherethenodalbasisresemblethestandard Lagrangebehaviorandthus,moree˚cienttocapturethesmoothsolution.Moreover,inthecase ofsingularsolution,wecanreadilycapturethesingularitybytunningtheinterpolationparameter suchthatthesingularityiscapturedbythefractionalpartofthenodalbasisandthesmoothpartis capturedbychoosingenoughnumberofinterpolationpoints.Inbothcases,decreasingthevalue of leadstoabetterconditionedresultinglinearsystem,seeTable.4.1. Table4.1:Pseudo-spectralmethod:conditionnumberoftheresultinglinearsystem,Case-I(left), Case-II(right). N = 1 10 4 = 9 š 10 = 5 š 10 = 1 š 10 5165.846118.41214.51923.5263 9582.666381.58943.1632122.499 131400.63853.817109.051358.571 172701.791607.76222.974796.262 N = 1 10 4 = 9 š 10 = 5 š 10 = 1 š 10 228.833422.27074.614923.03315 371.073756.06648.033636.21275 4119.13392.22876.8564612.7772 5184.823136.85815.93522.9091 Table4.2showstheconditionnumberoftheresultinglinearsystemforcase-I,constructedby fractionalcollocationmethodfordistributed-orderdi˙erentialequations,developedin[87]andthe currentpseudo-spectralmethod,andweobservethattheproposedschemewins. 103 Figure4.5: Pseudo-spectralmethod: L 2 -normerroroftheapproximatesolution,(left)CaseI(right)Case II Table4.2:Conditionnumberoftheresultinglinearsystem.Thecomparisonbetweenfractional collocationmethod(employingfractionalinterpolantsinthestrongsense[87]),andpseudo-spectral method(employingfractionalinterpolantsintheweaksense). = 9 š 10 = 5 š 10 = 1 š 10 NStrongWeakStrongWeakStrongWeak 6 60 : 3467118 : 41243 : 664914 : 51936 : 005623 : 5263 10 325 : 037381 : 589214 : 93543 : 1632202 : 826122 : 499 14 1076 : 14853 : 817685 : 166109 : 051713 : 002358 : 571 18 2665 : 321607 : 761661 : 16222 : 9743397 : 2796 : 262 4.4.2(1+1)-DimensionsSpaceDistributed-OrderBurgersEquation Let u : ! R ,where = 1 ; 1 ¼» 0 ; T ¼ .Weconsiderthefollowingspacedistributed-order fractionalBurgersequationwithnonlinearreactionterm @ u ¹ x ; t º @ t = D D ˚ u ¹ x ; t º u 3 ¹ x ; t º u 1 D ˙ x u ¹ x ; t º + f ¹ x ; t º ; (4.43) subjecttohomogeneousDirichletboundaryconditions u 1 ; t º = u ¹ 1 ; t º = 0 ,andinitialcondition u ¹ x ; 0 º = u ext ¹ x ; 0 º ,where = 10 4 isthedi˙usioncoe˚cient, and areconstantcoe˚cients. Thedistributed-orderoperator D D ˚ isde˝nedin(5.7),inwhich ˚ ¹ º : ¹ 1 ; 2 º! R .Thesingu- larexactsolutionis u ext ¹ x ; t º = e 0 : 2 t ¹ 1 + x º 3 + 1 2 ¹ 1 + x º 4 + ,andtheforceterm f ¹ x ; t º is computedbysubstitutingtheexactsolutionin(4.43),inwhichwetaketheintegralin -domain usingthesamequadratureruleasintheweakdistributeddi˙erentiationmatrix.Weusethepro- posedschemetoapproximatethesolutioninspaceandthen,byemployingpropertimeintegration method,wemarchintime.Tothisend,we˝rsttaketheweakformoftheproblembymultiplying testfunctionsandintegrateoverthespatialcomputationaldomain.Weformtwopropersetsof 104 N + 2 interpolationpointstoconstructthenodalbasis ¹ 1 º h j ¹ x º andtestfunctions ¹ 2 º h i ¹ x º ,as discussedinsection4.3.2.Then,similarto(4.19),weapproximatethesolutioninspaceas u N ¹ ˘; t º = N + 1 Õ j = 2 u j ¹ t º ¹ 1 º h j ¹ ˘ º ; (4.44) where u j ¹ t º = u ¹ ˘ j ; t º ,andweonlyneedtoconstruct N interpolantsduetozeroboundaryconditions. Thus,weobtainthelinearsystemofODEs M d dt U ¹ t º = D U ¹ t º N ¹ t º D ¹ t º + f ¹ t º ; (4.45) where U ¹ t º isthevectorof u j ¹ t º ; j = 2 ; 3 ; ; N + 1 , M ij istheinterpolationmatrix, D istheweak distributeddi˙erentiationmatrixin(4.25), f ¹ t º istheforcevector, N ¹ t º isthevectorassociated withthecubicnon-linearreactionterm,and D ¹ t º isthevectorassociatedwiththeweakformof non-linearterm.For i ; j = 2 ; 3 ; ; N + 1 ,weemployproperquadraturerulesbyfollowingRemark 4.4.1,andcomputetheentriesofeachmatricesandvectors,whicharede˝nedas: M ij = ¹ 1 1 ¹ 1 º h j ¹ x º ¹ 2 º h i ¹ x º dx ; (4.46) f i ¹ t º = ¹ 1 1 f ¹ x ; t º ¹ 2 º h i ¹ x º dx ; (4.47) N i ¹ t º = ~ a i N + 1 Õ q = 2 W q a 3 q ~ G i ¹ x q º u q ¹ t º 3 (4.48) D i ¹ t º = ~ a i N + 1 Õ q = 2 W q N + 1 Õ r = 2 u r ¹ t º a r L D ˙; x ; r ¹ x q º ! ~ G i ¹ x q º u q ¹ t º ; (4.49) where ~ a i , a q ,and ~ G i ¹ x º aregiveninsection4.3.3.Seealsosection4.5.2onderivationofvectors (4.48)and(4.49)correspondingtothenonlinearterms. Remark4.4.1. Thedi˚cultyinobtainingthesystemofODEsistoe˚cientlycomputetheintegral forthenon-linearterms N ¹ t º and D ¹ t º ,whichweovercomebyusingthepropertyoffractional LagrangeinterpolantsthatsatisfytheKroneckerdeltaattheinterpolationpoints.Thus,welet thequadraturepointsintakingtheintegralofnon-lineartermstocoincidewiththe˝rstsetof interpolationpoints,i.e. ¹ 1 º W ,whichweusetoconstructthenodalbasis.Therefore,forthecase 105 thatweonlyhavecubicnon-linearity,i.e. = 0 ,weshowthatGauss-Lobatto-Jacobiformulawith weights f ; 3 g istheproperquadratureruleintakingtheintegralofnon-linearterm.Hence, wechoose ¹ 1 º W tobethezerosof p ; 3 N + 2 ¹ ˘ º .Inthecasethat = 0 ,however,byconsideringthe orderoffractionalderivativeinthenon-linearterm,we˝ndthatGauss-Lobatto-Jacobiformula withweights f ; 2 ˙ g istheproperquadratureruleintakingtheintegralofnon-linearterm. Thus,wechoose ¹ 1 º W tobethezerosof p ; 2 ˙ N + 2 ¹ ˘ º . TheobtainedlinearsystemofODE(4.45)isthenintegratedintime,usinga 2 nd -orderAdams Bashforthmethodwithtimestep t = 10 4 fordi˙erentvaluesof , ,andleft-andright-biased distributionfunctions ˚ ¹ º ,asshowninFig.4.6(left).Weconsider = 1 and = 0 ,where werecoverthelinearsystemcorrespondingtothedistribute-orderdi˙usionreactionequation,for whichweshowtheconvergenceofnumericalsolutionin L 1 -norminFig.4.6(middle).Moreover, weconsider = 0 and = 1 ,andrecoverthelinearsystemcorrespondingtothedistribute-order Burgersequation,whereweshowtheconvergenceofnumericalsolutionin L 1 -norminFig.4.6 (right).Wenotethattheexactsolutioninspaceisapowerlawtypewithhighestorder 4 + andsingularityoforder ,towhichtheinterpolationparameteristunned.Theexactsolutionis well-approximateduptotheorderof 10 9 with N = 4 inthecaseofdistributed-orderdi˙usion reactionequation.However,inthecaseofdistributed-orderBurgersequation,thesingleorder fractionalderivativeshiftsthesingularityandthus,postponeconvergenceto N = 9 .Wealsonote thatinbothcases,weobserveaplateauintheconvergenceofsolutionafterreachingerrorlevel 10 9 ,whichisduetotheinaccuracyinintegrationsinvolvedwithGammafunctions. 4.4.3(1+2)-DimensionsTwo-SidedSpaceDistributed-OrderDi˙usionReactionEquation Let u ¹ t ; x ; y º : ! R ,where = » 0 ; T ¼ 1 ; 1 ¼ 1 ; 1 ¼ .Weusethede˝nitionoftwo-sided distributed-orderderivative,givenin(4.41)andconsiderthefollowing 1 + 2 -Dtwo-sidedspace 106 Figure4.6: (1+1)-Dspacedistributed-orderBurgersequation,pseudo-spectralinspaceand 2 nd -order AdamsBashforthintime.(left):Theleft-andright-biasednormaldistributionfunctions(mean 1 : 3 and 1 : 7 , respectivelywithvariance 0 : 1 )inthedistributed-orderoperator.(middle): L 1 -normerrorv.s. N for = 0 and = 1 .(right): L 1 -normerrorv.s. N for = 1 and = 0 . distributed-orderdi˙usionequationwithnonlinearreactionterm @ u @ t = ¹ max min ˚ x ¹ º l x 1 D x + r x x D 1 ud (4.50) + ¹ max min ˚ y ¹ º l y 1 D y + r y y D 1 ud u 3 + f subjecttoproperinitialandboundaryconditions,inwhich l x , r x , l y ,and r y arethedi˙usion coe˚cients.Weobtaintheweakformoftheproblembymultiplyingpropertestfunctionsand integratingoverthewholecomputationaldomain.Asdiscussedinsection4.3.6fortwo-sided operators,weformtwosetsof N + 2 interpolationpointsforeachdimensionandconstructthe correspondinginterpolants.Then,weapproximatethesolutionusingnodalexpansionas u N ¹ t ; x ; y º = N + 1 Õ j = 2 N + 1 Õ j = 2 u j ; s ¹ t º h j ¹ x º h s ¹ y º : (4.51) Byconsideringthetestfunction v ¹ x ; y º = h i ¹ x º h r ¹ y º andsubstituting(4.51)into(4.50),weobtain thecorrespondinglinearsystemas M x dU ¹ t º dt M y T = D x U ¹ t ºM y T + M x U ¹ t º D y T N¹ t º + F¹ t º ; (4.52) where U ¹ t º isthematrixof u js ¹ t º ; j ; s = 2 ; 3 ; ; N + 2 .Theweakdistributeddi˙erentiationmatrices in x and y directionsare D x and D y ,respectively,andhavesimilarstructures,givenin(4.42). Thediagonalmatrices M x and M y aretheinterpolationmatricesin x and y directions,de˝nedas ¯ 1 1 h j ¹ x º h i ¹ x º dx and ¯ 1 1 h s ¹ y º h r ¹ y º d y ,respectively. F¹ t º = ¯ 1 1 ¯ 1 1 f ¹ t ; x ; y º h i ¹ x º h r ¹ y º dx is 107 Figure4.7: ¹ 1 + 2 º -Dtwo-sidedspacedistributed-orderdi˙usionreactionequation,pseudo-spectralin spaceand 2 nd -orderAdamsBashforthintime.(left): L 1 -normerrorv.s. N .(right):Timeevolutionofthe solution. theforcetherm.Matrix N¹ t º ,associatedwiththeweakformofcubicnonlinearitycanbee˚ciently computedusingthepropertyofLagrangeinterpolantsasdiscussedinprevioussections.Thus, N ir ¹ t º = w i w r u ir ¹ t º 3 ,where w i and w r aretheJacobiquadratureweights. Toinvestigatetheperformanceofdevelopedschemeinhigherdimensionalproblems,wesolve (4.50),subjecttohomogeneousDirichletboundarycondition u ¹ t ; x ; y º = 0 @ andinitialcondition u ¹ 0 ; x ; y º = u ext ¹ 0 ; x ; y º ,forleft-andright-biased ˚ ¹ º (showninFig.4.6(left)),whereineach caseweconsidersimilardistributionsinbothdirections x and y ,i.e. ˚ x ¹ º = ˚ y ¹ º = ˚ ¹ º . Moreover,weconsiderisotropicdi˙usivitywith l x = r x = l y = r y = 10 4 ,andasmooth exactsolutionoftheform u ext ¹ t ; x ; y º = e 0 : 2 t sin ¹ ˇ 2 ¹ x 1 ºº sin ¹ ˇ 2 ¹ y 1 ºº .Toobtaintheforce term f ¹ t ; x ; y º ,wesubstitute u ext into(4.50)andtaketheintegralin -domainusingthesame quadratureruleasintheweakdistributeddi˙erentiationmatrices,whereweneedtotaketheright- andleft-sidedderivativesoftheexactsolutionforeachquadraturepointin -domain.Toobtain thesefractionalderivatives,we˝rstaccuratelyprojecttheexactsolutionineachdirection x and y ontheLegendrepolynomials P l + 1 ¹ x º P l 1 ¹ x º and P l + 1 ¹ y º P l 1 ¹ y º ,respectively,considering su˚cientnumberofpolynomials,andthen,byusing(3.1)with = 0 ,weobtainthederivatives. Wealsousea 2 nd -orderAdamsBashforthmethodwithtimestep t = 10 4 tomarchintime.In Fig.4.7,weshowtheconvergenceofnumericalsolutionin L 1 -norm(left),inwhichweobserve thatthechoiceofright-biaseddistributionfunctionresultsinslightlylargererror.Moreover,in Fig.4.7(right)weshowthetimeevolutionanddi˙usionofthesolution,wheretheforcetermis adjustedsuchthattheexponentialdecayintimeisoforder 0 : 2 . 108 4.5DetailedDerivations 4.5.1PolynomialsExpansionsInTermsOfJacobiPolynomials Inordertoobtainthecoe˚cient ¹ 1 º j n in(4.28),wemultiplythebothsideofequationby p ; k 1 and theproperweights ¹ 1 ˘ º and ¹ 1 + ˘ º .Then,weintegrateoverthewholedomaintoobtain ¹ 1 1 ¹ 1 ˘ º ¹ 1 + ˘ º G j ¹ ˘ º p ; k 1 d ˘ = N + 1 Õ n = 1 ¹ 1 º j n ¹ 1 1 ¹ 1 ˘ º ¹ 1 + ˘ º p ; k 1 p ; n 1 d ˘; (4.53) = N + 1 Õ n = 1 ¹ 1 º j n 2 2 n 1 ¹ n º ¹ n + º ¹ n º¹ n 1 º ! nk byorthogonalityofJacobipolynomials : Therefore, ¹ 1 º j k = 1 k ¯ 1 1 ¹ 1 ˘ º ¹ 1 + ˘ º G j ¹ ˘ º p ; k 1 d ˘; j = 2 ; 3 ; ; N + 1 ,inwhich k = 2 2 k 1 ¹ k º ¹ k + º ¹ k º¹ k 1 º ! . Similarly,toobtainthecoe˚cient ¹ 2 º i m in(4.29),wemultiplybothsideoftheequationby p ; k 1 andtheproperweights ¹ 1 ˘ º and ¹ 1 + ˘ º ,andthenintegrateoverthewholedomain. Thus, ¹ 1 1 ¹ 1 ˘ º ¹ 1 + ˘ º ~ G j ¹ ˘ º p ; k 1 d ˘ = N + 1 Õ m = 1 ¹ 2 º i m ¹ 1 1 ¹ 1 ˘ º ¹ 1 + ˘ º p ; k 1 p ; m 1 d ˘; (4.54) = N + 1 Õ m = 1 ¹ 2 º i m 2 2 m 1 ¹ m + º ¹ m º ¹ m º¹ m 1 º ! mk byorthogonalityofJacobipolynomials : Therefore, ¹ 2 º i k = 1 k ¯ 1 1 ¹ 1 ˘ º ¹ 1 + ˘ º ~ G j ¹ ˘ º p ; k 1 d ˘; i = 1 ; 2 ; ; N ,inwhich k hasthe samede˝nitionasabove. 109 4.5.2E˚cientComputationofNon-linearTerms Thevector N ¹ t º isassociatedwiththenon-linearreactiontermandiscomputedfollowingRemark 4.4.1as: N ¹ t º = u 3 N ; ¹ 2 º h i ¹ x º x i = 2 ; 3 ; ; N + 1 ; (4.55) = N + 1 Õ j = 2 N + 1 Õ r = 2 N + 1 Õ s = 2 u j u r u s ¹ 1 1 ¹ 1 º h j ¹ x º ¹ 1 º h r ¹ x º ¹ 1 º h s ¹ x º ¹ 2 º h i ¹ x º dx ; = N + 1 Õ j = 2 N + 1 Õ r = 2 N + 1 Õ s = 2 ¹ u j u r u s º¹ a j a r a s ~ a i º ¹ 1 1 ¹ 1 x º ¹ 1 + x º 3 G j ¹ x ºG r ¹ x ºG s ¹ x º ~ G i ¹ x º dx ; = N + 1 Õ q = 2 W q N + 1 Õ j = 2 N + 1 Õ r = 2 N + 1 Õ s = 2 ¹ u j u r u s º¹ a j a r a s ~ a i ºG j ¹ x q ºG r ¹ x q ºG s ¹ x q º ~ G i ¹ x q º ; = N + 1 Õ q = 2 W q N + 1 Õ j = 2 N + 1 Õ r = 2 N + 1 Õ s = 2 ¹ u j u r u s º¹ a j a r a s ~ a i º jq rq sq ~ G i ¹ x q º ; = N + 1 Õ q = 2 W q ¹ a q u q º 3 ~ a i ~ G i ¹ x q º i = 2 ; 3 ; ; N + 1 ; where f x q ; W q gj N + 1 q = 2 aretheGauss-Lobatto-Jacobipointsandweightsassociatedtoweightfunctions ¹ 1 x º and ¹ 1 + x º 3 .Wenotethatsince u 1 = u N + 2 = 0 ,wedonotconsider q = 1 ; N + 2 inthe abovequadraturerule. 110 Moreover,followingthesameRemark4.4.1andusing(4.35),wecomputethevector D ¹ t º as: D ¹ t º = u ¹ x ; t º 1 D ˙ x u ¹ x ; t º ; ¹ 2 º h i ¹ x º x i = 2 ; 3 ; ; N + 1 ; (4.56) = N + 1 Õ j = 2 N + 1 Õ r = 2 u j u r ¹ 1 1 ¹ 1 º h j ¹ x º 1 D ˙ x ¹ 1 º h r ¹ x º ¹ 2 º h i ¹ x º dx ; = N + 1 Õ j = 2 N + 1 Õ r = 2 u j u r a j ~ a i ¹ 1 1 ¹ 1 x º ¹ 1 + x º 2 ˙ G j ¹ x º L D ˙; x ; r ¹ x º ~ G i ¹ x º dx ; = N + 1 Õ q = 2 W q N + 1 Õ j = 2 N + 1 Õ r = 2 u j u r a j ~ a i G j ¹ x q º L D ˙; x ; r ¹ x q º ~ G i ¹ x q º ; = N + 1 Õ q = 2 W q N + 1 Õ j = 2 N + 1 Õ r = 2 u j u r a j ~ a i jq L D ˙; x ; r ¹ x q º ~ G i ¹ x q º ; = N + 1 Õ q = 2 W q u q ~ a i N + 1 Õ r = 2 u r a r L D ˙; x ; r ¹ x q º ! ~ G i ¹ x q º ; i = 2 ; 3 ; ; N + 1 ; where f x q ; W q gj N + 1 q = 2 aretheGauss-Lobatto-Jacobipointsandweightsassociatedtoweightfunctions ¹ 1 x º and ¹ 1 + x º 2 ˙ .Wenotethatsince u 1 = u N + 2 = 0 ,wedonotconsider q = 1 ; N + 2 in theabovequadraturerule. 111 CHAPTER5 TEMPORALLY-DISTRIBUTEDFRACTIONALPDES:PETROV-GALERKIN SPECTRALMETHOD 5.1Background WeconstructPetrov-Galerkinspectralmethodswithauni˝edfastsolverforaclassoftemporally- distributedFPDEswithconstantcoe˚cientssubjecttoDirichletboundary/initialconditions.We developthefastlinearsolverbasedontheeigensolutionsofthecorrespondingtemporal/spatial massandsti˙nessmatrices.WecarryoutthediscretestabilityanderroranalysisofthePGmethod forthetwo-dimensionalcase.Eventually,weillustratethespectralconvergenceandthee˚ciency ofthemethodbyperformingseveralnumericalsimulations. Thischapterisorganizedasfollows:insection5.2,weintroducethepreliminariesonfractional calculus,de˝nethedistributedfractionalSobolevspaces,theproblem,andthecorresponding variationalform.Insection5.3,weconstructthePGmethods,formulatethefastsolver,andcarry outthediscretestabilityanderroranalysis.Insection5.4,weprovidesomenumericaltests. 5.2De˝nitions Following[118],wedenotetheleft-andright-sidedReimann-Liouvillefractionalderivatives by RL a D x f ¹ x º and RL x D b g ¹ x º ,respectively,inwhich g ¹ x º2 C n » a ; b ¼ .Werecallfrom[5]that RL a D x g ¹ x º = C a D x g ¹ x º = a D x g ¹ x º , 2¹ 0 ; 1 º ,whenhomogeneousDirichletinitialandboundary conditionsareenforced.Following[87],weanalyticallyobtainthefractionalderivativesofthe Jacobipoly-fractonomials[194],whicharelaterusedindevelopingthenumericalscheme,as RL 1 D ˙ ˘ n ¹ 1 + ˘ º P ; n 1 ¹ ˘ º o = ¹ n + º ¹ n + ˙ º ¹ 1 + ˘ º ˙ P + ˙; ˙ n 1 ¹ ˘ º (5.1) RL ˘ D ˙ 1 n ¹ 1 ˘ º P ; n 1 ¹ ˘ º o = ¹ n + º ¹ n + ˙ º ¹ 1 ˘ º ˙ P ˙; + ˙ n 1 ¹ ˘ º (5.2) 112 inwhich ;˙> 0 and P ; n 1 ¹ ˘ º isthestandardJacobipolynomialoforder n 1 .Similarly,the -thorderfractionalderivativesoftheLegendrepolynomialsaregivenas 1 D x P n ¹ x º = ¹ n + 1 º ¹ n + 1 º P ; n ¹ x º¹ 1 + x º ; x D 1 P n ¹ x º = ¹ n + 1 º ¹ n + 1 º P ; n ¹ x º¹ 1 x º ; inwhich P n ¹ x º representstheLegendrepolynomialofordern. 5.2.1DistributedFractionalSobolevSpaces Accordingto[101],theusualSobolevspaceassociatedwiththerealindex 1 onboundedinterval 1 = ¹ a 1 ; b 1 º ,isdenotedby H 1 ¹ 1 º .DuetoLemma2.6in[101], kk H 1 ¹ 1 º kk c H 1 ¹ 1 º , where kk c H 1 ¹ 1 º = k x 1 D 1 b 1 ¹ºk 2 L 2 ¹ 1 º + k a 1 D 1 x 1 ¹ºk 2 L 2 ¹ 1 º + kk 2 L 2 ¹ 1 º 1 2 : Let i = ¹ a i ; b i º i 1 for i = 2 ; ; d ,and X 1 = H 1 0 ¹ 1 º ,whichisassociatedwiththenorm kk c H 1 ¹ 1 º . Therefore, X d isconstructedsuchthat X d = H d 0 ¹ a d ; b d º ; L 2 ¹ d 1 º \ L 2 ¹ I ; X d 1 º ,associated withnorm kk X d = ˆ kk 2 L 2 ¹ d º + Í d i = 1 k x i D i b i ¹ºk 2 L 2 ¹ d º + k a i D i x i ¹ºk 2 L 2 ¹ d º ˙ 1 2 ,where X d 1 = H d 1 0 ¹ a d 1 ; b d 1 º ; L 2 ¹ d 2 º \ L 2 ¹ I ; X d 2 º ; : : : X 2 = H 2 0 ¹ a 2 ; b 2 º ; L 2 ¹ 1 º \ L 2 ¹ I ; X 1 º : (5.3) Following[87],wedenoteby H ' ¹ R º the distributed fractionalSobolevspaceon R ,whichis endowedwiththefollowingnorm kk H ' ¹ R º = ¯ 2 1 ' ¹ ºk¹ 1 + j ! j 2 º 2 F¹º¹ ! ºk 2 L 2 ¹ R º d 1 2 ; where ' 2 L 1 ¹» 1 ; 2 ¼º , 0 1 < 2 .Subsequently,wedenoteby H ' ¹ I º the distributed fractionalSobolevspaceonthe˝niteclosedinterval I = ¹ 0 ; T º ,whichisde˝nedas H ' ¹ I º = f v 2 L 2 ¹ I ºj9 ~ v 2 H ' ¹ R º s : t : ~ v j I = v g ; withthetheequivalentnorms kk l H ' ¹ I º and kk r H ' ¹ I º in[87], where kk l H ' ¹ I º = kk 2 L 2 ¹ I º + ¹ 2 1 ' ¹ ºk RL 0 D t ¹ºk 2 L 2 ¹ I º d ! 1 2 ; kk r H ' ¹ I º = kk 2 L 2 ¹ I º + ¹ 2 1 ' ¹ ºk RL t D T ¹ºk 2 L 2 ¹ I º d ! 1 2 : (5.4) 113 Let = I d .Wede˝ne l 0 H ' I ; L 2 ¹ d º : = n u jk u ¹ t ; ºk L 2 ¹ d º 2 H ' ¹ I º ; u j t = 0 = u j x = a i = u j x = b i = 0 ; i = 1 ; ; d o ; whichisequippedwiththenorm k u k l H ˝ ¹ I ; L 2 ¹ d ºº = k u ¹ t ; ºk L 2 ¹ d º l H ' ¹ I º = k u k 2 L 2 ¹ º + ¹ 2 1 ' ¹ ºk RL 0 D t ¹ u ºk 2 L 2 ¹ º d ! 1 2 : Similarly, r 0 H ' I ; L 2 ¹ d º : = n v jk v ¹ t ; ºk L 2 ¹ d º 2 H ' ¹ I º ; v j t = T = v j x = a i = v j x = b i = 0 ; i = 1 ; ; d o ; whichisequippedwiththenorm k v k r H ' ¹ I ; L 2 ¹ d ºº = k v ¹ t ; ºk L 2 ¹ d º r H ' ¹ I º = k v k 2 L 2 ¹ º + ¹ 2 1 ' ¹ ºk RL t D T ¹ v ºk 2 L 2 ¹ º d ! 1 2 : Wede˝nethesolutionspace B '; 1 ; ; d ¹ º : = l 0 H ˝ I ; L 2 ¹ d º \ L 2 ¹ I ; X d º ; endowedwiththe norm k u k B '; 1 ; ; d = n k u k 2 l H ' ¹ I ; L 2 ¹ d ºº + k u k 2 L 2 ¹ I ; X d º o 1 2 : Therefore, k u k B '; 1 ; ; d = n k u k 2 L 2 ¹ º + ¹ 2 1 ' ¹ ºk RL 0 D t ¹ u ºk 2 L 2 ¹ º d + d Õ i = 1 k x i D i b i ¹ u ºk 2 L 2 ¹ º + k a i D i x i ¹ u ºk 2 L 2 ¹ º o 1 2 : (5.5) Likewise,wede˝nethetestspace B '; 1 ; ; d ¹ º : = r H ' I ; L 2 ¹ d º \ L 2 ¹ I ; X d º ; endowedwith thenorm k v k B ˝; 1 ; ; d = n k v k 2 r H ˝ ¹ I ; L 2 ¹ d ºº + k v k 2 L 2 ¹ I ; X d º o 1 2 : Therefore, k v k B '; 1 ; ; d = n k v k 2 L 2 ¹ º + ¹ 2 1 ' ¹ ºk RL t D T ¹ v ºk 2 L 2 ¹ º d + d Õ i = 1 k x i D i b i ¹ v ºk 2 L 2 ¹ º + k a i D i x i ¹ v ºk 2 L 2 ¹ º o 1 2 : (5.6) Wenotethatingeneral, ' canbede˝nedinanypossiblesubsetoftheinterval » 1 ; 2 ¼ andthus arbitrarilycon˝nesthedomainofintegration,wherethetheoreticalframeworkoftheproblem remainsinvariantwhilerequiringthesolutiontohavelessregularity.Thefollowinglemmais usefulinconstructionoftheproposednumericalscheme. 114 Lemma5.2.1. [99]:Forall 0 < 1 ,if u 2 H 1 ¹» a ; b ¼º suchthat u ¹ a º = 0 ,and w 2 H š 2 ¹» a ; b ¼º , then ¹ a D s u ; w º = ¹ a D š 2 s u ; s D š 2 b w º ,where ¹ ; º representsthestandardinnerproduct in = » a ; b ¼ . 5.2.2ProblemDe˝nition Let 7! ' ¹ º beacontinuousmappingin » 1 ; 2 ¼ .Then,wede˝nethedistributedorderfractional derivativeas D D ' u ¹ t ; x º = ¹ 2 1 ' ¹ º a D t u ¹ t ; x º d ; t > a ; (5.7) where a D t denotestheRiemann-Liouvillefractionalderivativeoforder .Next,Let u 2 B '; 1 ; ; d ¹ º forsomepositiveinteger d and = » 0 ; T ¼» a 1 ; b 1 ¼» a 2 ; b 2 ¼» a d ; b d ¼ , where D D ' u + d Õ j = 1 c l j a j D 2 j x j u + c r j x j D 2 j b j u d Õ j = 1 l j a j D 2 j x j u + r j x j D 2 j b j u + u = f ; (5.8) inwhichallthecoe˚cients ; c l j ; c r j ; l j ; and r j areconstant, 2 j 2¹ 0 ; 1 º , 2 j 2¹ 1 ; 2 º for j = 1 ; 2 ; ; d ,and 0 < 1 < 2 1 .Problem(5.8)issubjecttotheDirichletinitialandboundary conditions,i.e. u j t = 0 = 0 and u j x j = a j = u j x j = b j = 0 for j = 1 ; 2 ; ; d .Accordingto(5.5),the normassociatedwith B '; 1 ; ; d ¹ º canbereducedto k u k B '; 1 ; ; d ¹ º = n ¹ 2 1 ' ¹ ºk 0 D t ¹ u ºk 2 L 2 ¹ º | {z } U ' I d + d Õ j = 1 h k a j D j x j ¹ u ºk 2 L 2 ¹ º | {z } U j II + k x j D j b j ¹ u ºk 2 L 2 ¹ º | {z } U j III io 1 š 2 ; andsimilarly,thenorm,associatedwith B '; 1 ; ; d ¹ º ,in(5.6)isequivalentto k v k B '; 1 ; ; d ¹ º = n ¹ 2 1 ' ¹ ºk t D T ¹ v ºk 2 L 2 ¹ º | {z } V ' I d + d Õ j = 1 h k x j D j b j ¹ v ºk 2 L 2 ¹ º | {z } V j II + k a j D j x j ¹ v ºk 2 L 2 ¹ º | {z } V j III io 1 š 2 : Inordertoobtainthevariationalformofproblem,wemultiply(5.8)byapropertestfunction v andintegrateoverthecomputationaldomain.Thecorrespondingcontinuousbilinearform 115 a : B '; 1 ; ; d ¹ º B '; 1 ; ; d ¹ º! R takestheform a ' ¹ u ; v º = ¹ 2 1 ' ¹ º¹ 0 D š 2 t u ; t D š 2 T v º d + d Õ j = 1 c l j ¹ a j D j x j u ; x j D j b j v º + c r j ¹ x j D j a j u ; a j D j x j v º d Õ j = 1 l j ¹ a j D j x j u ; x j D j b j v º + r j ¹ x j D j b j u ; a j D j x j v º + ¹ u ; v º ; (5.9) where ¹ ; º representstheusual L 2 -product.Thus,theproblemreadsas:˝nd u 2B '; 1 ; ; d ¹ º suchthat a ' ¹ u ; v º = ¹ f ; v º ; 8 v 2 B '; 1 ; ; d ¹ º : (5.10) Next,wechooseproper˝nite-dimensionalsubspacesof U N ˆB '; 1 ; ; d ¹ º and V N ˆ B '; 1 ; ; d ¹ º ; thus,thediscreteproblemreadsas:˝nd u N 2 U N suchthat a ' ¹ u N ; v N º = ¹ f ; v N º ; 8 v N 2 V N : (5.11) 5.3PetrovGalerkinMathematicalFormulation WeconstructaPetrov-Galerkinspectralmethodforthediscreteproblem u N 2 U N ,satisfying theweakform(5.11).We˝rstde˝netheproper˝nite-dimensionalbasis/testspacesandthen implementthenumericalscheme. 5.3.1SpaceofBasis( U N )andTest( V N )Functions WeemploytheLegendrepolynomialsasthespatialbasis,giveninthestandarddomain ˘ 2 1 ; 1 ¼ as ˚ m ¹ ˘ º = ˙ m P m + 1 ¹ ˘ º P m 1 ¹ ˘ º ; m = 1 ; 2 ; .Wealsoemploythepoly-fractonomialof ˝rstkind[183,194]asthetemporalbasisfunction,giveninthestandarddomain 2 1 ; 1 ¼ as ˝ n ¹ º = ˙ n ¹ 1 + º ˝ P ˝;˝ n 1 ¹ º ; n = 1 ; 2 ; .Thecoe˚cients ˙ m arede˝nedas ˙ m = 2 + 1 º m . Therefore,weconstructthetrialspaceas U N = span n ˝ n ¹ t º d Ö j = 1 ˚ m j ˘ j ¹ x j º : n = 1 ; 2 ; ; N ; m j = 1 ; 2 ; ; M j o ; 116 where ¹ t º = 2 t š T 1 and ˘ j ¹ s º = 2 s a j b j a j 1 .Thetemporalandspatialbasisfunctionsnaturally satisfytheinitialandboundaryconditions,respectively.Moreover,wede˝nethetemporaland spatialtestfunctionsinthestandarddomainas ˝ r ¹ º = e ˙ r ¹ 1 º ˝ P ˝; ˝ r 1 ¹ º ; r = 1 ; 2 ; (poly- fractonomialofsecondkind)and k ¹ ˘ º = e ˙ k P k + 1 ¹ ˘ º P k 1 ¹ ˘ º ; k = 1 ; 2 ; ,respectively.The coe˚cients e ˙ k arede˝nedas e ˙ k = 2 1 º k + 1 .Hence,weconstructthecorrespondingtestspace as V N = span n ˝ r ¹ t º d Ö j = 1 k j ˘ j ¹ x j º : r = 1 ; 2 ; ; N ; k j = 1 ; 2 ; ; M j o : 5.3.2ImplementationofPGSpectralMethod Werepresentthesolutionof(5.11)asalinearcombinationofelementsofthesolutionspace U N . Therefore, u N ¹ x ; t º = N Õ n = 1 M 1 Õ m 1 = 1 M d Õ m d = 1 ^ u n ; m 1 ; ; m d h ˝ n ¹ t º d Ö j = 1 ˚ m j ¹ x j º i (5.12) in .Bysubstitutingtheexpansion(7.38)into(5.11)andchoosing v N = ˝ r ¹ t º Î d j = 1 k j ¹ x j º , r = 1 ; 2 ;:::; N , k j = 1 ; 2 ;:::; M j ,weobtainthefollowingLyapunovsystem S ' ˝ M 1 M 2 M d + d Õ j = 1 » M ˝ M 1 M j 1 S Tot j M j + 1 M d ¼ + M ˝ M 1 M 2 M d U = F ; (5.13) inwhich representstheKroneckerproduct, F denotesthemulti-dimensionalloadmatrixwhose entriesaregivenas F r ; k 1 ; ; k d = ¹ f ¹ t ; x 1 ; ; x d º ˝ r ¹ t º d Ö j = 1 k j ˘ j ¹ x j º d ; (5.14) and S Tot j = c l j S j ; l + c r j S j ; r l j S j r j S j ; r .Thematrices S ' ˝ and M ˝ denote thetemporalsti˙nessandmassmatrices,respectively; S j , S j and M j denotethespatialsti˙ness andmassmatrices,respectively.Theentriesofspatialmassmatrix M j arecomputedanalytically, 117 whileweemployproperquadraturerulestoaccuratelycomputetheentriesofspatialsti˙ness S j , S j andtemporalmassmatrices M ˝ .Wenotethatduetothechoicesofbasis/testfunctions,the obtainedmassandsti˙nessmatricesaresymmetric.Moreover,weaccuratelycomputetheentries oftemporalsti˙nessmatrix, S ' ˝ ,usingtheorem(3.1)in[87]. 5.3.3Uni˝edFastFPDESolver Wedevelopauni˝edfastsolverintermsofthegeneralizedeigensolutionsinordertoformulatea closed-formsolutiontotheLyapunovsystem(7.39). Theorem5.3.1. Let f ® e j m j ; j m j g M j m j = 1 bethesetofgeneraleigen-solutionsofthespatialsti˙ness matrix S Tot j withrespecttothemassmatrix M j .Moreover,let f ® e ˝ n ; ˝ n g N n = 1 bethesetofgeneral eigen-solutionsofthetemporalmassmatrix M ˝ withrespecttothesti˙nessmatrix S ' ˝ .Thenthe matrixofunknowncoe˚cients U isexplicitlyobtainedas U = N Õ n = 1 M 1 Õ m 1 = 1 M d Õ m d = 1 n ; m 1 ; ; m d ® e ˝ n ® e 1 m 1 ® e d m d ; (5.15) where n ; m 1 ; ; m d isgivenby n ; m 1 ; ; m d = ¹ ® e ˝ n ® e 1 m 1 ® e d m d º F h ¹ ® e ˝ T n S ' ˝ ® e ˝ n º Î d j = 1 ¹ ® e j T m j M j ® e j m j º i n ; m 1 ; ; m d ; (5.16) inwhichthenumeratorrepresentsthestandardmulti-dimensionalinnerproduct,and n ; m 1 ; ; m d isobtainedintermsoftheeigenvaluesofallmassmatricesas n ; m 1 ; ; m d = h ¹ 1 + ˝ n º + ˝ n Í d j = 1 ¹ j m j º i : Proof. Considerthefollowinggeneralisedeigenvalueproblemsas S Tot j ® e j m j = j m j M j ® e j m j ; m j = 1 ; 2 ; ; M j ; j = 1 ; 2 ; ; d ; (5.17) M ˝ ® e ˝ n = ˝ n S ' ˝ ® e ˝ n ; n = 1 ; 2 ; ; N : (5.18) Havingthespatialandtemporaleigenvectorsdeterminedinequations(5.18)and(5.17),wecan representtheunknowncoe˚cientmatrix U in(7.38)intermsoftheaforementionedeigenvectors 118 as U = Í N n = 1 Í M 1 m 1 = 1 Í M d m d = 1 n ; m 1 ; ; m d ® e ˝ n ® e 1 m 1 ® e d m d ; where n ; m 1 ; ; m d isobtained asfollows.Following[145],wesubstitute U inthecorrespondingLyapunovequationandthen, taketheinnerproductofbothsidesofequationby ® e ˝ q ® e 1 p 1 ® e d p d .Therefore,byrearrangingthe terms,weobtain n ; m 1 ; ; m d = ¹ ® e ˝ n ® e 1 m 1 ® e d m d º F h ¹ ® e ˝ T n S ' ˝ ® e ˝ n º Î d j = 1 ¹ ® e j T m j M j ® e j m j º i h ¹ 1 + ˝ n º + ˝ n Í d j = 1 ¹ j m j º i : SincethespatialMass M j andtemporalsti˙nessmatrices S ' ˝ arediagonal,wehave ¹ ® e ˝ T q S ' ˝ ® e ˝ n º = 0 if q , n ,andalso ¹ ® e j T p j M j ® e j m j º = 0 if p j , m j ,whichcompletestheproof. 5.3.4StabilityAnalysis Thefollowingtheoremprovidesthediscretestabilityanalysisoftheschemefor(1+1)-dimensional temporally-distributedfractionaldi˙usionproblem.Suchastabilityanalysiscanbeextendedto theproblemof(1+d)-dimensionalwithboth-sidedderivatives,whichwewillbecarriedoutinour futurework. Theorem5.3.2. ThePetrov-Gelerkinspectralmethodfor(1+1)-Dtemporally-distributedand space-fractionaldi˙usionproblem a ' ¹ u ; v º = l ¹ v º isstable,i.e.,thediscrete inf - sup condition inf u N 2 U N u N , 0 sup v N 2 V N v N , 0 j a ¹ u N ; v N ºj k v N k B '; 1 ; ; d ¹ º k u N k B '; 1 ; ; d ¹ º > 0 ; (5.19) holdswith > 0 andindependentof N . Proof. Seesection(5.5.1). 5.3.5ErrorAnalysis Kharazmietal.[87]performedtheerroranalysisofthedistributedorderdi˙erentialequations, wheretheyemployedJacobipolyfractonomialsof˝rstkindasthebasisfunction.Following 119 similarsteps,wecanshowthattheprojectionerrorintimeandspacetakesthesameform.Let D ¹ r º u = @ r u @ t r 0 @ x 1 r 1 @ x d r d ,where r = Í d i = 0 r i .Thus,if D ¹ r º u 2 U forsomeinteger r 1 ,thatis, ¯ 2 1 ' ¹ ºk 0 D 2 t ¹D ¹ r º u ºk L 2 d < 1 ,and u N denotestheprojectionoftheexactsolution u ,then k u u N k U M r n kD ¹ r º u k 2 L 2 ¹ º + ¹ 2 1 ' ¹ ºk RL 0 D t ¹D ¹ r º u ºk 2 L 2 ¹ º d + d Õ i = 1 k x i D i b i ¹D ¹ r º u ºk 2 L 2 ¹ º + k a i D i x i ¹D ¹ r º u ºk 2 L 2 ¹ º o 1 2 (5.20) Sincethe inf-sup conditionholdsintheorem5.3.2,bytheBanach-Ne£as-Babu²katheoremin[51], theerrorinthenumericalschemeislessthanorequaltoaconstanttimestheprojectionerror. 5.4NumericalSimulations Weprovidenumericalexamplesofthespectralschemewehaveproposed.Weconsiderthe exactsolutionoftheform u ext = u t Î d j = 1 u ˘ j with˝niteregularity,where u t = t p 1 + ˝ , t 2» 0 ; T ¼ , and u ˘ j = ¹ 1 + ˘ j º p 2 + ¹ 1 ˘ j º p 3 + , ˘ j 2 1 ; 1 ¼ .Weobtaintheforcefunctionbysubstituting u ext into(5.8),wheretheadvectionanddi˙usioncoe˚cientsareconsideredtobeunityinall dimensions. Figure5.1showstheconvergenceoferrorviaspatialandtemporal p -re˝nementfor(1+2)- Dproblem.Intheleftsub-˝gure, u ext = t 3 + 1 š 2 Î 2 j = 1 ¹ 1 + ˘ j º 4 + 1 š 2 ¹ 1 ˘ j º 4 + 1 š 2 ,forwhich wechoose N = 4 tocontroltheerrorintimeandperform p -re˝nementinspacefordi˙erent valuesoffractionalorders f 2 ; 2 g = f 0 : 5 ; 1 : 1 g and f 2 ; 2 g = f 0 : 5 ; 1 : 9 g .Theresultsshow theexpectedspectralconvergence.Intherightsub-˝gure,weperform p -re˝nementintimefor u ext = t 3 + ˝ Î 2 j = 1 ¹ 1 + ˘ j º 4 ¹ 1 ˘ j º 4 ,where ˝ = 0 : 1 ; 0 : 9 andwechoose M 1 = M 2 = 8 tocontrol theerrorinspace.Thechoiceofploy-fractonomialsasthetemporalbasisenabletheschemeto accuratelycapturethesingularityintime.Theobtainedresultsshowtheconvergenceoferrorto machineprecisionwith N = 4 .Moreover,inTable5.1,weshowtheCPUtime(whichincludes theconstructionofthelinearsystemandloadvector)aswellasthecomputed L 2 -normerrorfor theproblemsof(1+1)-to(1+3)-dimensions,where p 1 = 3 ;˝ = 0 : 5 ; p 2 = p 3 = 4 ; = 0 : 5 ; 2 = 0 : 5 ; 2 = 1 : 5 . 120 Figure5.1:PGspectralmethod,temporalandspatial p -re˝nementfor(1+2)-Dproblem Table5.1:PGspectralmethod,CPUtime(inmin)and L 2 -normerrorformulti-dimensional problems. (1+1)-D(1+2)-D(1+3)-D N = M 1 = M 2 = M 3 L 2 -normErrorCPUTime L 2 -normErrorCPUTime L 2 -normErrorCPUTime 2 6 : 2067 10 1 0.6 5 : 9428 10 1 1 5 : 1307 10 1 1.7 6 2 : 7852 10 2 1 2 : 9233 10 2 1.5 2 : 6720 10 2 4 10 6 : 7506 10 5 3.13 7 : 089 10 5 4.5 6 : 4714 10 5 27.9 14 1 : 7541 10 6 20.3 1 : 8463 10 6 27.5 1 : 6876 10 6 149 5.5ProofofLemmasandTheorems 5.5.1ProofofTheorem (5.3.2) Proof. Let ˝ n ¹ º = ¹ 1 + º ˝ P ˝;˝ n ¹ º , ˝ n ¹ º = ¹ 1 º ˝ P ˝; ˝ n ¹ º ,and u N = Í N n = 1 Í M + 1 m = 0 u n ; m ˝ n ¹ t º P m ¹ x º ,where u N 2 U N .Hence, U ' I = ¹ + 1 1 ¹ T 0 N Õ n = 1 M + 1 Õ m = 0 N Õ k = 1 M + 1 Õ r = 0 u k ; r u n ; m 0 D š 2 t ˝ n ¹ t º 0 D š 2 t ˝ k ¹ t º P m ¹ x º P r ¹ x º dtdx = N Õ n = 1 M + 1 Õ m = 0 N Õ k = 1 M + 1 Õ r = 0 u k ; r u n ; m ¹ + 1 1 P m ¹ x º P r ¹ x º dx | {z } C 0 ; 0 m m ; r ¹ T 2 º 1 2 ˝ 1 ˝ 1 ; ˝ 1 n 1 ˝ 1 ; ˝ 1 k 1 ¹ + 1 1 ¹ 1 + º ˝ 1 P ˝ 1 ;˝ 1 n 1 ¹ º¹ 1 + º ˝ 1 P ˝ 1 ;˝ 1 k 1 ¹ º d ; (5.21) 121 where ˝ 1 ; ˝ 1 n 1 = ˝ 1 ;˝ 1 n 1 = ¹ n + ˝ 1 º ¹ n º and ˝ 1 = ˝ 2 .Take P ˝ 1 ;˝ 1 n ¹ º = Í n q = 0 a ˝ 1 ; n q P 0 ; 2 ˝ 1 q ¹ º then, U ' I = N Õ n = 1 N Õ k = 1 M + 1 Õ m = 0 n Õ q 3 = 1 r Õ q 4 = 1 u k ; m u n ; m C 0 ; 0 m ¹ T 2 º 1 2 ˝ 1 ˝ 1 ; ˝ 1 n 1 ˝ 1 ; ˝ 1 k 1 a ˝ 1 ; n q 3 a ˝ 1 ; r q 4 ¹ + 1 1 ¹ 1 + º 2 ˝ 1 P 0 ; 2 ˝ 1 q 3 ¹ º P 0 ; 2 ˝ 1 q 4 ¹ º d | {z } C 0 ; 2 ˝ 1 q 3 q 3 ; q 4 = M + 1 Õ m = 0 N Õ q 3 = 1 ¹ 1 º u 2 q 3 ; m C 0 ; 2 ˝ 1 q 3 C 0 ; 0 m ¹ T 2 º 1 2 ˝ 1 = M + 1 Õ m = 0 N Õ n = 1 ¹ 1 º u 2 n ; m ¹ T 2 º 1 2 ˝ 1 C 0 ; 2 ˝ 1 n C 0 ; 0 m ; inwhich ¹ 1 º u n ; m = Í M + 1 q q = 0 u q ; m a ˝ 1 ; q n ˝ 1 ; ˝ 1 q 1 .Besides, U 1 II = ¹ + 1 1 ¹ T 0 N Õ n = 1 M + 1 Õ m = 0 N Õ k = 1 M + 1 Õ r = 0 u k ; r u n ; m ˝ n ¹ t º ˝ k ¹ t º 1 D x P m ¹ x º 1 D x P r ¹ x º dtdx = N Õ n = 1 M + 1 Õ m = 0 N Õ k = 1 M + 1 Õ r = 0 u k ; r u n ; m ¹ T 2 º ¹ + 1 1 ¹ 1 + º 2 ˝ P ˝;˝ n 1 ¹ º P ˝;˝ k 1 ¹ º d ¹ + 1 1 ¹ 1 + x º 2 m r P ; m ¹ x º P ; r ¹ x º dx ; (5.22) where m = m + 1 m + 1 .Bysubstituting P ; i ¹ x º = Í i q = 0 b 2 ; i q P 2 ; 0 q ¹ x º and P ˝;˝ n ¹ º = Í n q = 0 a ˝; n q P 0 ; 2 ˝ q ¹ º into(5.22)andreorganizing,weobtain U 1 II = N Õ n = 1 N Õ k = 1 M + 1 Õ m = 0 n Õ q 3 = 1 k Õ q 4 = 1 ¹ 2 º u n ; m ¹ 2 º u k ; m C 2 ; 0 m ¹ T 2 º a ˝; n q 3 a ˝; k q 4 ¹ + 1 1 ¹ 1 + º 2 ˝ P 0 ; 2 ˝ q 3 ¹ º P 0 ; 2 ˝ q 4 ¹ º d | {z } C 0 ; 2 ˝ q 3 q 3 ; q 4 = M + 1 Õ m = 0 N Õ q 3 = 1 ¹ L º u 2 q 3 ; m C 0 ; 2 ˝ q 3 C 2 ; 0 m ¹ T 2 º = M + 1 Õ m = 0 N Õ n = 1 ¹ L º u 2 n ; m ¹ T 2 º C 0 ; 2 ˝ n C 2 ; 0 m ; (5.23) where ¹ 2 º u n ; m = Í M + 1 q q = 0 u q b 2 ; q m q and ¹ L º u n ; m = Í N n q = 1 ¹ 2 º u q ; m a ˝; q n .Let v N = N Õ k = 1 M + 1 Õ n = 0 u k ; r 1 º k + r ˝ k ¹ t º P r ¹ x º : 122 Followingthesamestepsasin U ' I ,forthenormofthetestfunctionwehave V ' I = ¹ + 1 1 ¹ T 0 N Õ n = 1 M + 1 Õ m = 0 N Õ k = 1 M + 1 Õ r = 0 u k ; r u n ; m 1 º n + k t D š 2 T ˝ n ¹ t º t D š 2 T ˝ k ¹ t º P m ¹ x º P r ¹ x 1 º r + m dtdx = M + 1 Õ m = 0 N Õ n = 1 ¹ 1 º v 2 n ; m ¹ T 2 º 1 2 ˝ 1 C 0 ; 2 ˝ 1 n C 0 ; 0 m ; (5.24) inwhichweemploy P ˝ 1 ; ˝ 1 n ¹ º = Í n q = 0 a ˝ 1 ; n q P 2 ˝ 1 ; 0 q ¹ º and ¹ 1 º v n ; m = Í M + 1 q q = 0 u n ; q a ˝ 1 ; q n ˝ 1 ; ˝ 1 q . Besides, V 1 II = ¹ + 1 1 ¹ T 0 N Õ n = 1 M + 1 Õ m = 0 N Õ k = 1 M + 1 Õ r = 0 u k ; r u n ; m 1 º n + k ˝ n ¹ t º ˝ k ¹ t º 1 º r + m 1 D x P m ¹ x º 1 D x P r ¹ x º dtdx = N Õ n = 1 M + 1 Õ m = 0 N Õ k = 1 M + 1 Õ r = 0 u k ; r u n ; m ¹ T 2 1 º n + k ¹ + 1 1 ¹ 1 º 2 ˝ P ˝; ˝ n 1 ¹ º P ˝; ˝ k 1 ¹ º d 1 º m + r ¹ + 1 1 ¹ 1 + x º 2 m r P ; m ¹ x º P ; r ¹ x º dx ; = N Õ n = 1 N Õ k = 1 M + 1 Õ m = 0 n Õ q 3 = 1 r Õ q 4 = 1 ¹ 2 º v n ; m ¹ 2 º v k ; m C 2 ; 0 m ¹ T 2 º a ˝; n q 3 a ˝; r q 4 1 º n + k ¹ + 1 1 ¹ 1 º 2 ˝ P 2 ˝; 0 q 3 ¹ º P 2 ˝; 0 q 4 ¹ º d | {z } C 2 ˝; 0 q 3 q 3 ; q 4 = C 0 ; 2 ˝ q 3 q 3 ; q 4 = M + 1 Õ m = 0 N Õ q 3 = 1 ¹ L º v 2 q 3 ; m C 0 ; 2 ˝ q 3 C 2 ; 0 m ¹ T 2 º = N Õ n = 1 M + 1 Õ m = 0 ¹ L º v 2 n ; m ¹ T 2 º C 0 ; 2 ˝ n C 2 ; 0 m ; (5.25) where ¹ 2 º v n ; m = Í M + 1 m q = 0 1 º q u n ; q b 2 ; q m q , ¹ L º v n ; m = Í N n i = 1 ¹ 2 º v i ; m a ˝; i n 1 º i ,and P ˝; ˝ n ¹ º = Í n q = 0 a ˝; n q P 2 ˝; 0 q ¹ º .Let A ' I = ¹ 0 D š 2 t u N ; t D š 2 T v N º and A II = l ¹ 1 D x u N ; x D 1 u N º : By employing P ˝ 1 ; ˝ 1 n 1 ¹ x º = Í n 1 q = 0 a 2 ˝ 1 ; n q P ˝ 1 ;˝ 1 q ¹ x º and P ˝ 1 ;˝ 1 k 1 ¹ x º = Í k 1 q = 0 1 º q + k a 2 ˝ 1 ; k q P ˝ 1 ;˝ 1 q ¹ x º ,we 123 obtain A ' I = ¹ T 0 ¹ + 1 1 N Õ n = 1 N Õ k = 1 M + 1 Õ m = 0 M + 1 Õ r = 0 u n ; m u k ; r 1 º k 0 D š 2 t ˝ n ¹ t º t D š 2 T ˝ k ¹ t 1 º r P m ¹ x º P r ¹ x º dxdt = N Õ n = 1 N Õ k = 1 M + 1 Õ m = 0 M + 1 Õ r = 0 u n ; m u k ; r 1 º r ¹ + 1 1 P m ¹ x º P r ¹ x º dx | {z } C 0 ; 0 m m ; r 1 º k ¹ T 2 º 1 2 ˝ 1 ˝ 1 ; ˝ 1 n 1 ˝ 1 ; ˝ 1 k 1 ¹ 1 1 ¹ 1 2 º ˝ 1 P ˝ 1 ;˝ 1 n 1 ¹ º P ˝ 1 ; ˝ 1 k 1 ¹ º d = N Õ n = 1 M + 1 Õ m = 0 ¹ 3 º u 2 n ; m 1 º m + k ¹ T 2 º 1 2 ˝ 1 C 0 ; 0 m C ˝ 1 ;˝ 1 n ; (5.26) where ¹ 3 º u n ; m = Í N q = 1 a 2 ˝ 1 ; q n ˝ 1 ; ˝ 1 q 1 u q ; m .Moreover,basedon P ˝; ˝ n 1 ¹ º = Í n 1 q = 0 a 2 ˝; n q P ˝;˝ q ¹ º , P ˝;˝ k 1 ¹ º = Í k 1 q = 0 1 º q + k a 2 ˝; k q P ˝;˝ q ¹ º , P ; i ¹ x º = Í i q = 0 b 2 ; i q P 2 ; 0 q ¹ x º ,and P ; i ¹ x º = Í i q = 0 1 º i + q b 2 ; i q P 2 ; 0 q ¹ x º ,weget A II = ¹ T 0 ¹ + 1 1 N Õ n = 1 N Õ k = 1 M + 1 Õ m = 0 M + 1 Õ r = 0 u n ; m u k ; r ˝ n ¹ t º ˝ k ¹ t 1 º r + k 1 D x P m ¹ x º x D 1 P r ¹ x º dxdt = N Õ n = 1 N Õ k = 1 M + 1 Õ m = 0 n Õ q 3 = 1 r Õ q 4 = 1 ¹ 1 º ~ u n ; m ¹ 1 º ~ u k ; m 1 º m C 2 ; 0 m ¹ T 2 º a 2 ˝; n q 3 a 2 ˝; k q 4 1 º q 4 ¹ + 1 1 ¹ 1 + º 2 ˝ P 0 ; 2 ˝ q 3 ¹ º P 0 ; 2 ˝ q 4 ¹ º d | {z } C 0 ; 2 ˝ q 3 q 3 ; q 4 ; whichcanbesimpli˝edto A II = Í M + 1 m = 0 Í N n = 1 ¹ L º ~ u 2 n ; m 1 º n + m ¹ T 2 º C 0 ; 2 ˝ n C 2 ; 0 m ,where ¹ L º ~ u n ; m = Í N n q 3 = 1 ¹ 1 º ~ u q 3 ; m a ˝; q 3 n and ¹ 1 º ~ u n ; m = Í M + 1 i = 0 u n ; i b 2 ; q .Ontheotherhand,wehave j a ¹ u N ; v N ºj c ¯ 2 1 ' ¹ ºj A ' I j + l j A II j : Tocompare j a ¹ u N ; v N ºj with k u N k B '; 1 ; ; d ¹ º k v N k B '; 1 ; ; d ¹ º , j A ' I j = j N Õ n = 1 M + 1 Õ m = 0 1 º m + k ¹ 3 º u 2 n ; m ¹ ˝ 1 ; ˝ 1 n 1 º 2 C 0 ; 0 m C ˝ 1 ;˝ 1 n ¹ 1 º u 2 n ; m C 0 ; 2 ˝ 1 n C 0 ; 0 m | {z } ¹ 1 º ~ n ; m ¹ T 2 º 1 2 ˝ 1 ¹ 1 º u 2 n ; m C 0 ; 2 ˝ 1 n C 0 ; 0 m j 1 ¹ 1 º ~ U ' I 124 and j A II j = j M + 1 Õ m = 0 N Õ n = 1 1 º n + m ¹ L º ~ u 2 n ; m ¹ T 2 º C ˝;˝ n C ; m ¹ L º u 2 n ; m ¹ T 2 º C 0 ; 2 ˝ n C 2 ; 0 m | {z } ¹ 2 º ~ n ; m ¹ L º u 2 n ; m ¹ T 2 º C 0 ; 2 ˝ n C 2 ; 0 m j 2 ¹ 2 º ~ U II ; where ¹ 2 º ~ = min f ¹ 2 º ~ n ; m g .Besides,wecanhave ¹ º V I = M + 1 Õ m = 0 ¹ 1 º v 2 m ¹ 1 º u 2 m ¹ 1 º u 2 m C 2 ; 0 m ¹ T 2 º C 0 ; 2 ˝ 1 n = M + 1 Õ m = 0 ¹ 1 º m ¹ 1 º u 2 m C 2 ; 0 m ¹ T 2 º C 0 ; 2 ˝ 1 n ¹ 1 º U ' I ; V II = M + 1 Õ m = 0 ¹ R º v 2 n ; m ¹ R º u 2 n ; m ¹ R º u 2 n ; m ¹ T 2 º C 0 ; 2 ˝ n C 2 ; 0 m = M + 1 Õ m = 0 ¹ 2 º n ; m ¹ R º u 2 n ; m C 2 ; 0 m ¹ T 2 º C 0 ; 2 ˝ n ¹ 2 º U II ; where ¹ 1 º = max f ¹ 1 º m g and ¹ 2 º = max f ¹ 2 º n ; m g .Thisresultsin k v N k 2 B '; 1 ; ; d ¹ º max f ¹ 2 º ; ¹ 1 º g | {z } ~ 2 k u N k 2 B '; 1 ; ; d ¹ º : u 2 U , A ' I ,and A II has˝nitevalues,therefore j a ¹ u N ; v N ºj j A ' I j + l j A II jj 1 ¹ 1 º ~ U ' I + 2 ¹ 2 º ~ l U II (5.27) min f 1 ¹ 1 º ~ ; 2 ¹ 2 º ~ l g | {z } ~ k u N k 2 B '; 1 ; ; d ¹ º ~ ~ k u N k B '; 1 ; ; d ¹ º k v N k B '; 1 ; ; d ¹ º ; whichshowsthatdiscrete inf-sup conditionholdsforthetime-dependentfractionaldi˙usionprob- lem. 125 CHAPTER6 FRACTIONALSENSITIVITYEQUATIONMETHOD:APPLICATIONSTO FRACTIONALMODELCONSTRUCTION 6.1Background Theexcellenceoffractionaloperatorinaccuratepredictionofnon-localityandmemorye˙ects istheinherentnon-localnatureofsingularpower-lawkernel,whoseorderisde˝nedasfractional derivativeorder,i.e.fractionalindex.However,thekeychallengesofsuchmodelsaretheexcessive computationalcostinnumericallyintegratingtheconvolutionoperation,andmoreimportantly, introducingfractionalderivativeordersasextramodelparameters,whosevaluesareessentially obtainedfromexperimentaldata.Thesensitivityassessmentoffractionalmodelswithrespect tofractionalindeciscanbuildabridgebetweenexperimentsandmathematicalmodelstogear observabledataviaproperoptimizationtechniques,andthus,systematicallyimprovetheexisting modelsinbothanalysisanddesignapproaches.Weformulateamathematicalframeworkby developinga fractionalsensitivityequationmethod ,whereweinvestigatetheresponsesensitivity offractionaldi˙erentialequationswithrespecttomodelparametersincludingderivativeorders, andfurtherconstructaniterativealgorithminordertoexploittheobtainedsensitivity˝eldin parameterestimation. FractionalSensitivityAnalysis .Sensitivityassessmentapproachesarecommonlycategorizedas, ˝nitedi˙erence,continuumanddiscretederivatives,andcomputationalorautomaticdi˙erentia- tion,wherethesensitivitycoe˚cientsaregenerallyde˝nedaspartialderivativeofcorresponding functions(modeloutput)withrespecttodesign/analysisparametersofinterest.Finitedi˙er- enceschemesusea˝rstorderTaylorseriesexpansiontoapproximatethesensitivitycoe˚cients, whereaccuracydependsstronglyonstepincrement[117,152].Continuumanddiscretederiva- tivetechniqueshowever,di˙erentiatethesystemresponsewithrespecttoparameters,wherethe former,whichisalsoknownassensitivityequationmethod(SEM,see[107,193]andreferences 126 therein),directlycomputesthederivativesandobtainasetof(coupled)adjointcontinuumsensi- tivityequations;whilethelatterperformsdi˙erentiationafterdiscretizationoforiginalequation [155].Automaticdi˙erentiationmethodalsoreferstoadi˙erentiationofthecomputercode 31].Fig.2in[166]providesadescriptiveschematicofthesedi˙erentapproaches.Weextendthe continuumderivativetechniquetodevelopa fractionalsensitivityequationmethod (FSEM)inthe contextoffractionalpartialdi˙erentialequations(FPDEs).Toformulatethesensitivityanalysis framework,welet q beasetofmodelparametersincludingfractionalindicesandobtaintheadjoint fractionalsensitivityequations (FSEs)bytakingthepartialderivativeofFPDEwithrespectto q . Theseadjointequationsintroduceanewfractionaloperator,associatedwiththe logarithmic-power law kernel,whichtobestofourknowledgehasbeenpresentedforthe˝rsttimehereinthecontextof fractionalsensitivityanalysis.ThekeypropertyofderivedFSEsisthattheypreservethestructure oforiginalFPDE.Thus,similardiscretizationschemeandforwardsolvercanbereadilyapplied withaminimalrequiredchanges. ModelConstruction:EstimationofFractionalIndices .Severalnumericalmethodshavebeen developedtosolveinverseproblemofmodelconstructionfromavailableexperimentalobservations orsyntheticdata.Theytypicallyconverttheproblemofmodelparameterestimationintoan optimizationproblem,andthen,formulateasuitableestimatorbyminimizinganobjectivefunction. Thesemethodsarestretchedoverbutnolimitedtoperturbationmethods[173],weightedleast squaresapproach[37,40,82],nonlinearregression[102],andLevenberg-Marquardtmethod [38,64,180,181].Wedevelopabi-levelFSEM-basedparameterestimationmethodinorderto constructfractionalmodels,inasensethatthemethodobtainsmodelcoe˚cientsinonelevel,and thensearchesforestimateoffractionalindicesinthenextlevel.Weformulatetheoptimization problembyde˝ningobjectivefunctionsastwotypesofmodelerrorthatmeasuresthedi˙erence incomputedoutput/inputoffractionalmodelwithtrueoutput/inputinan L 2 -normsense.We furtherformulateagradient-basedminimizer,employingdevelopedFSEM,andproposeatwo- stagesearchalgorithm,namely,coarsegridsearchingandnearbysolution.The˝rststageconstruct acrudemanifoldofmodelerroroveracoarsediscretizationofparameterspacetolocatealocal 127 neighborhoodofminimum,andthesecondstageusesthegradientdecentmethodinorderto convergetotheminimumpoint. DiscretizationScheme .Theiterativenatureofparameterestimatorsinstructsimulationoffrac- tionalmodelateachiterationstepofmodelparameters.Therefore,oneofthemajortasksin computationalmodelconstructionistodevelopnumericalmethodsthatcane˚cientlydiscretize thephysicaldomainandaccuratelysolvethefractionalmodel.Thesensitivityframeworkad- ditionallyraisethecomplicationbyrenderingcoupledsystemsofFPDEandadjointFSEs,and thus,demandingmoreversatileschemes.Inadditiontonumerous˝nitedi˙erencemethods [33,65,103,159,167,172,192,196],recentworkshaveelaboratede˚cientspectralschemes,for discretizingFPDEsinphysicaldomain,seee.g.,[28,39,83,84,99,100,103,136,170].Morere- cently,Zayernourietal.[183,186]developedtwonewspectraltheoriesonfractionalandtempered fractionalSturm-Liouvilleproblems,andintroducedexplicitcorrespondingeigenfunctions,namely Jacobipoly-fractonomialsof˝rstandsecondkind.Theseeignefunctionsarecomprisedofsmooth andfractionalparts,wherethelattercanbetunnedtocapturesingularitiesoftruesolution.Theyare successfullyemployedinconstructingdiscretesolution/testfunctionspacesanddevelopingaseries ofhigh-orderande˚cientPetrov-Galerkinspectralmethods,see106,160].We formulateanumericalschemeinsolvingcoupledsystemofFPDEandadjointFSEsbyextending themathematicalframeworkin[145]andaccommodatingextrarequiredregularityintheunderly- ingfunctionspaces.WeemployJacobipoly-fractonomialsandLegendrepolynomialsastemporal andspatialbasis/testfunctions,respectively,todevelopaPetrov-Galerkin(PG)spectralmethod. Thesmartchoiceofcoe˚cientsinspatialbasis/testfunctionsyieldssymmetricpropertyinthe resultingmass/sti˙nessmatrices,whichisthenexploitedtoformulateafastsolver.Following similarprocedureasin[145],wealsoshowthatthecoupledsystemismathematicallywell-posed, andtheproposednumericalschemeisstable. Therestofchapterisorganizedasfollows.Insection6.2,werecallsomepreliminaryde˝nitions infractionalcalculus,de˝netheproblemusingfractionalmodels,introducepropersolution/test spaces,providesomeusefullemmas,andthenobtaintheweakformoftheproblem.Weconstruct 128 aPetrov-Galerkinspectralnumericalschemeinsection6.3andcarryoutdiscretestabilityanalysis. WedevelopFSEMinsection6.4forFIVP/FPDEandde˝netheunderlyingmathematicalframe workforthecoupledsystemofFPDEandFSEs.Moreover,wedeveloptheFSEMbasedmodel constructionalgorithminsection6.5and˝nally,providethenumericalresultsinsection6.6. 6.2De˝nitions De˝nition6.2.1. Wede˝nethefollowingleft-andright-sidedintegro-di˙erentialoperatorwith logarithmic-powerlawkernel,namely Log-Powintegro-di˙erentialoperator ,givenas, RL LP a D ˙ x u ¹ x º = 1 ¹ n ˙ º d n dx n ¹ x a log ¹ x s º u ¹ s º ¹ x s º ˙ n + 1 ds ; (6.1) RL LP x D ˙ b u ¹ x º = 1 ¹ n ˙ º d º n dx n ¹ b x log ¹ s x º u ¹ s º ¹ s x º ˙ n + 1 ds ; (6.2) C LP a D ˙ x u ¹ x º = 1 ¹ n ˙ º ¹ x a log ¹ x s º u ¹ n º ¹ s º ¹ x s º ˙ n + 1 ds ; (6.3) C LP x D ˙ b u ¹ x º = 1 ¹ n ˙ º ¹ b x log ¹ s x º u ¹ n º ¹ s º ¹ s x º ˙ n + 1 ds ; (6.4) where RL LP and C LP standforLog-Powintegro-di˙erentialoperator,whichpartiallyresemble thefractionalderivativeinRiemann-LiouvilleandCaputosense,respectively.Thefollowinglemma showsausefulrelationbetweenthetwoaforementionedoperators. Lemma6.2.2. Let x 2» a ; b ¼ .Then,thefollowingrelationholds. PartA: ˙ 2¹ 0 ; 1 º RL LP a D ˙ x u ¹ x º = u ¹ a º ¹ 1 ˙ º log ¹ x a º ¹ x a º ˙ + C LP a D ˙ x u ¹ x º : (6.5) PartB: ˙ 2¹ 1 ; 2 º RL LP a D ˙ x u ¹ x º = u ¹ a º ¹ 2 ˙ º 1 + ¹ 1 ˙ º log ¹ x a º ¹ x a º ˙ + u 0 ¹ a º ¹ 2 ˙ º log ¹ x a º ¹ x a º ˙ 1 + C LP a D ˙ x u ¹ x º : (6.6) Proof. Seesection6.7.1forproof. 129 6.2.1ProblemDe˝nition Let = ¹ 0 ; T ¼¹ a 1 ; b 1 º¹ a 2 ; b 2 º¹ a d ; b d º bethecomputationaldomainforsomepositive integer d .Wede˝ne u ¹ t ; x ; q º : Q ! R ,where q = f ; 1 ; 2 ; ; d ; k 1 ; k 2 ; ; k d g isthevectorofmodelparameterscontainingthefractionalindicesandmodelcoe˚cients,and Q = » 0 ; 1 ¼» 1 ; 2 ¼ d R d + isthespaceofparameters.Thus,forany q 2 Q ,thetransport˝eld u ¹ t ; x ; q º : ! R .WeconsidertheFPDEofstrongform L q ¹ u º = f ,subjecttoDirichletinitial andboundaryconditions,where L isalineartwo-sidedfractionaloperator,givenasfollows 0 D t u ¹ t ; x ; q º d Õ j = 1 k j a j D j x j + x j D j b j u ¹ t ; x ; q º = f ¹ t ; x ; q º ; (6.7) u j t = 0 = 0 ; (6.8) u j x = a j = u j x = b j = 0 ; (6.9) inwhich 2¹ 0 ; 1 º , j 2¹ 1 ; 2 º , k j arerealpositiveconstantcoe˚cients,andthefractional derivativesaretakenintheRiemann-Liouvillesense.Intherestofthischapter,wedropthe pre-superscript RL forthesakeofsimplicityandabbreviation;westatethetypeofderivativeif needbe. 6.2.2MathematicalFramework:FractionalSobolevSpaces Wede˝nesomefunctionalspacesandtheirassociatednorms[87,99].By H ˙ ¹ R º = u ¹ t ºj u 2 L 2 ¹ R º ; ¹ 1 + j ! j 2 º ˙ 2 F¹ u º¹ ! º2 L 2 ¹ R º , ˙ 0 ,wedenotethefractionalSobolevspaceon R , endowedwithnorm k u k H ˙ R = k¹ 1 + j ! j 2 º ˙ 2 F¹ u º¹ ! ºk L 2 ¹ R º ,where F¹ u º representstheFourier transformof u .Subsequently,wedenoteby H ˙ ¹ º = u 2 L 2 ¹ ºj9 ~ u 2 H ˙ ¹ R º s : t : ~ u j = u , ˙ 0 ,thefractionalSobolevspaceonany˝niteclosedinterval,e.g. = ¹ a ; b º ,withnorm 130 k u k H ˙ ¹ º = inf ~ u 2 H ˙ R ; ~ u j = u k ~ u k H ˙ ¹ R º .Wede˝nethefollowingusefulnormsas: kk l H ˙ ¹ º = k a D ˙ x ¹ºk 2 L 2 ¹ º + kk 2 L 2 ¹ º 1 2 ; kk r H ˙ ¹ º = k x D ˙ b ¹ºk 2 L 2 ¹ º + kk 2 L 2 ¹ º 1 2 ; kk c H ˙ ¹ º = k x D ˙ b ¹ºk 2 L 2 ¹ º + k a D ˙ x ¹ºk 2 L 2 ¹ º + kk 2 L 2 ¹ º 1 2 ; wheretheequivalenceof kk l H ˙ ¹ º and kk r H ˙ ¹ º areshownin[53,99,100].Weshowthe equivalenceofthesetwonormswith kk c H ˙ ¹ º inthefollowinglemma. Lemma6.2.3. Let ˙ 0 and ˙ , n 1 2 .Then,thenorms kk l H ˙ ¹ º and kk r H ˙ ¹ º areequivalent to kk c H ˙ ¹ º . Proof. Seesection6.7.2forproof. Wealsode˝ne C 1 0 ¹ º asthespaceofsmoothfunctionswithcompactsupportin ¹ a ; b º .We denoteby l H ˙ 0 ¹ º , r H ˙ 0 ¹ º ,and c H ˙ 0 ¹ º astheclosureof C 1 0 ¹ º withrespecttothenorms kk l H ˙ ¹ º , kk r H ˙ ¹ º ,and kk c H ˙ ¹ º . Lemma6.2.4 ([53,100]) . TheSobolevspaces l H ˙ 0 ¹ º , r H ˙ 0 ¹ º ,and c H ˙ 0 ¹ º areequalandtheir seminormsareequivalentto jj H ˙ ¹ º = a D ˙ x ¹º ; x D ˙ b ¹º 1 2 BasedonLemma6.2.4,andassumingthat ¹ a D ˙ x u ; x D ˙ b v º > 0 and ¹ x D ˙ b u ; a D ˙ x v º > 0 ,wecanprovethat ¹ a D ˙ x u ; x D ˙ b v º 1 j u j l H ˙ ¹ º j v j r H ˙ ¹ º and ¹ x D ˙ b u ; a D ˙ x v º 2 j u j r H ˙ ¹ º j v j l H ˙ ¹ º ,where 1 and 2 arepositiveconstants.Following[145],wede˝ne thecorrespondingsolutionandtestspacesofourproblem.Thus,byletting 1 = ¹ a 1 ; b 1 º , j = ¹ a j ; b j º j 1 for j = 2 ; ; d ,wede˝ne X 1 = H 1 2 0 ¹ 1 º ,whichisassociatedwiththenorm 131 kk c H 1 2 ¹ 1 º ,andaccordingly, X j ; j = 2 ; ; d as X 2 = H 2 2 0 ¹ a 2 ; b 2 º ; L 2 ¹ 1 º \ L 2 ¹¹ a 2 ; b 2 º ; X 1 º ; (6.10) : : : X d = H d 2 0 ¹ a d ; b d º ; L 2 ¹ d 1 º \ L 2 ¹¹ a d ; b d º ; X d 1 º ; (6.11) associatedwithnorms kk X j = ˆ kk 2 H j 2 0 ¹ a j ; b j º ; L 2 ¹ j 1 º + kk 2 L 2 ¹ a j ; b j º ; X j 1 ˙ 1 2 ; j = 2 ; 3 ; ; d . Lemma6.2.5. Let j 0 and j , n 1 2 for j = 1 ; 2 ; ; d .Then, kk 2 X j j Õ i = 1 k x i D i š 2 b i ¹ºk 2 L 2 ¹ j º + k a i D i š 2 x i ¹ºk 2 L 2 ¹ j º + kk 2 L 2 ¹ j º : Proof. Seesection6.7.3forproof. Moreover,byletting 0 C 1 ¹ I º and C 1 0 ¹ I º bethespaceofsmoothfunctionswithcompactsupport in ¹ 0 ; T ¼ and » 0 ; T º ,respectively,wede˝ne l H s ¹ I º and r H s ¹ I º astheclosureof 0 C 1 ¹ I º and C 1 0 ¹ I º withrespecttothenorms kk l H s ¹ I º and kk r H s ¹ I º .Wealsode˝ne l 0 H 2 I ; L 2 ¹ d º = n u k u ¹ t ; ºk L 2 ¹ d º 2 H 2 ¹ I º ; u j t = 0 = u j x = a j = u j x = b j = 0 ; j = 1 ; 2 ; ; d o ; r 0 H 2 I ; L 2 ¹ d º = n v k v ¹ t ; ºk L 2 ¹ d º 2 H 2 ¹ I º ; v j t = T = v j x = a j = v j x = b j = 0 ; j = 1 ; 2 ; ; d o ; equippedwithnorms k u k l H 2 ¹ I ; L 2 ¹ d ºº and k u k r H 2 ¹ I ; L 2 ¹ d ºº ,respectively,whichtakethefollow- ingforms k u k l H 2 ¹ I ; L 2 ¹ d ºº = k u ¹ t ; ºk L 2 ¹ d º l H 2 ¹ I º = k 0 D 2 t ¹ u ºk 2 L 2 ¹ º + k u k 2 L 2 ¹ º 1 2 ; (6.12) k u k r H 2 ¹ I ; L 2 ¹ d ºº = k u ¹ t ; ºk L 2 ¹ d º r H 2 ¹ I º = k t D 2 T ¹ u ºk 2 L 2 ¹ º + k u k 2 L 2 ¹ º 1 2 : (6.13) SolutionandTestSpaces .Wede˝nethesolutionspace U andtestspace V ,respectively,as U = l 0 H 2 I ; L 2 ¹ d º \ L 2 ¹ I ; X d º ; V = r 0 H 2 I ; L 2 ¹ d º \ L 2 ¹ I ; X d º ; (6.14) 132 endowedwithnorms k u k U = n k u k 2 l H 2 ¹ I ; L 2 ¹ d ºº + k u k 2 L 2 ¹ I ; X d º o 1 2 ; k v k V = n k v k 2 r H 2 ¹ I ; L 2 ¹ d ºº + k v k 2 L 2 ¹ I ; X d º o 1 2 ; (6.15) UsingLemma7.3.2,wecanshowthat k u k L 2 ¹ I ; X d º = k u ¹ t ;: ºk X d L 2 ¹ I º = n k u k 2 L 2 ¹ º + d Õ j = 1 k x j D j 2 b j u k 2 L 2 ¹ º + k a j D j 2 x j u k 2 L 2 ¹ º o 1 2 : (6.16) Therefore,by(7.24)wewrite(7.23)as k u k U = n k u k 2 L 2 ¹ º + k 0 D 2 t u k 2 L 2 ¹ º + d Õ j = 1 k x j D j 2 b j u k 2 L 2 ¹ º + k a j D j 2 x j u k 2 L 2 ¹ º o 1 2 ; (6.17) k v k V = n k v k 2 L 2 ¹ º + k t D 2 T v k 2 L 2 ¹ º + d Õ j = 1 k x j D j 2 b j v k 2 L 2 ¹ º + k a j D j 2 x j v k 2 L 2 ¹ º o 1 2 : (6.18) Thefollowinglemmashelpusobtaintheweakformulationofourproblem,constructthe numericalschemeandfurtherprovethestabilityofourmethod. Lemma6.2.6 ([99]) . Forall 2¹ 0 ; 1 º ,if u 2 H 1 ¹» 0 ; T ¼º suchthat u ¹ 0 º = 0 ,and v 2 H š 2 ¹» 0 ; T ¼º , then ¹ 0 D t u ; v º = ¹ 0 D š 2 t u ; t D š 2 T v º ,where ¹ ; º representsthestandardinnerproductin = » 0 ; T ¼ . Lemma6.2.7 ([87]) . Let 1 << 2 , a and b bearbitrary˝niteorin˝niterealnumbers.Assume u 2 H ¹ a ; b º suchthat u ¹ a º = 0 ,also x D š 2 b v isintegrablein = ¹ a ; b º suchthat v ¹ b º = 0 .Then, ¹ a D x u ; v º = ¹ a D š 2 x u ; x D š 2 b v º . WegeneralizeLemma7.3.4tothetwo-sided ¹ 1 + d º -dimensionalcase(seesection6.7.4for proof). Lemma6.2.8. Let 1 < j < 2 for j = 1 ; 2 ; ; d ,and u ; v 2X d .Then, a j D j x j u ; v d = a j D j 2 x j u ; x j D j 2 b j v d ; x j D j b j u ; v d = x j D j 2 b j u ; a j D j 2 x j v d : 133 6.2.3WeakFormulation Foranysetofmodelparameter q ,weobtaintheweaksystem,i.e.thevariationalformofthe problem(6.7)subjecttothegiveninitial/boundaryconditions,bymultiplyingtheequationwith propertestfunctionsandintegrateoverthewholecomputationaldomain .Therefore,using Lemmas7.3.3-7.3.5,thebilinearformcanbewrittenas a ¹ u ; v º = ¹ 0 D 2 t u ; t D 2 T v º d Õ j = 1 k j h ¹ a j D j 2 x j u ; x j D j 2 b j v º + ¹ x j D j 2 b j u ; a j D j 2 x j v º i ; (6.19) andthus,byletting ~ U and ~ V bethepropersolution/testspaces,theproblemreadsas:˝nd u 2 ~ U suchthat a ¹ u ; v º = ¹ f ; v º ; 8 v 2 ~ V : (6.20) 6.3Petrov-GalerkinSpectralMethod Wede˝nethefollowing˝nitedimensionalsolutionandtestspaces.WeemployLegendre polynomials ˚ m j ¹ ˘ º ; j = 1 ; 2 ; ; d ,andJacobipoly-fractonomialof˝rstkind ˝ n ¹ º [183,186], asthespatialandtemporalbases,respectively,givenintheircorrespondingstandarddomainas ˚ m j ¹ ˘ º = ˙ m j P m j + 1 ¹ ˘ º P m j 1 ¹ ˘ º ;˘ 2 1 ; 1 ¼ m j = 1 ; 2 ; ; (6.21) ˝ n ¹ º = ˙ n ¹ 1 º P ˝ n ¹ º = ˙ n ¹ 1 + º ˝ P ˝;˝ n 1 ¹ º ; 2 1 ; 1 ¼ n = 1 ; 2 ; ; (6.22) inwhich ˙ m j = 2 + 1 º m j .Therefore,byperforminga˚nemappings = 2 t T 1 and ˘ = 2 x a j b j a j 1 fromthecomputationaldomaintothestandarddomain,weconstructthesolutionspace U N as U N = span n ˝ n ¹ t º d Ö j = 1 ˚ m j ˘ ¹ x j º : n = 1 ; 2 ; ; N ; m j = 1 ; 2 ; ; M j o : (6.23) Wenotethatthechoiceoftemporalandspatialbasisfunctionsnaturallysatisfytheinitialand boundaryconditions,respectively.Theparameter ˝ inthetemporalbasisfunctionsplaysaroleof ˝netunningparameter,whichcanbechosenproperlytocapturethesingularityofexactsolution. 134 Moreover,weemployLegendrepolynomials r j ¹ ˘ º ; j = 1 ; 2 ; ; d ,andJacobipoly-fractonomial ofsecondkind ˝ k ¹ º ,asthespatialandtemporaltestfunctions,respectively,givenintheircorre- spondingstandarddomainas r j ¹ ˘ º = e ˙ r j P r j + 1 ¹ ˘ º P r j 1 ¹ ˘ º ;˘ 2 1 ; 1 ¼ r j = 1 ; 2 ; ; (6.24) ˝ k ¹ º = e ˙ k ¹ 2 º P ˝ k ¹ º = e ˙ k ¹ 1 º ˝ P ˝; ˝ k 1 ¹ º ; 2 1 ; 1 ¼ k = 1 ; 2 ; ; (6.25) where e ˙ r j = 2 1 º r j + 1 .Therefore,bysimilara˚nemappingweconstructthetestspace V N as V N = span n ˝ k ¹ t º d Ö j = 1 r j ˘ j ¹ x j º : k = 1 ; 2 ; ; N ; r j = 1 ; 2 ; ; M j o : (6.26) Wecanshowthatourchoiceofbasis/testfunctionssatisfytheextraregularityimposedbythe Log-Powintegro-di˙erentialoperator.Thus,since U N ˆ ~ U ˆ U and V N ˆ ~ V ˆ V ,theproblems (6.59)and(6.60)readas:˝nd u N 2 U N suchthat a h ¹ u N ; v N º = l ¹ v N º ; 8 v N 2 V N ; (6.27) where l ¹ v N º = ¹ f ; v N º ;and˝nd Su N 2 U N suchthat a h ¹ Su N ; w N º = l ¹ w N º ; 8 w N 2 V N ; (6.28) where l ¹ w N º = ¹ f q i ; w N º .Also,thediscretebilinearform a h ¹ u N ; v N º canbewrittenas a h ¹ u N ; v N º = ¹ 0 D 2 t u N ; t D 2 T v N º (6.29) d Õ j = 1 k j h ¹ a j D j 2 x j u N ; x j D j 2 b j v N º + ¹ x j D j 2 b j u N ; a j D j 2 x j v N º i : Weexpandtheapproximatesolution u N 2 U N ,satisfyingthediscretebilinearform(7.37),inthe followingform u N ¹ t ; x º = N Õ n = 1 M 1 Õ m 1 = 1 M d Õ m d = 1 ^ u n ; m 1 ; ; m d h ˝ n ¹ t º d Ö j = 1 ˚ m j ¹ x j º i ; (6.30) 135 andobtainthecorrespondingLyapunovsystembysubstituting(7.38)into(7.37)bychoosing v N ¹ t ; x º = ˝ k ¹ t º Î d j = 1 r j ¹ x j º , k = 1 ; 2 ;:::; N , r j = 1 ; 2 ;:::; M j .Therefore, h S T M 1 M 2 M d + d Õ j = 1 M T M 1 M j 1 S j M j + 1 M d i U = F ; (6.31) inwhich representstheKroneckerproduct, F denotesthemulti-dimensionalloadmatrixwhose entriesaregivenas F k ; r 1 ; ; r d = ¹ f ¹ t ; x º ˝ k ¹ t º d Ö j = 1 r j ˘ j ¹ x j º d ; (6.32) and U isthematrixofunknowncoe˚cients.Thematrices S T and M T denotethetemporalsti˙ness andmassmatrices,respectively;andthematrices S j and M j denotethespatialsti˙nessandmass matrices,respectively.Weobtaintheentriesofspatialmassmatrix M j analyticallyandemploy properquadraturerulestoaccuratelycomputetheentriesofothermatrices S T , M T and S j . Wenotethatthechoicesofbasis/testfunctions,employedindevelopingthePGschemeleadsto symmetricmassandsti˙nessmatrices,providingusefulpropertiestofurtherdevelopafastsolver. ThefollowingTheorem7.3.6providesauni˝edfastsolver,developedintermsofthegeneralized eigensolutionsinordertoobtainaclosed-formsolutiontotheLyapunovsystem(7.39). Theorem6.3.1 (Uni˝edFastFPDESolver[143,145]) . Let f ® e m j ; m j g M j m j = 1 bethesetofgeneral eigen-solutionsofthespatialsti˙nessmatrix S j withrespecttothemassmatrix M j .Moreover,let f ® e ˝ n ; ˝ n g N n = 1 bethesetofgeneraleigen-solutionsofthetemporalmassmatrix M T withrespectto thesti˙nessmatrix S T .Then,thematrixofunknowncoe˚cients U isexplicitlyobtainedas U = N Õ n = 1 M 1 Õ m 1 = 1 M d Õ m d = 1 n ; m 1 ; ; m d ® e ˝ n ® e m 1 ® e m d ; (6.33) where n ; m 1 ; ; m d isgivenby n ; m 1 ; ; m d = ¹ ® e ˝ n ® e m 1 ® e m d º F h ¹ ® e ˝ T n S T ® e ˝ n º Î d j = 1 ¹ ® e T m j M j ® e m j º i n ; m 1 ; ; m d ; (6.34) 136 inwhichthenumeratorrepresentsthestandardmulti-dimensionalinnerproduct,and n ; m 1 ; ; m d isobtainedintermsoftheeigenvaluesofallmassmatricesas n ; m 1 ; ; m d = h ¹ 1 + ˝ n º + ˝ n Í d j = 1 ¹ m j º i : 6.3.1StabilityAnalysis Weshowthewell-posednessofde˝nedproblemandprovethestabilityofproposednumerical scheme. Lemma6.3.2. Let 2¹ 0 ; 1 º , = I d ,and u 2 l 0 H š 2 ¹ I ; L 2 ¹ d ºº .Then, 0 D š 2 t u ; t D š 2 T v k u k l H š 2 ¹ I ; L 2 ¹ d ºº k v k r H š 2 ¹ I ; L 2 ¹ d ºº ; 8 v 2 r 0 H š 2 ¹ I ; L 2 ¹ d ºº : Proof. Seesection6.7.5forproof. Byequivalenceoffunctionspaces l H ˙ 0 ¹ º , r H ˙ 0 ¹ º ,and c H ˙ 0 ¹ º andalsotheirassociated norms kk l H ˙ ¹ º , kk r H ˙ ¹ º ,and kk c H ˙ ¹ º ;andalsobyfollowingsimilarstepsasinLemma 7.3.7,wecanalsoprovethat j a d D d š 2 x d u ; x d D d š 2 b d v d jj u j c H d š 2 ¹ a d ; b d º ; L 2 ¹ d 1 º j v j c H d š 2 ¹ a d ; b d º ; L 2 ¹ d 1 º ; (6.35) j x d D d š 2 b d u ; a d D d š 2 x d v d jj u j c H d š 2 ¹ a d ; b d º ; L 2 ¹ d 1 º j v j c H d š 2 ¹ a d ; b d º ; L 2 ¹ d 1 º : (6.36) Lemma6.3.3 (Continuity) . Thebilinearform (7.28) iscontinuous,i.e., 8 u 2 U ; 9 > 0 ; s.t. j a ¹ u ; v ºj k u k U k v k V ; 8 v 2 V : (6.37) Proof. Theproofdirectlyconcludesfrom(7.43),(7.44)andLemma7.3.7. Theorem6.3.4 (Stability) . Thefollowinginf-supconditionholdsforthebilinearform (7.28) ,i.e., inf 0 , u 2 U sup 0 , v 2 V j a ¹ u ; v ºj k v k V k u k U > 0 ; (6.38) where = I d and sup u 2 U j a ¹ u ; v ºj > 0 . 137 Proof. Seesection6.7.6forproof. Theorem6.3.5 (well-posedness) . Forall 0 << 1 , , 1 ,and 1 < j < 2 ,and j = 1 ; ; d , thereexistsauniquesolutionto (7.29) ,continuouslydependenton f ,whichbelongstothedual spaceof U . Proof. Lemmas7.3.8(continuity)and7.3.9(stability)yieldthewell-posednessofweakform(7.29) in(1+d)-dimensionduetothegeneralizedBabu²ka-Lax-Milgramtheorem. Sincethede˝nedbasisandtestspacesareHilbertspaces,and U N ˆ U and V N ˆ V ,wecan provethatthedevelopedPetrov-Gelerkinspectralmethodisstableandthefollowingcondition holds inf 0 , u N 2 U N sup 0 , v 2 V N j a ¹ u N ; v N ºj k v N k V k u N k U > 0 ; (6.39) with > 0 andindependentof N ,where sup u N 2 U N j a ¹ u N ; v N ºj > 0 ; 8 v N 2 V N . WerecallagainherethattheadjointFSEshavesimilarbilinearform;andsince ~ U ˆ U and ~ V ˆ V ,theobtainedresultsarealsoapplicabletothem. 6.4FractionalSensitivityEquationMethod(FSEM) Wede˝nethesensitivitycoe˚cientsasthepartialderivativeoftransport˝eld u withrespect tothemodelparameters q i ,i.e. S u ; q i = @ u @ q i ; i = 1 ; 2 ; ; 2 d + 1 ; (6.40) assumingthatthepartialderivativeiswell-de˝ned.Toobtainthegoverningequationofevolution ofsensitivity˝elds,i.e.FSEs,we˝rsttakethepartialderivativeofleft-andright-sidedfractional derivative(1.16)and(1.17)withrespecttotheirorders.Therefore,byletting ˙ 2¹ n 1 ; n ¼ , x 2» a ; b ¼ , A n ¹ ˙ º = ¹ n ˙ º @ @˙ 1 ¹ n ˙ º ,wehave @ @˙ ¹ a D ˙ x u º = a D ˙ x S u ;˙ + A n ¹ ˙ º a D ˙ x u LP a D ˙ x u ; (6.41) @ @˙ ¹ x D ˙ b u º = x D ˙ b S u ;˙ + A n ¹ ˙ º x D ˙ b u LP x D ˙ b u : (6.42) 138 Figure6.1:SchematicofstrategiesinderivingtheweakformofFSEs.(I-1):˝rsttake @ @ q and thenobtaintheweakformulation,fedbystrongsolution u s .(I-2):˝rsttake @ @ q andthenobtainthe weakformulation,fedbyweaksolution u w .(II):˝rstobtaintheweakformulationandthentake @ @ q ,fedbyweaksolution u w . Thepre-superscriptLPstandsforthe Log-Powintegro-di˙erentialoperator ,givenin(6.1)-(6.4), whichweintroducehere,forthe˝rsttimeinthecontextofFSEs. Remark6.4.1. Inthesequel,weonlyusethe RL LP operatorandthus,forthesakeofsimplicity, wedropthepre-superscript RL and C andonlyusethemwhenitisnecessarytodistinguish betweenthetwosensesofderivatives. WederivetheadjointFSEsbypursuingtwodi˙erentstrategiesIandII,shownschematically inFig.6.1.Weadoptthenotationof u s and u w todistinguishthesolutiontostrongandweakform oftheproblemforeaseofdescribingthetwofollowingstrategies.Inthe˝rststrategy,we˝rsttake thepartialderivativeofFPDEwithrespecttothemodelparameters q ,andthen,obtaintheweak formofproblem.If u s isknown,thenwefollowI-1(left˝gure),otherwiseweformulateandsolve theweakformofFPDEtoobtainweaksolution u w andfollowI-2(middle˝gure). I-1: L q ¹ u s º = f @ @ q ! L q ¹ S s u s ; q º = f q ¹ u s º weakform ! a ¹ S w u s ; q ; v º = ¹ f q ¹ u s º ; v º (6.43) I-2: L q ¹ u s º = f @ @ q ! L q ¹ S s u s ; q º = f q ¹ u s º weakform ! a ¹ ~ S w u s ; q ; v º = ¹ f q ¹ u w º ; v º (6.44) Viaproperconstructionofthecorrespondingsubspaces,wediscretizeandsolve a ¹ S w u s ; q ; v º = ¹ f q ¹ u s º ; v º and a ¹ ~ S w u s ; q ; v º = ¹ f q ¹ u w º ; v º inI-1andI-2,respectively.Wecanshowthat k ~ S w u s ; q 139 S w u s ; q k L 2 ! 0 as u w ! u s bystability/erroranalysisofemployednumericalscheme,wherethe solutionspacehastheextraregularityrequiredbytheLog-Powintegro-di˙erentialoperatorinf q . Remark6.4.2. ThesolutiontostrongformofFPDE,i.e. u s canbeanalytically/numerically computed(byLaplacetransformand˝nitedi˙erencemethodforexample),ormaybeavailableas priorexperimentaldata,andthus,canbefeddirectlytoconstruct f q inFSEs(seeleftsub-˝gurein Fig.6.1).Thisisusedinparameterestimationformodelconstruction,section6.5. Inthesecondstrategy,we˝rstobtaintheweakformofFPDE,andthentakethepartialderivative withrespecttothemodelparameters q .Inthiscase,weprocure ¹ h ¹ u w º ; v º astherighthandsideof weakformulation,whichisfedbytheweaksolution u w .Inthiscase,thefunction h requiresless regularityforthesolutionspaceduetotheLog-Powintegro-di˙erentialoperator,sincetheorder ofkernelislesscomparetothe˝rststrategy. II: L q ¹ u s º = f weakform ! a ¹ u w ; v º = ¹ f ; v º @ @ q ! a ¹ S w u w ; q ; v º = ¹ h ¹ u w º ; v º (6.45) Inthenextsubsection,weadoptthetwostrategiestoderiveadjointFSEtoafractionalinitial valueproblem,whereweshowthecorrespondingright-hand-sideandtheimposedextraregularity ineachcase.Wethen,extendthederivationtothecaseFPDE,inwhichweadoptstrategyI-2. 6.4.1FSEM(FIVP) Let = ¹ 0 ; T ¼ bethecomputationaltimedomainandde˝ne u ¹ t ; º : ¹ 0 ; 1 º! R .Weconsider thecaseoffractionalinitialvalueproblem(FIVP)bylettingthecoe˚cients k j 'stobezeroin(6.7), andthusobtainthefollowingFIVP,subjecttoDirichletinitialcondition,as 0 D t u ¹ t ; º = f ¹ t ; º , u ¹ 0 º = 0 .Bytakingthepartialderivativewithrespectto ,weobtaintheadjointFSEinthestrong formas 0 D t S u ; = f , S u ; q j ¹ t = 0 º = 0 ,wheref = S f ; A 1 ¹ º 0 D t u + LP 0 D t u .Following strategyI,weobtain a ¹ S u ; ; v º = ¹ f ; v º ; (6.46) ¹ f ; v º = ¹ S f ; ; v º A 1 ¹ º¹ 0 D 2 t u ; t D 2 T v º + ¹ LP 0 D t u ; v º : (6.47) 140 Inthiscase,constructingtheright-hand-sideimposesextrastrongregularityof k LP 0 D t u k L 2 < 1 tothesolutionofFIVP.However,byfollowingstartegyII,weobtain a ¹ S u ; ; v º = h ¹ v º ; (6.48) h ¹ v º = ¹ S f ; ; v º + ¹ f ; S v ; º A 1 ¹ 2 º¹ 0 D 2 t u ; t D 2 T v º ¹ 0 D 2 t u ; t D 2 T S v ; º (6.49) + 1 2 ¹ LP 0 D 2 t u ; t D 2 T v º + 1 2 ¹ 0 D 2 t u ; LP t D 2 T v º ; where,thefunction h imposesextraweakregularityof k LP 0 D 2 t u k L 2 < 1 and k LP t D 2 T v k L 2 < 1 tothesolution.Wecomputationallystudyandmakesurethatthesolutionto(6.46)convergesto (6.48). 6.4.2FSEM(FPDE) Weconsidertheproblem(6.7)-(7.3).WeadoptstrategyI-2andderivetheadjointFSEsandtheir correspondingweakform,wheretoconstructtheright-hand-side,wealsoobtaintheweakformof FPDE.Thus,wesolveacoupledsystemofFPDEandFSEs.Bytakingthepartialderivativesof (6.7)withrespecttomodelparameters q i ; i = 1 ; 2 ; ; 2 d + 1 ,weobtainthecorrespondingadjoint FSEsas L q S u ; = f ; L q S u ; j = f j ; L q S u ; k j = f k j ; j = 1 ; 2 ; ; d ; (6.50) inwhich L q ¹º = 0 D t ¹º d Õ j = 1 k j a j D j x j + x j D j b j ¹º (6.51) f = S f ; A 1 ¹ º 0 D t u + LP 0 D t u (6.52) f j = S f ; j + k j A 2 ¹ j º a k D j x j + x j D j b k u k j LP a k D j x j + LP x j D j b k u ; (6.53) f k j = S f ; k j + a j D j x j + x j D j b j u : (6.54) Moreover,bytakingthepartialderivativeofinitialandboundaryconditions(7.2)and(7.3),respec- tively,withrespecttomodelparameters,weobtainthefollowingconditionsfor i = 1 ; 2 ; ; 2 d + 1 , 141 as S u ; q i t = 0 = @ S u ; q i @ t t = 0 = 0 ; S u ; q i x = a j = S u ; q i x = b j = 0 ; j = 1 ; 2 ; ; d : (6.55) 6.4.3MathematicalFramework:CoupledSystemofTheFPDEandDerivedFSEs Weextendthesolution/testspaces,de˝nedin(7.22)byimposingthextraregularities"dueto theright-hand-sideofadjointFSEs(6.50),andde˝netheproperunderlyingspacesforsolvingthe coupledsystemofadjointFSEsandFPDE. Solution/TestSpaces .Let H j 2 0 ¹ j º = n u 2 H j 2 0 ¹ j º s k LP a j D j x j u k 2 L 2 ¹ j º + k LP x j D j b j u k 2 L 2 ¹ j º < 1 o ; j = 1 ; 2 ; ; d ; associatedwiththenorm kk c H j 2 ¹ j º .Wede˝ne X 1 = H 1 2 0 ¹ 1 º ,andaccordingly, X j ; j = 2 ; ; d as X 2 = H 2 2 0 ¹¹ a 2 ; b 2 º ; L 2 ¹ 1 ºº\ L 2 ¹¹ a 2 ; b 2 º ; X 1 º ; (6.56) : : : X d = H d 2 0 ¹¹ a d ; b d º ; L 2 ¹ d 1 ºº\ L 2 ¹¹ a d ; b d º ; X d 1 º ; (6.57) associatedwiththesimilarnorm kk X d .Thus,wede˝nethecorrespondingspace" ~ U andtspace" ~ V ,respectively,as ~ U = l 0 H 2 I ; L 2 ¹ d º \ L 2 ¹ I ; X d º ; ~ V = r 0 H 2 I ; L 2 ¹ d º \ L 2 ¹ I ; X d º ; (6.58) endowedwithsimilarnorms(7.26)and(7.27),where l 0 H 2 I ; L 2 ¹ d º = n u k u ¹ t ; ºk L 2 ¹ d º 2 H 2 ¹ I º ; k LP 0 D t u k L 2 ¹ I º < 1 ; u j t = 0 = u j x = a j = u j x = b j = 0 ; j = 1 ; 2 ; ; d o ; r 0 H 2 I ; L 2 ¹ d º = n v k v ¹ t ; ºk L 2 ¹ d º 2 H 2 ¹ I º ; k LP t D T u k L 2 ¹ I º < 1 ; v j t = T = v j x = a j = v j x = b j = 0 ; j = 1 ; 2 ; ; d o ; 142 equippedwithnorms k u k l H 2 ¹ I ; L 2 ¹ d ºº and k u k r H 2 ¹ I ; L 2 ¹ d ºº ,respectively. WeakFormulation .SincederivedFSEs(6.50)preservethestructureofFPDE(6.7),thebilinear formofcorrespondingweakformulationtakesthesameformas(7.28).Therefore,Byletting ~ U and ~ V bethesolution/testspaces,de˝nedin(6.58),theproblemreadsas:˝nd u 2 ~ U suchthat a ¹ u ; v º = ¹ f ; v º ; 8 v 2 ~ V ; (6.59) and˝nd S u ; q i 2 U ; 1 = 1 ; 2 ; ; 2 d + 1 suchthat a ¹ S u ; q i ; w º = ¹ f q i ; w º 8 w 2 V ; (6.60) where U and V arede˝nedin(7.22). 6.5FractionalModelConstruction WeemploythedevelopedFSEMinordertoconstructaniterativealgorithmtoestimatemodel parametersfromknownsolution(oravailablesetsofdata).Weformulatetheiterativealgorithm byminimizinganobjectivemodelerrorfunction.Werecallagainherethatinourfractionalmodel, thesetofmodelparametersis q = f ; 1 ; 2 ; ; d ; k 1 ; k 2 ; ; k d g ,andhere,wemainlyfocus onestimationoffractionalindices.Thus,assumingthemodelcoe˚cients f k 1 ; k 2 ; ; k d g tobe given/known,wereducethemodelparametersetto q = f ; 1 ; 2 ; ; d g2 Q ˆ R 1 + d . 6.5.1ModelError ThefractionalmodelcanbesimplyvisualizedasFig.6.2,where L q u = f .Wedenotebythe superscript ¹ º astheexactvaluesofquantities.Therefore, L q u = f ,where u , f aretheexact solutionandforcefunctions,respectively,and q isthesetofexactmodelparameters.Obviously, bychoosingdi˙erentvaluesofmodelparameters(fractionalindices),thefractionalmodelobserves theinputdi˙erently,andthus,resultsinadi˙erentoutput.Thisleadstotwotypesof modelerror , namely,type-Iandtype-II,describedasfollows.Wenotethattheintroducedmodelerrorsarezero attheexactvalues q ,byde˝nition. 143 Figure6.2:Schematicoffractionalmodel (a) (b) Figure6.3:Schematicofvariationoffractionalmodelbasedon(a)modelerrortype-Iand(b) modelerrortype-II. 6.5.1.1ModelError:Type-I Inmodelerrortype-I,weconsidertheoutputofmodeltobe˝xed,i.e, f = f ,however,changing parametersmakesthefractionalmodeltoobservethevariatedinput u q asopposedto u .Therefore, wede˝nethemodelerrorasthedi˙erencebetweenvariatedandexactinputs,i.e. E ¹ q º = jj u q u jj L 2 .TheschematicofvariationofmodelfromtheexactmodelisshowninFig.6.3 (a).Foreachvariatedmodel,weaccuratelycomputethenumericalapproximation, u q N ,bysolving (6.7),wherebyincreasingthenumberoftermsintheapproximatesolution,wemakesurethatthe function E ¹ q º = jj u q N u jj L 2 solelydescribesthemodelerrorwithminimumdiscretizationerror. Theproposediterativealgorithm,aswillbediscussedlater,involvesthegradientofmodelerror withrespecttothemodelparameters.Thus,wetakethepartialderivativeof E withrespectto q ,as S E ; q = @ E @ q = ¯ S u q ; q ¹ u q N u º d E (6.61) where S u q ; q denotesthesensitivity˝elds,whichisnumericallyobtainedbysolvingFSEs(6.50). Wenotethatinthiscase,since f is˝xedandtherefore,notsensitivetoanyparameter,weexclude the˝rstterminthede˝nitionofforcefunctionsf andf . 144 6.5.1.2ModelError:type-II Inmodelerrortype-II,weconsidertheinputofmodeltobe˝xed,i.e, u = u ,however,changing parametersmakesthefractionalmodeltoresultinthevariatedoutput f q asopposedto f . Therefore,wede˝nethemodelerrorasthedi˙erencebetweenvariatedandexactoutputs,i.e. E ¹ q º = jj f q f jj L 2 .TheschematicofvariationofmodelfromtheexactmodelisshowninFig. 6.3(b).Inthiscase,unlikemodelerrortype-I,themodelerroranditsgradientcanbeexpressed analytically.Therefore,theydonotcontainanydiscretizationerror. 6.5.2ModelErrorMinimization:IterativeAlgorithm Weminimizethemodelerrorbyformulatingatwo-stagesalgorithm.Sincewedonothaveprior informationaboutthevariatedsolution/forcefunction,itisdi˚culttoanalyticallypredictthe behaviorofintroducedmodelerror.However,ineveryexample,wenumericallystudythebehavior ofalowresolutionmodelerrormanifoldonacoarsegrid,andthen,performthelocalminimization. Theminimizationproblemiswrittenas: min q 2 Q E ¹ q º ; (6.62) inwhich E ¹ q º : Q ! R ,andweassumethattheproblemissolvable,i.e.thereexistaminimum point q 2 Q .Properchoiceofinitialguessinlocalminimizationisofgreatimportance,wherea wronginitialguess,notfallingwithinsmallenoughadjacencyofminimum,mayneverconverge. Therefore,theiterativeconvergenceinahypercubespaceofparametersishighlyconnectedtoan optimalinitialguessforeachparameter.Inthesequel,wedelineatethetwostagesofouralgorithm, namely,stageI: coarsegrid searching,andstageII: nearbysolution . InstageI,weprogressivelydividethehypercubeparameterspaceintosubspacestonarrow downtheobjectivesearchregionintoasmallerregion.Thisdivisionprocessisnotnecessarily uniqueandcanbedoneindi˙erentways,amongwhichwediscusstheeasy-to-implementonehere, whereineachprogressionstep,wechoosethesubspacewithminimumerroratitscorner.Wecarry outthecoarsegridsearchingtillwereachasmallenoughregion,inwhichthenearbysolution(stage 145 Figure6.4:Iterativealgorithm:coarsegridsearchingfor ¹ 1 + 1 º -Dparameterspace,where = 0 : 3 and = 0 : 8 . II)isvalid.Asanexample,weconsidera ¹ 1 + 1 º -Dfractionalmodelwith q = f ; g = f 0 : 3 ; 0 : 8 g astheexactfractionalindicesintheparametersurface,showninFig.6.4.Wedividetheparameter spaceintofourequalsubspacesandbycomputingtheerroratcornerpointsofeachsubsurface (blackdots),weshrinkthesearchregion(tothelabeledsubsurface 3 ).Weprogressfurtheronce againinasimilarfashion,dividethesubsurface,andcomputetheerroratcornerpoints(reddots). We˝nally,narrowdownthesearchregionintolabeledsubsurface 31 .Weseethatinthiscase,with computingtheerroronlyat14points,wecane˚cientlynarrowdowntheparameterspaceintoa smallenoughsearchregion,inwhichwecanperformstageIIofthealgorithm. InstageIIofthealgorithm,weemployagradientdecentmethod,inwhichbystartingfrom aninitialguess q 0 = f 0 ; 0 1 ; 0 2 ; ; 0 d g intheobtainedsearchregionfromstageI,weproducea minimizingsequence q i ; i = 1 ; 2 ; ,where q i + 1 = q i + q i ; (6.63) andtheincrement q i = s i p i containsboththestepsize s i andnormalizedstepdirection p i .The superscript i indicatestheiterationindex.Weobtainthenormalizeddirection p i bycomputing thegradientofmodelerrorwithrespecttotheparameters.Thestepsizeisusuallycomputedby performingalinesearchsuchthat E ¹ q i + sp i º isminimizedover 8 s 2 R .However,inourcase themethoddoesnotproducewell-scaledsearchdirections,andweneedtoapproximatethecurrent 146 stepsize,usingthepreviousone.Thus, p i = r E ¹ q i º kr E ¹ q i ºk ; s i = s i 1 r E ¹ q i 1 º T p i 1 r E ¹ q i º T p i ; (6.64) wherethe˝rstiterationsizeisobtained,usingtheTaylorexpansionofmodelerrorabout q 0 . 6.5.3FractionalModelConstruction:FSEM-basedIterativeAlgorithm Let = » 0 ; T ¼ 1 ; 1 ¼ bethecomputationaldomain.Weconsiderthe ¹ 1 + 1 º -DcaseofFPDE (6.7),subjecttotheinitialandboundaryconditions(7.2)and(7.3),respectively,wheretheadjoint FSEsaregivenin(6.50).Assumingthattheexacttransport˝eld u ¹ t ; x º andforcefunction f ¹ t ; x º aregiven,then, 0 D t u k 1 D x + x D 1 u = f (6.65) inwhich f ; g aretheexactfractionalindicesandthecoe˚cient k isknown. Byconsideringthetwotypesofmodelerror,weusethedevelopediterativeformulationand followthestepsbelowtoobtaintheoptimalmodelparameters.Ineachiteration,theincrements areobtained,using(6.64). Foreachmodelerrorwefollowdi˙erentsteps,givenbelow.BasedonthemodelerrorI,we follow 1. Initialguess q 0 = f 0 ; 0 g 2. Do i = 0 ; 1 ; 3. Solvefor u q i N :FPDE 4. Computethemodelerror E = jj u q i N u jj L 2 5. If E < tolerance, Then Break, Otherwise Continue 6. Solveforsensitivity˝elds:FSEs 7. Computethemodelerrorgradientusingsensitivity˝eld 8. Computetheiterationincrement q i 9. Marchinparameterspace q i + 1 = q i + q i 10. End 147 andbasedonthemodelerrorII,wefollow 1. Initialguess q 0 = f 0 ; 0 g 2. Do i = 0 ; 1 ; 3. Computethemodelerror E = jjL q i u L q u jj L 2 4. If E < tolerance, Then Break, Otherwise Continue 5. Computethemodelerrorgradientusingsensitivity˝eld(analyticallyavailable) 6. Computetheiterationincrement q i 7. Marchinparameterspace q i + 1 = q i + q i 8. End Remark6.5.1. Inthe˝rstiteration,wecomputethestepsize,usingtheTaylorexpansionofthe modelerrorabouttheinitialguess f 0 ; 0 g ,whichweseparateintotwodirectionsas E f ; g ˇ E f 0 ; 0 g + S E ; f 0 ; 0 g ¹ 0 º ; E f ; g ˇ E f 0 ; 0 g + S E ; f 0 ; 0 g ¹ 0 º : (6.66) Knowingthat E f ; g = 0 ,weobtaintheparametersatnextiterationsas 1 = 0 + 0 and 1 = 0 + 0 ,inwhich 0 ˇ E f 0 ; 0 g S E ; f 0 ; 0 g ; 0 ˇ E f 0 ; 0 g S E ; f 0 ; 0 g : (6.67) 6.6NumericalResults Inthe˝rstpartofnumericalresults,weinvestigatetheperformanceofdevelopedPGscheme insolvingFPDEandtheadjointFSEs.Weconsiderthecoupled ¹ 1 + 1 º -dFPDEandFSEswith one-sidedfractionalderivativeand k = 1 ,as 0 D t u 1 D x u = f ; (6.68) 0 D t S u ; 1 D x S u ; = S f ; A 1 ¹ º 0 D t u + LP 0 D t u ; (6.69) 0 D t S u ; 1 D x S u ; = S f ; + A 2 ¹ º 1 D x u LP 1 D x u : (6.70) Weconsidertwocasesofexactsolutionas 148 Figure6.5:PlotofexactfunctionsforcaseIwith š 2 = 0 : 25 and š 2 = 0 : 75 :exactsolution u ext (left),exactsensitivity˝eld S u ext ; = @ u ext @ (middle),exactsensitivity˝eld S u ext ; = @ u ext @ (right). Figure6.6:PlotofexactfunctionsforcaseIIwith š 2 = 0 : 25 and š 2 = 0 : 75 :exactsolution u ext (left),exactsensitivity˝eld S u ext ; (middle),exactsensitivity˝eld S u ext ; (right). ‹ CaseI: u ext ¹ t ; x º = t 3 + š 2 ¹ 1 + x º 3 + š 2 1 2 ¹ 1 + x º 4 + š 2 , ‹ CaseII: u ext ¹ t ; x º = t 3 + š 2 ¹ t 0 : 4 º¹ t 0 : 9 º ¹ 1 + x º 3 + š 2 1 2 ¹ 1 + x º 4 + š 2 . where š 2 = 0 : 25 ,and š 2 = 0 : 75 .Theexactsolutionandsensitivity˝elds,obtainedbytaking @ @ and @ @ oftheexactsolutions,areshowninFig.6.5and6.6forthetwocasesIandII,respectively. WeemploythedevelopedPGmethodtosolveFPDE(6.68)andobtain u N ,whichweuseto constructtherighthandsideofadjointFSEs.Then,weagainemploythedevelopedPGmethod tosolveFSEs(6.69)and(6.70)andobtainthenumericalsensitivity˝elds, S N u ; ; S N u ; .Westudy the L 2 -normconvergenceofourproposedmethodbyincreasingthenumberofbasisfunctions,as showninFig.6.7. FractionalModelConstruction. Thesecondpartofnumericalresultsisdedicatedto studythee˚ciencyofdevelopediterativealgorithminobtainingthesetofmodelparameters q 149 Figure6.7:PGspectralmethod, L 2 -normconvergencestudy: ¹ 1 + 1 º -dFPDEadjointtocorre- spondingFSEswithone-sidedfractionalderivative, k = 1 , š 2 = 0 : 25 ,and š 2 = 0 : 75 ,forCase I(left)andCaseII(right),where N = M . (fractionalindices)andthus,constructthefractionalmodel.Wetestourdevelopedschemeby methodoffabricatedsolution,assumingagivensetofinput(exactsolution)andoutput(force term)forourfractionalmodel. WebeginwithafractionalIVPoftheform 0 D t u ¹ t º = f ¹ t º , 2¹ 0 ; 1 º ,andassumethatthe exactsolutionandforcefunctionaregivenas, u ¹ t º = sin ¹ 5 ˇ š 2 º t 3 + š 2 ; f ¹ t º = sin ¹ 5 ˇ š 2 º ¹ 4 + š 2 º ¹ 4 š 2 º t 3 š 2 ; andthefractionalorder istheunknownmodelparameter.Westartfromaninitialguess 0 and usethedevelopediterativealgorithmtoconvergetothetruevalueoffractionalindex .Wealso considerafractionalBVPoftheform 1 D x u ¹ x º = f ¹ x º , 2¹ 1 ; 2 º ,andassumethattheexact solutionandforcefunctionaregivenas, u ¹ x º = ¹ 1 + x º 3 + š 2 1 2 ¹ 1 + x º 4 + š 2 f ¹ x º = ¹ 4 + š 2 º ¹ 4 š 2 º ¹ 1 + x º 3 + š 2 1 2 ¹ 5 + š 2 º ¹ 5 š 2 º ¹ 1 + x º 4 + š 2 andthefractionalorder istheunknownmodelparameter.Weagainusethedevelopediterative algorithmtocapturethetruevalueoffractionalindex ,startingfromaninitialguess 0 . 150 Table6.1:FractionalmodelconstructionforthetwocasesoffractionalIVP. IterationIndex 0 D t u ¹ t º = f ¹ t º i i i initialguess 0 : 3000000 : 900000 1 0 : 9719800 : 320520 2 0 : 9004180 : 300068 3 0 : 9000000 : 300000 TrueValue 0 : 90 : 3 Table6.2:FractionalmodelconstructionforthetwocasesoffractionalBVP. IterationIndex 1 D x u ¹ x º = f ¹ x º i i i initialguess 1 : 1000001 : 9000000 1 1 : 8820201 : 228040 2 1 : 7081201 : 106096 3 1 : 7000201 : 100016 4 1 : 7000001 : 100000 TrueValue 1 : 71 : 1 Tables6.1and6.2showtwoexamplesforeachcaseoffractionalIVPandBVP,wherethetrue valuesoffractionalordersare = 0 : 3 , = 0 : 9 , = 1 : 1 ,and = 1 : 7 .Weobservethat theproposediterativeformulationconvergesaccuratelytotheexactvalueswithinfewnumbersof iterations.WenotethatinthecaseoffractionalIVPandBVP,thesearchregionisalreadysmall enoughsothatthenearbysolutionisvalid,andtherefore,weonlyneedtoperformthesecondstage ofiterativealgorithm. Moreover,weconsiderFPDEoftheform 0 D t u k 1 D x u = f .Weassumetheexactsolution u = t 1 + š 2 ¹ 1 + x º 3 + š 2 1 2 ¹ 1 + x º 4 + š 2 andplugitintotheFPDEwithgiven f ; g to obtaintheexactforcefunction f .Westudytheexample,inwhich, f ; g = f 0 : 1 ; 1 : 64 g .We performthetwostagesofiterativealgorithm,whereinthe˝rststage,weshrinkdownthesearch region16timesmallerthantheoriginalsize,bycomputingthemodelerrorat8points(SeeFig.6.8, right).Then,inthenextstage,westartfromtheinitialguess f 0 ; 0 g = f 0 : 125 ; 1 : 75 g ,andobserve thatthedevelopediterativemethodconvergestoacloseneighborhoodoftruevalues f 0 : 1 ; 1 : 64 g within 10 3 tolerance. 151 Figure6.8:FractionalmodelconstructionforthecaseFPDE,usingFSEMbasediterativealgorithm. Thetruevaluesoffractionalindicesare f ; g = f 0 : 1 ; 1 : 64 g . Thedevelopedmodelconstructionmethodcanalsobeappliedinformulatingfractionalmodels tostudycomplextime-varyingnonlinear˛uid-solidinteractionphenomena[2,3,9]andalsothe e˙ectofdampinginstructuralvibrations[182]. 152 6.7ProofofLemmasandTheorems 6.7.1ProofofLemma6.2.2 PartA: ˙ 2¹ 0 ; 1 º .Westartfromthe RL PL de˝nition,givenin(6.1). RL LP a D ˙ x u = 1 ¹ 1 ˙ º d dx ¹ x a ¹ x s º ˙ log ¹ x s º u ¹ s º ds ; (integratebyparts)(6.71) = 1 ¹ 1 ˙ º d dx n u ¹ s º¹ x s º 1 ˙ ˙ + 1 º 2 ¹ 1 ˙ + 1 º log ¹ x s ºº s = x s = a ¹ x a ¹ x s º ˙ + 1 ˙ + 1 º 2 ¹ 1 ˙ + 1 º log ¹ x s ºº u 0 ¹ s º ds o ; = 1 ¹ 1 ˙ º d dx n u ¹ a º¹ x a º 1 ˙ ˙ + 1 º 2 ¹ 1 ˙ + 1 º log ¹ x a ºº ¹ x a ¹ x s º ˙ + 1 ˙ + 1 º 2 ¹ 1 ˙ + 1 º log ¹ x s ºº u 0 ¹ s º ds o ; = u ¹ a º ¹ 1 ˙ º log ¹ x a º ¹ x a º ˙ + 1 ¹ 1 ˙ º ¹ x a log ¹ x s º ¹ x s º ˙ u 0 ¹ s º ds ; (byLeibnitzrule) = u ¹ a º ¹ 1 ˙ º log ¹ x a º ¹ x a º ˙ + C LP a D ˙ x u 153 PartB: ˙ 2¹ 1 ; 2 º .Similarly,westartfromthe RL PL de˝nition,givenin(6.1). RL LP a D ˙ x u = 1 ¹ 2 ˙ º d 2 dx 2 ¹ x a ¹ x s º ˙ + 1 log ¹ x s º u ¹ s º ds ; (integratebypartstwice) (6.72) = 1 ¹ 2 ˙ º d 2 dx 2 n u ¹ s º¹ x s º ˙ + 2 ˙ + 2 º 2 ¹ 1 ˙ + 2 º log ¹ x s ºº s = x s = a u 0 ¹ s º¹ x s º ˙ + 3 ˙ + 2 º 2 ˙ + 3 º 2 ¹ 1 2 ˙ + 3 º + ˙ + 3 ˙ + 2 º log ¹ x s º º s = x s = a + ¹ x a ¹ x s º ˙ + 3 ˙ + 2 º 2 ˙ + 3 º 2 ¹ 1 2 ˙ + 3 º + ˙ + 3 ˙ + 2 º log ¹ x s º º u 00 ¹ s º ds o ; = 1 ¹ 2 ˙ º d 2 dx 2 n u ¹ a º¹ x a º ˙ + 2 ˙ + 2 º 2 ¹ 1 ˙ + 2 º log ¹ x a ºº u 0 ¹ a º¹ x a º ˙ + 3 ˙ + 2 º 2 ˙ + 3 º 2 ¹ 1 2 ˙ + 3 º + ˙ + 3 ˙ + 2 º log ¹ x a º º + ¹ x a ¹ x s º ˙ + 3 ˙ + 2 º 2 ˙ + 3 º 2 ¹ 1 2 ˙ + 3 º + ˙ + 3 ˙ + 2 º log ¹ x s º º u 00 ¹ s º ds o ; = u ¹ a º ¹ 2 ˙ º 1 + ˙ + 1 º log ¹ x a º ¹ x a º ˙ + u 0 ¹ a º ¹ 2 ˙ º log ¹ x a º ¹ x a º ˙ 1 + 1 ¹ 2 ˙ º ¹ x a ¹ x s º ˙ + 1 log ¹ x s º u 00 ¹ s º ds ; (byLeibnitzrule) = u ¹ a º ¹ 1 ˙ º 1 + ˙ + 1 º log ¹ x a º ¹ x a º ˙ + u 0 ¹ a º ¹ 1 ˙ º log ¹ x a º ¹ x a º ˙ 1 + C PL a D ˙ x u : 6.7.2ProofofLemma7.3.1 InLemma2.1in[100]andalsoin[53],itisshownthat kk l H ˙ ¹ º and kk r H ˙ ¹ º areequivalent. Therefore,for u 2 H ˙ ¹ º ,thereexistpositiveconstants C 1 and C 2 suchthat k u k H ˙ ¹ º C 1 k u k l H ˙ ¹ º ; k u k H ˙ ¹ º C 2 k u k r H ˙ ¹ º ; (6.73) 154 whichleadsto k u k 2 H ˙ ¹ º C 2 1 k u k 2 l H ˙ ¹ º + C 2 2 k u k 2 r H ˙ ¹ º ; = C 2 1 k a D ˙ x ¹ u ºk 2 L 2 ¹ º + C 2 2 k x D ˙ b ¹ u ºk 2 L 2 ¹ º + ¹ C 2 1 + C 2 2 ºk u k 2 L 2 ¹ º ; ~ C 1 k u k 2 c H ˙ ¹ º ; (6.74) where ~ C 1 isapositiveconstant.Similarly,wecanshowthat k u k 2 c H ˙ ¹ º ~ C 2 k u k H ˙ ¹ º ,where ~ C 2 isapositiveconstant. 6.7.3ProofofLemma7.3.2 X 1 isendowedwiththenorm kk X 1 ,where kk X 1 kk c H 1 š 2 ¹ 1 º byLemma7.3.1.Moreover, X 2 isassociatedwiththenorm kk X 2 ˆ kk 2 c H 2 š 2 0 ¹ a 2 ; b 2 º ; L 2 ¹ 1 º + kk 2 L 2 ¹ a 2 ; b 2 º ; X 1 ˙ 1 2 ; (6.75) where k u k 2 c H 2 š 2 0 ¹ a 2 ; b 2 º ; L 2 ¹ 1 º (6.76) = ¹ b 1 a 1 ¹ b 2 a 2 j a 2 D 2 š 2 x 2 u j 2 dx 2 + ¹ b 2 a 2 j x 2 D 2 š 2 b 2 u j 2 dx 2 + ¹ b 2 a 2 j u j 2 dx 2 dx 1 = ¹ b 1 a 1 ¹ b 2 a 2 j a 2 D 2 š 2 x 2 u j 2 dx 2 dx 1 + ¹ b 1 a 1 ¹ b 2 a 2 j x 2 D 2 š 2 b 2 u j 2 dx 2 dx 1 + ¹ b 1 a 1 ¹ b 2 a 2 j u j 2 dx 2 dx 1 = k x 2 D 2 š 2 b 2 ¹ u ºk 2 L 2 ¹ d º + k a 2 D 2 š 2 x 2 ¹ u ºk 2 L 2 ¹ d º + k u k 2 L 2 ¹ d º ; (6.77) and k u k 2 L 2 ¹ a 2 ; b 2 º ; X 1 = ¹ b 2 a 2 ¹ b 1 a 1 j a 1 D 1 š 2 x 1 u j 2 dx 1 + ¹ b 1 a 1 j x 1 D 1 š 2 b 1 u j 2 dx 1 + ¹ b 1 a 1 j u j 2 dx 1 dx 2 = ¹ b 2 a 2 ¹ b 1 a 1 j a 1 D 1 š 2 x 1 u j 2 dx 1 dx 2 + ¹ b 2 a 2 ¹ b 1 a 1 j x 1 D 1 š 2 b 1 u j 2 dx 1 dx 2 + ¹ b 2 a 2 ¹ b 1 a 1 j u j 2 dx 1 dx 2 = k x 1 D 1 š 2 b 1 u k 2 L 2 ¹ 2 º + k a 1 D 1 š 2 x 1 u k 2 L 2 ¹ 2 º + k u k 2 L 2 ¹ 2 º : (6.78) 155 Weusethemathematicalinductiontocarryouttheproof.Therefore,weassumethefollowing equalityholds kk X k 1 ˆ k 1 Õ i = 1 k x i D i š 2 b i ¹ºk 2 L 2 ¹ k 1 º + k a i D i š 2 x i ¹ºk 2 L 2 ¹ k 1 º + kk 2 L 2 ¹ k 1 º ˙ 1 2 : (6.79) Since, k u k 2 c H k š 2 0 ¹ a k ; b k º ; L 2 ¹ k 1 º = ¹ k 1 ¹ b k a k j a k D k š 2 x k u j 2 dx k + ¹ b k a k j x k D k š 2 b k u j 2 dx k + ¹ b k a k j u j 2 dx k d k 1 = ¹ k 1 ¹ b k a k j a k D k š 2 x k u j 2 dx k d k 1 + ¹ k 1 ¹ b k a k j x k D k š 2 b k u j 2 dx k d k 1 + ¹ k 1 ¹ b k a k j u j 2 dx k d k 1 = k x k D k š 2 b k ¹ u ºk 2 L 2 ¹ k º + k a k D k š 2 x k ¹ u ºk 2 L 2 ¹ k º + k u k 2 L 2 ¹ k º ; and k u k 2 L 2 ¹ a k ; b k º ; X k 1 = ¹ b k a k k 1 Õ i = 1 ¹ k 1 j a i D i š 2 x i u j 2 d k 1 + ¹ k 1 j x i D i š 2 b i u j 2 d k 1 ! + ¹ k 1 j u j 2 d k 1 ! dx k = k 1 Õ i = 1 ¹ k j a i D i š 2 x i u j 2 d k + ¹ k j x i D i š 2 b i u j 2 d k + ¹ k j u j 2 d k = k 1 Õ i = 1 k x i D i š 2 b i u k 2 L 2 ¹ k º + k a i D i š 2 x i u k 2 L 2 ¹ k º + k u k 2 L 2 ¹ k º ; wecanshowthat kk X k ˆ k Õ i = 1 k x i D i š 2 b i ¹ºk 2 L 2 ¹ k º + k a i D i š 2 x i ¹ºk 2 L 2 ¹ k º + kk 2 L 2 ¹ k º ˙ 1 2 : (6.80) 156 6.7.4ProofofLemma7.3.5 Accordingto[87],wehave a i D i x i u = a i D i š 2 x i ¹ a i D i š 2 x i u º and x i D i š 2 b i u = x i D i š 2 b i ¹ x i D i š 2 b i u º . Let u = a i D i š 2 x i u .Then, ¹ a i D i x i u ; v º d = ¹ a i D i š 2 x i u ; v º d = ¹ d 1 ¹ 1 i š 2 º d dx i ¹ x i a i u ¹ s º ds ¹ x i s º i š 2 v d d = n v ¹ 1 i š 2 º ¯ x i a i uds ¹ x i s º i š 2 o b i x i = a i ¹ d 1 ¹ 1 i š 2 º ¹ x i a i u ¹ s º ds ¹ x i s º i š 2 d v dx i d d : (6.81) Basedonthehomogeneousboundaryconditions, n v ¹ 1 i š 2 º ¯ x i a i uds ¹ x i s º i š 2 o b i x i = a i = 0 : Therefore, ¹ a i D i x i u ; v º d = ¹ i 1 ¹ 1 i š 2 º ¹ x i a i u ¹ s º ds ¹ x i s º i š 2 d v dx i d i : (6.82) Moreover,we˝ndthat d ds ¹ b i a i u ¹ x i s º i š 2 dx i = d ds n f v ¹ x i s º 1 i š 2 1 i š 2 g b i x i = s i 1 1 i š 2 ¹ b i s d v dx i ¹ x i s º 1 i š 2 dx i o = 1 1 i š 2 ¹ b i s d v dx i ¹ x i s º 1 i š 2 dx i = ¹ b i s d v dx i ¹ x i s º i š 2 dx i : (6.83) Therefore,weget ¹ a i D i š 2 x i u ; v º d = ¹ d 1 ¹ 1 º i u ¹ s º d ds ¹ b i s v ¹ x i s º i š 2 dx i ds = ¹ u ; x i D i š 2 b i v º d : 6.7.5ProofofLemma7.3.7 Weknowthat 0 D š 2 t u ; t D š 2 T v = ¹ d ¹ T 0 j 0 D š 2 t u t D š 2 T v j 2 dtd d 1 2 : 157 Therefore,byHölderinequality 0 D š 2 t u ; t D š 2 T v ¹ d ¹ T 0 j 0 D š 2 t u j 2 dtd d 1 2 ¹ d ¹ T 0 j t D š 2 T v j 2 dtd d 1 2 ¹ d ¹ T 0 j 0 D š 2 t u j 2 dtd d + ¹ d ¹ T 0 j u j 2 dtd d 1 2 ¹ d ¹ T 0 j t D š 2 T v j 2 dtd d + ¹ d ¹ T 0 j v j 2 dtd d 1 2 = k 0 D š 2 t u k L 2 ¹ º k t D š 2 T v k L 2 ¹ º = k u k l H š 2 ¹ I ; L 2 ¹ d ºº k v k r H š 2 ¹ I ; L 2 ¹ d ºº : Moreover,byequivalenceof jj H s ¹ I º jj H s ¹ I º = jj 1 š 2 l H s ¹ I º jj 1 š 2 r H s ¹ I º wehave j¹ 0 D š 2 t u ; t D š 2 T v º I j = ¹ T 0 j 0 D š 2 t u t D š 2 T v j 2 dt ¹ T 0 j 0 D š 2 t u j 2 dt ¹ T 0 j t D š 2 T v j 2 dt (6.84) ~ 1 k u k l H s ¹ I º k v k r H s ¹ I º ; where 0 < ~ 1 1 .Therefore, j¹ 0 D š 2 t u ; t D š 2 T v º j 2 = ¹ d ¹ T 0 j 0 D š 2 t u t D š 2 T v j 2 dtd d ¹ d ¹ T 0 j 0 D š 2 t u j 2 dt ¹ T 0 j t D š 2 T v j 2 dt d d ¹ d ¹ T 0 j 0 D š 2 t u j 2 dtd d ¹ d ¹ T 0 j t D š 2 T v j 2 dt d ~ 2 k u k l H s ¹ I º k v k r H s ¹ I º ; (6.85) where 0 < ~ 2 1 and 0 < . 158 6.7.6ProofofTheStabilityTheorem7.3.9 PartA : d = 1 .Itisevidentthat u and v areinHilbertspaces(see[53,100]).For 0 < ~ 1 ,we have j a ¹ u ; v ºj = j¹ 0 D š 2 t ¹ u º ; t D š 2 T ¹ v ºº + ¹ a 1 D 1 š 2 x 1 ¹ u º ; x 1 D 1 š 2 b 1 ¹ v ºº (6.86) + ¹ a 1 D 1 š 2 x 1 ¹ u º ; x 1 D 1 š 2 b 1 ¹ v ºº + ¹ u ; v º j ~ j¹ 0 D š 2 t ¹ u º ; t D š 2 T ¹ v ºº j + j¹ a 1 D 1 š 2 x 1 ¹ u º ; x 1 D 1 š 2 b 1 ¹ v ºº j (6.87) + j¹ a 1 D 1 š 2 x 1 ¹ u º ; x 1 D 1 š 2 b 1 ¹ v ºº j + j¹ u ; v º j ; since sup u 2 U j a ¹ u ; v ºj > 0 .Next,byequivalenceofspacesandtheirassociatednorms,(7.43),and (7.44),weobtain j¹ 0 D š 2 t ¹ u º ; t D š 2 T ¹ v ºº j C 1 k 0 D š 2 t u k L 2 ¹ º k t D š 2 T v k L 2 ¹ º ; j¹ a 1 D 1 š 2 x 1 ¹ u º ; x 1 D 1 š 2 b 1 ¹ v ºº j C 2 k a 1 D 1 š 2 x 1 u k L 2 ¹ º k x 1 D 1 š 2 b 1 v k L 2 ¹ º ; and j¹ x 1 D 1 š 2 b 1 ¹ u º ; a 1 D 1 š 2 x 1 ¹ v ºº j C 3 k x 1 D 1 š 2 b 1 u k L 2 ¹ º k a 1 D 1 š 2 x 1 v k L 2 ¹ º ; (6.88) where C 1 , C 2 ,and C 3 arepositiveconstants.Therefore, j a ¹ u ; v ºj ~ C ~ n k 0 D š 2 t u k L 2 ¹ º k t D š 2 T v k L 2 ¹ º + k a 1 D 1 š 2 x 1 u k L 2 ¹ º k x 1 D 1 š 2 b 1 v k L 2 ¹ º + k a 1 D 1 š 2 x 1 u k L 2 ¹ º k x 1 D 1 š 2 b 1 v k L 2 ¹ º o ; (6.89) where ~ C is min f C 1 ; C 2 ; C 3 g .Also,thenorm k u k U k v k V isequivalenttotherighthandsideof inequality(6.89).Therefore, j a ¹ u ; v ºj C k u k U k v k V . PartB : d > 1 .Similarly,wehave j a ¹ u ; v ºj (6.90) j¹ 0 D š 2 t ¹ u º ; t D š 2 T ¹ v ºº j + d Õ i = 1 j¹ a i D i š 2 x i ¹ u º ; x i D i š 2 b i ¹ v ºº j + j¹ a i D i š 2 x i ¹ u º ; x i D i š 2 b i ¹ v ºº j ; 159 where 0 < 1 .Recallingthatasthedirectconsequencesof(7.43),weobtain j¹ a i D i š 2 x i ¹ u º ; x i D i š 2 b i ¹ v ºº jk a i D i š 2 x i ¹ u ºk L 2 ¹ º k x i D i š 2 b i ¹ v ºk L 2 ¹ º ; j¹ x i D i š 2 b i ¹ u º ; a i D i š 2 x i ¹ v ºº jk x i D i š 2 b i ¹ u ºk L 2 ¹ º k a i D i š 2 x i ¹ v ºk L 2 ¹ º : Thus, d Õ i = 1 j¹ a i D i š 2 x i ¹ u º ; x i D i š 2 b i ¹ v ºº j + j¹ x i D i š 2 b i ¹ u º ; a i D i š 2 x i ¹ v ºº j ; (6.91) ~ C d Õ i = 1 k a i D i š 2 x i ¹ u ºk L 2 ¹ º k x i D i š 2 b i ¹ v ºk L 2 ¹ º + k x i D i š 2 b i ¹ u ºk L 2 ¹ º k a i D i š 2 x i ¹ v ºk L 2 ¹ º ; ~ C 1 ~ d Õ i = 1 k a i D i š 2 x i ¹ u ºk L 2 ¹ º + k x i D i š 2 b i ¹ u ºk L 2 ¹ º d Õ j = 1 k x j D j b j ¹ v ºk L 2 ¹ º ; + k a j D j x j ¹ v ºk L 2 ¹ º ; for u ; v 2 L 2 ¹ I ; X d º ,where 0 < ~ C and 0 < ~ 1 .Furthermore,Lemma7.3.7yields j¹ 0 D š 2 t ¹ u º ; t D š 2 T ¹ v ºº jk u k r H š 2 ¹ I ; L 2 ¹ d ºº k v k l H š 2 ¹ I ; L 2 ¹ d ºº : (6.92) Therefore,from(6.91)and(6.92)wehave j a ¹ u ; v ºj k u k r H š 2 ¹ I ; L 2 ¹ d ºº k v k l H š 2 ¹ I ; L 2 ¹ d ºº + k u k L 2 ¹ I ; X d º k v k L 2 ¹ I ; X d º ; (6.93) where k u k r H š 2 ¹ I ; L 2 ¹ d ºº k v k l H š 2 ¹ I ; L 2 ¹ d ºº + k u k L 2 ¹ I ; X d º k v k L 2 ¹ I ; X d º ~ C 2 k u k r H š 2 ¹ I ; L 2 ¹ d ºº + k u k L 2 ¹ I ; X d º k v k l H š 2 ¹ I ; L 2 ¹ d ºº + k v k L 2 ¹ I ; X d º (6.94) for u 2 U , v 2 U and 0 < ~ C 2 1 .Byconsidering(6.93)and(6.94),weget j a ¹ u ; v ºj C k u k U k v k V : (6.95) 160 CHAPTER7 OPERATOR-BASEDUNCERTAINTYQUANTIFICATIONFORSTOCHASTIC FRACTIONALPDES 7.1Background Signi˝cantapproximationsasinherentpartofassumptionsuponwhichthemodelisbuilt,lack ofinformationabouttruevaluesofparameters(incompletedata),andrandomnatureofquantities beingmodeledpervadeuncertaintyinthecorrespondingmathematicalformulations[41,142].In thiswork,wedevelopanuncertaintyquanti˝cation(UQ)frameworkinthecontextofstochastic fractionalpartialdi˙erentialequations(SFPDEs),inwhichwecharacterizedi˙erentsourcesof uncertaintiesandfurtherpropagatetheassociatedrandomnesstothesystemresponsequantityof interest(QoI).Theintentionofthisworkisnottointroducenewmathematicaltheoriesormethods forUQ,butrathertobringforwardpracticalsolutionsusingexistingtheoriesinanattemptto overcomethecomputationalchallengesofUQinfractionalmodels. TypesandSourcesofUncertainty .Themodeluncertaintiesareingeneralbeingclassi˝ed asaleatoryandepistemicaccordingtotheirfundamentalessence.Itisimportanttoretainthe separationbetweenthesetwosourcesinordertoassessthepredictivee˚ciencyofmodel[45,125]. Aleatoryuncertaintyimpactsoutputofinterestduetonaturalvariationofinputsandparameters;itis irreducibleandcommonlytreatedwithprobabilitytheory.Epistemicuncertainty,however,results fromlackofknowledgeaboutthesystemofinterestandcanbereducedbyobtainingadditional information.Theepistemicuncertaintiesarebroadlycharacterizedas i) modeluncertainties, occurringinmodelinputs,numericalapproximationerrors,andmodelformuncertainty;and ii) data uncertaintiesduetomeasurementinaccuracyandsparseorimprecisedata.Themodeluncertainty encompassesallmodelparameterscomingfromgeometry,constitutivelaws,and˝eldsequation, whilealsopertainingsurroundinginteractions,suchasboundaryconditionsandrandomforcing sources(noise).Numericalapproximations,whichareanessenceofdi˙erentialequationssince 161 theygenerallydonotlendthemselvestoanalyticalsolutions,introduceuncertaintybyimposing di˙erentsourcesofdiscretizationerror,iterativeconvergenceerror,androundo˙error.Inthis work,weonlyconsidertheepistemicuncertaintyinourfractionalmodelandthus,introducethe fractionalderivativeordersasnewsetofmodelparametersinadditiontomodelcoe˚cients.We notethatthevaluesofthesenewparametersarestronglytiedtothedistributionofunderlying stochasticprocessandtheirstatisticsareestimatedfromexperimentalobservationsinpractice,see e.g.[18,26]. UncertaintyFramework .ConventionalapproachesinparametricUQofdi˙erentialequationsis basedaroundMonteCarlosampling(MCS)[54],whichperformsensembleofforwardcalculations tomaptheuncertaininputspacetotheuncertainoutputspace.Thismethodenjoysfrombeing embarrassinglyparallelizableandcanbeimplementquitereadilyonhighdimensionalrandom spaces.However,thekeyissueistheslowrateofconvergence ˘ 1 š p N with N numberof realization,whichconsequentlyimposesexhaustivelysomanyoperationsofforwardsolver,makes itnotpracticalforexpensivesimulations.OthermethodssuchassequentialMCS[43]andmultilevel sequentialMSC[27]arealsodevelopedandrecentlyusedin[77]toimprovetheparametric uncertaintyassessmentinellipticnonlocalequations.AnalternativetoexpensiveMCSistobuild surrogatemodels.Anextensivecomparisonoftwowidelyusedones,namelypolynomialchaos andGaussianprocess,areprovidedinarecentwork[133].Polynomialchaos,inwhichtheoutput ofstochasticmodelisrepresentedasaseriesexpansionofinputparameters,wasinitiallyapplied in[63],andlaterextendedandusedin[92,127,165,176,177].Itisalsogeneralizedandused inconstructingstochasticGalerkinmethods[16,17,94,95]forproblemswithhigher-dimensional randomspaces.Othernon-samplingnumericalmethods,includingbutnotlimitedtoperturbation method[13,149,163,174]andmomentequationmethod[108,109]arealsodeveloped,however theirapplicationsarerestrictedtostochasticsystemswithrelativelylow-dimensionalrandomspace. Theseso-called intrusive "approachestypicallydonottreattheforwardsolverasablack-box,rather requiresomeknowledgeandreformulationofthegoverningequationsandthus,maynotbepractical inmanyproblemswithcomplexcodes. 162 Awiderangeof non-intrusive "techniquesmostlystretchoversampling,quadrature,andre- gression,see[133]andreferencestherein.Morerecently,high-orderprobabilisticcollocation methods(PCM),employingtheideaofinterpolation/collocationintherandomspaces,aredevel- opedin[14,129,175].Thesemethodslimitthesamplepointstoane˚cientsubsetofrandom space,whileadequatelysamplingthenecessaryrange.TheexcellenceinuseofPCMistwofold;it hasthebene˝tofeasilysamplingatnodalpointsthatnaturallyleadstoindependentrealizationsof theproblemasinMCS,andtheadvantageoffastconvergencerate.Thechallengingpostprocessing ofsolutionstatistics,whichcanessentiallybedescribedasahigh-dimensionalintegrationproblem, canalsoberesolvedbyadoptingsparsegridgenerators,suchasSmolyakalgorithm[129,151].The useofsparsegrids,asopposedtofulltensorproductconstructionfromone-dimensionalquadrature rules,wille˙ectivelyreducethenumberofsampling,whilepreservingafastconvergencerateto highlevelofaccuracy. ForwardSolver .AcoretaskincomputationalforwardUQistoformane˚cientnumerical method,whichforeachrealizationsofrandomvariablescanaccuratelysolvesandsimulatesthe deterministiccounterpartofstochasticmodelinthephysicaldomain.Suchnumericalmethodis usuallycalled forwardsolver "or simulator ".InthecaseofFPDEs,theexcessivecostofnumerical approximationsduetotheinherentnonlocalnatureoffractionaldi˙erentialequationsadditionally becomemorechallengingwhengenerallymostofuncertaintypropagationtechniquesinstruct operationsofforwardsolvermanytimes.Thisrequiresimplementationofmoree˚cientnumerical schemes,whichcanmanageincreasingcomputationalcostswhilemaintainingsu˚cientlylow errorlevelinmitigatingthecorrespondinguncertainties.Inadditiontonumerous˝nitedi˙erence methodsforsolvingFPDEs[33,65,103,159,167,172,192,196],recentworkshaveelaborated e˚cientspectralschemes,fordiscretizingFPDEsinphysicaldomain,seee.g.,[28,39,83,84, 99,100,103,136,170].Morerecently,Zayernourietal.[183,186]developedtwonewspectral theoriesonfractionalandtemperedfractionalSturm-Liouvilleproblems,andintroducedexplicit correspondingeigenfunctions,namely Jacobipoly-fractonomials of˝rstandsecondkind.These eignefunctionsarecomprisedofsmoothandfractionalparts,wherethelattercanbetunnedto 163 capturesingularitiesoftruesolution.Theyaresuccessfullyemployedinconstructingdiscrete solution/testfunctionspacesanddevelopingaseriesofhigh-orderande˚cientPetrov-Galerkin spectralmethods,see106,160]. Themainfocusofthisworkistodevelopanoperator-basedcomputationalforwardUQframe- workinthecontextofstochasticfractionalpartialdi˙erentialequation.Assumingthatthemath- ematicalmodelunderconsiderationiswell-posedandaccountsinprincipleforallfeaturesof underlyingphenomena,weidentifythreemainsourcesofuncertainty, i )parametricuncertainty, includingfractionalindicesasnewsetofrandomparametersappearedintheoperator, ii )addi- tivenoises,whichincorporatesallintrinsic/extrinsicunknownforcingsourcesaslumpedrandom inputs,and iii )numericalapproximations.Computationalchallengesarisewhentheadmissible spaceofrandominputsisin˝nite-dimensional,e.g.problemssubjecttoadditivenoise[137],and thus,theframeworkinvolvesuncertaintyparametrizationviaa˝nitenumberofrandomspacebasis. Unliketheclassicalapproachinmodelingrandominputs,whichconsidersidealizeduncorrelated processes(whitenoises),wemodeltherandominputsasmore/fullycorrelatedrandomprocesses (colorednoises),andparametrizethemviaKarhunen-Loève(KL)expansionbyassuming˝nite- dimensionalnoiseassumption.Thisyieldstheproblemin˝nitedimensionalrandomspace.We then,propagatetheparametricuncertaintiesintothesystemresponsebyapplyingPCM.Weobtain thecorrespondingdeterministicFPDEforeachrealizationofrandomvariables,usingtheSmolyak sparsegridgeneratorsforlowtomoderatelyhighdimensions.Inordertoformulatetheforward solver,wefollow[145]anddevelopahigh-orderPetrov-Galerkin(PG)spectralmethodtosolvefor eachrealizationofSFPDEinthephysicaldomain,employingJacobipoly-fractonomialsinaddi- tiontoLegendrepolynomialsastemporalandspatialbasis/testfunctions,respectively.Thesmart choiceofcoe˚cientsinconstructionofspatialbasis/testfunctionsyieldssymmetricpropertiesin theresultingmass/sti˙nessmatrix,whichisthenexploitedtoformulateane˚cientfastsolver.We alsoshowthatforeachrealizationofrandomvariables,thedeterministicproblemismathematically well-posedandtheproposedforwardsolverisstable.Byadoptingsu˚cientnumberofbasisinthe physicaldomain,weeliminatetheepistemicuncertaintyassociatedwithnumericalapproximation 164 andisolatetheimpactofparametricuncertaintyonsystemresponseQoI. Theorganizationofthischapterisasfollows.Weformulatethestochasticsysteminsection 7.2,andparametrizetherandominputs.Wealsodevelopthestochasticsampling,namelyPCM andMCSforourstochasticproblem.Wefurtherconstructthedeterministicsolverinsection7.3, andprovidethenumericalresultsinsection7.4. 7.2ForwardUncertaintyFramework 7.2.1FormulationofStochasticFPDE Let D = » 0 ; T ¼» a 1 ; b 1 ¼» a 2 ; b 2 ¼» a d ; b d ¼ bethephysicalcomputationaldomainforsome positiveinteger d andstochasticfunction u ¹ t ; x ; ! º : D ! R ,where ! 2 denotestherandom inputsofthesysteminaproperlyde˝nedcompleteprobabilityspace ¹ ; F ; P º .Weconsider thefollowingSFPDE,subjecttocertainhomogeneousDirichletinitial/boundaryconditionsand stochasticprocessasadditionalforcefunction,givenas L q ¹ ! º u ¹ t ; x ; ! º = F ¹ t ; x ; ! º (7.1) u ? ? t = 0 = 0 ; (7.2) u ? ? x = a j = u ? ? x = b j = 0 ; (7.3) suchthatfor P -almosteverywhere ! 2 theequationholds.Thestochasticfractionaloperator andforceterm,aregivenrespectivelyas: L q ¹ ! º = 0 D ¹ ! º t d Õ j = 1 k j a j D j ¹ ! º x j + x j D j ¹ ! º b j (7.4) F ¹ t ; x ; ! º = h ¹ t ; x º + f ¹ t ; ! º ; (7.5) wherethefractionalindices ¹ ! º2¹ 0 ; 1 º and j ¹ ! º2¹ 1 ; 2 º ; j = 1 ; 2 ; d aremutuallyinde- pendentrandomvariables, k j arerealpositiveconstantcoe˚cients,andthefractionalderivatives aretakenintheRiemann-Liouvillesense.Weassumethatthedrivingterms h and f areproperly posed,suchthatEqns.(7.1)-(7.3)iswell-posed P -a.e. ! 2 ,andalsothesolutioninphysical 165 domain D issmoothenoughsuchthatwecanconstructanumericalschemetosolveeachrealization ofSFPDE.Asanextensiontofutureworks,thestochasticoperatorEqn.(7.4)canbeextendedto ¹ ! º2¹ 1 ; 2 º forthecaseofwaveequations,andthusappliedinformulatingfractionalmodels tostudycomplextime-varyingnonlinear˛uid-solidinteractionphenomena[2,3,9]andalsothe e˙ectofdampinginstructuralvibrations[182]. 7.2.2RepresentationoftheNoise:DimensionReduction Weapproximatetheadditionalrandomforcingtermbyrepresenting f ¹ t ; ! º intoits˝nitedimen- sionalversionandthus,reducethein˝nite-dimensionalprobabilityspacetoa˝nite-dimensional space.ThisisachievedviatruncatingKarhunen-Loève(KL)expansionwiththedesiredaccuracy[110]. Let ¹ ; F ; P º beacompleteprobabilityspace,where isthespaceofevents, Fˆ 2 denotesthe ˙ -algebraofsetsin ,and P istheprobabilitymeasure.Therandom˝eld f ¹ t ; ! º hastheensemblemean E f f ¹ t ; ! ºg = f ¹ t º ,˝nitevariance E f» f ¹ t ; ! º f ¹ t º¼ 2 g andcovariance C f ¹ t 1 ; t 2 º = E f» f ¹ t 1 ; ! º f ¹ t 1 º¼» f ¹ t 2 ; ! º f ¹ t 2 º¼g .TheKLexpansionof f ¹ t ; ! º takestheform f ¹ t ; ! º = f ¹ t º + 1 Õ k = 1 p k k ¹ t º Q k ¹ ! º ; (7.6) where Q ¹ ! º = f Q k ¹ ! ºg k = 1 k = 1 isasetofmutuallyuncorrelatedrandomvariableswithzeromean andunitvariance,while k ¹ t º and k aretheeigenfunctionandeigenvaluesofthecovariance kernel C f ¹ t 1 ; t 2 º .Weobtainthecovariancekernel C f anditseigenvaluesandeigenfunctions, following[156]andbysolvingastochasticHelmholtzequation 4 f ¹ t ; ! º m 2 f ¹ t ; ! º = g ¹ t ; ! º ; (7.7) wheretherandomforcing g ¹ t ; ! º isawhite-noiseprocesswithzeromeanandunitvariance.The eigenvaluesandeigenvectorsofEqn.(7.7)formaFourierseries,sothattheKLexpansionEqn.(7.6) isreplacedwithitssineFourierseriesversion f ¹ t ; ! º = f ¹ t ; ! º + 1 Õ k = 1 a k sin 2 k ˇ t T Q k ¹ ! º ; (7.8) 166 inwhichtherandomvariables Q k ¹ ! º arechosentobe uniformly distributedwithprobabilitydensity function ˆ k ¹ q k º . T isthelengthoftheprocessalongthe t -axis,andthecoe˚cients a k = 2 p T ` 2 " 1 + 2 ˇ k T ` 2 # 1 ; (7.9) where ` = T š A and A isthecorrelationlengthof f ¹ t ; ! º .Toensurethattherandomvariables Q k ¹ ! º havezeromeanandunitvariance,wede˝nethemon Q k ¹ ! º2 p 3 ; p 3 ¼ .Wenotethat thisprocessisconsistenttothezero-DirichletinitialconditiongiveninEqn.(7.2).Next,in ordertorenderEqn.(7.8)computable,wetruncatethein˝niteserieswithaprescribed( ˇ 90% ) fractionoftheenergyoftheprocess,followingthe˝nite-dimensionalnoiseassumptioninstochastic computations.Tothisend,weset T = 1 ,thecorrelationlength A = T š 2 ,andconsideronlythe ˝rstfourtermsintheKLexpansion.Let f M ¹ t ; ! º = 1 Í M k = 1 a k sin 2 k ˇ t T Q k ¹ ! º denotethe normalizedtruncatedexpansion,assuming f M ¹ t ; ! º = 0 ,where = max t ˙ f M and ˙ f M isthe standarddeviationof f M ¹ t ; ! º .Thus,werepresenttherandomprocesstobeemployedinEqn.(7.1) as f ¹ t ; ! º = f M ¹ t ; ! º (7.10) where istheamplitudeofprocess. Therefore,theformulationofSFPDEEqn.(7.1)canbeposedasfollows:Find u ¹ t ; x ; ! º : D ! R suchthat 8 t ; x 2 D 0 D ¹ ! º t u ¹ t ; x ; ! º d Õ j = 1 k j a j D j ¹ ! º x j + x j D j ¹ ! º b j u ¹ t ; x ; ! º = h ¹ t ; x º + f ¹ t ; Q 1 ¹ ! º ; Q 2 ¹ ! º ; ; Q M ¹ ! ºº (7.11) holds P -a.s.for ! 2 ,subjecttothehomogeneousinitialandboundaryconditions. 167 7.2.3InputParametrization Let Z : ! R N bethesetof N = 1 + d + M independentrandomparameters,givenas Z = n Z i o N i = 1 = n ¹ ! º ; 1 ¹ ! º ; 2 ¹ ! º ; ; d ¹ ! º ; Q 1 ¹ ! º ; Q 2 ¹ ! º ; ; Q M ¹ ! º o withprobabilitydensityfunctions ˆ i : i ! R ; i = 1 ; 2 ; ; N ,wheretheirimages i Z i ¹ º are boundedintervalsin R .Thejointprobabilitydensityfunction(PDF) ˆ ¹ ˘ º = N Ö i = 1 ˆ i ¹ Z i º ; 8 ˘ 2 (7.12) withthesupport = Î N i = 1 i ˆ R N constitutesamappingofthesamplespace ontothetarget space .Therefore,arandomvector ˘ = ¹ ˘ 1 ;˘ 2 ;:::;˘ N º2 denoteanarbitrarypointinthe parametricspace. AccordingtotheDoob-Dynkinlemma[132],thesolution u ¹ t ; x ; ! º canbeexpressedas u ¹ t ; x ; ˘ º , whichprovidesaveryusefultooltoworkinthetargetspaceratherthantheabstractsamplespace. Thus,theformulationofSFPDEEqn.(7.1)canbeposedas:Find u ¹ t ; x ; ˘ º : D ! R suchthat 8 t ; x 2 D 0 D ¹ ˘ º t u ¹ t ; x ; ˘ º d Õ j = 1 k j a j D j ¹ ˘ º x j + x j D j ¹ ˘ º b j u ¹ t ; x ; ˘ º = h ¹ t ; x º + f ¹ t ; ˘ º (7.13) holds ˆ -a.s.for ˘ ¹ ! º2 and 8 t ; x 2 D ,subjecttoproperinitialandboundaryconditions. 7.2.4StochasticSampling Weexpoundthetwosamplingmethods,MCSandPCMtosamplefromrandomspaceand,then propagatetheassociateduncertaintiesbycomputingthestatisticsofstochasticsolutionsviapost processing. MonteCarloSampling:MCS. ThegeneralprocedureinstatisticalMonteCarlosamplingis thethreefollowingsteps. 168 1. Generatingasetofrandomvariables ˘ i , i = 1 ; 2 ; ; K foraprescribednumberofrealizations K . 2. SolvingthedeterministicproblemEqn.(7.13)andobtainingthesolution u i = u ¹ t ; x ; ˘ i º for each i = 1 ; 2 ; ; K . 3. Computingthesolutionstatistics,e.g. E » u ¼ = 1 M Í M i = 1 u i . Wenotethatstep1and3arepre-andpost-processingsteps,respectively.Step2requiresrepetitive simulationofdeterministiccounterpartoftheproblem,whichweobtainbydevelopingaPetrov- Galerkinspectralmethodinthephysicaldomain.AlthoughMCSisrelativelyeasytoimplement onceadeterministicforwardsolverisdeveloped,itrequirestoomanysamplingsforthesolution statisticstoconverge,andyettheextranumericalcostduetonon-localityandmemorye˙ectin fractionaloperatorsarestillremained.Inaddition,thenumberofrequiredsamplingalsogrows rapidlyasthedimensionofproblemincreases,resultinginanexhaustivelylongruntimeforthe statisticstoconverge. ProbabilityCollocationMethod:PCM. Weemployahigh-orderstochasticdiscretization intherandomspacefollowing[56,175]inordertoconstructaprobabilisticcollocationmethod (PCM),whichyieldsahighconvergenceratewithmuchfewernumberofsampling.Theideaof PCMisbasedonpolynomialinterpolation,howeverintherandomspace.Let N = ˘ i J i = 1 be asetofprescribedsamplingpoints.ByemployingtheLagrangeinterpolationpolynomials L i ,the polynomialapproximation I ofthestochasticsolution u intherandomspacecanbeexpressedas: ^ u ¹ t ; x ; ˘ º = I u ¹ t ; x ; ˘ º = J Õ i = 1 u ¹ t ; x ; ˘ i º L i ¹ ˘ º : (7.14) Therefore,thecollocationprocedureofsolvingEqn.(7.13)toobtainthestochasticsolution u is: R ¹ ^ u ¹ t ; x ; ˘ º º ˘ i = L q ¹ ˘ º ^ u ¹ t ; x ; ˘ ºº F ¹ t ; x ; ˘ º ˘ i = 0 ; (7.15) 169 for i = 1 ; 2 ; ; J ,where L q isgiveninEqn.(7.4).ByusingthepropertyofLagrangeinterpolants thatsatisfytheKroneckerdeltaattheinterpolationpoints,weobtain: L q ¹ ˘ i º u ¹ t ; x ; ˘ i ºº = F ¹ t ; x ; ˘ i º ; i = 1 ; 2 ; ; J ; (7.16) subjecttoproperinitial/boundaryconditions.Thus,theprobabilisticcollocationprocedureisequiv- alenttosolving J deterministicproblemsEqn.(7.16)withconditionsEqn.(7.2)andEqn.(7.3). Oncethedeterministicsolutionsareobtainedateachsamplingpoint,thenumericalstochastic solutionisinterpolated,usingEqn.(7.14)toconstructaglobalapproximate ^ u ¹ t ; x ; ˘ º .Wethen obtainthesolutionstatisticsas E » ^ u ¼ = ¹ ^ u ¹ t ; x ; ˘ º ˆ ¹ ˘ º d ˘ ;˙ » u ¼ = q E » ^ u 2 ¼ E » ^ u ¼ 2 : (7.17) Theaboveintegralscanbecomputede˚cientlybylettingtheinterpolation/collocationpointsto bethesameasasetofcubaturerules N = ˘ i J i = 1 ontheparametricspacewithintegration weights f w i g J i = 1 ,whichareemployedincomputingtheintegral.BypropertyofKroneckerdelta ofLagrangeinterpolantanduseofanyquadratureruleovertheaboveintegralyields E » ^ u ¹ t ; x : ˘ º¼ˇ J Õ i = 1 w i u ¹ t ; x ; ˘ i º : (7.18) ChoiceofCollocation/InterpolationPoints. Anaturalchoiceofthesamplingpointsisthe tensor-productofone-dimensionalsets,whichise˚cientsforlow-dimensionalrandomspaces. However,inhigh-dimensionalmultivariatecase,where N > 6 ,thetensor-productinterpolation operatorsarecomputationallyexpensiveduetotheincreasingnestedsummationloops.Inaddition, thetotalnumberofsamplingpointsgrowsrapidlybyincreaseofdimensionby J N ,where J is thenumberofpointsineachdirection. Anotherchoicethatprovidesanalternativetothemorecostlyfulltensorproductruleisthe isotropicSmolyaksparsegridoperator A ¹ w ; Nº [129,151]withtwoinputparametersdimension size N andthelevelofgrid w .TheSmolyakalgorithmsigni˝cantlyreducesthetotalnumberof samplingpoints;seeFig.7.1forcomparisonof A ¹ 2 ; 2 º , A ¹ 4 ; 2 º ,and A ¹ 6 ; 2 º withfulltensorproduct ruleforatwo-dimensionalrandomspaces.Thetotalnumberofsamplingpointsforeachcaseis 170 (a) (b) (c) (d) Figure7.1:Illustrationofsamplingnodalpointsintwo-dimensionalrandomspace,usingSmolyak sparsegridgenerator(a) A ¹ 2 ; 2 º ,(b) A ¹ 4 ; 2 º ,(c) A ¹ 6 ; 2 º ;and(d)fulltensorproductrulewith 50pointsineachdirection.Thetotalnumberofpointsineachcaseis,25,161,837,and2500, respectively. Spacedimensionality Fulltensorproduct Smolyaksparsegridgenerator A ¹ w ; Nº N w = 2 w = 4 w = 6 w = 8 w = 10 2 10 2 25 161 837 4105 19469 5 10 5 131 3376 45458 440953 3542465 15 10 15 1066 197176 15480304 25 10 25 2901 1445975 55 10 55 87780 Table7.1:Thetotalnumberofnodalpointsinrandomspacesampling,usingSmolyaksparsegrid generatorandfulltensorproductwith10pointsineachdirection. alsolistedinTab.7.1.Moreresearchhasalsobeendevotedtotheanalysisandconstructionof Smolyaksparsegrids[24,130,131,175]. 7.3ForwardDeterministicSolver Foreachrealizationofrandomvariablesintheemployedsamplingmethods,thestochastic modelyieldsadeterministicFPDE,lefttobesolvedinthephysicaldomain.Werecallthatfor every ˘ i ; i = 1 ; 2 ; inSFPDEEqn.(7.13),thedeterministicproblemisrecastas: 0 D t u ¹ t ; x º d Õ j = 1 k j a j D j x j + x j D j b j u ¹ t ; x º = h ¹ t ; x º + f ¹ t º ; (7.19) subjecttothesameinitial/boundaryconditionsasEqn.(7.2)andEqn.(7.3).Inthesequel,we developaPetrov-Galerkinspectralmethodtonumericallysolvethedeterministicprobleminthe physicaldomain.Wealsoshowthewellposednessofdeterministicprobleminaweaksenseand 171 furtherinvestigatethestabilityofproposednumericalscheme. 7.3.1MathematicalFramework Wede˝netheusefulfunctionalspacesandtheirassociatednorms[87,99].By H ˙ ¹ R º = u ¹ t ºj u 2 L 2 ¹ R º ; ¹ 1 + j ! j 2 º ˙ 2 F¹ u º¹ ! º2 L 2 ¹ R º , ˙ 0 ,wedenotethefractionalSobolevspaceon R , endowedwithnorm k u k H ˙ R = k¹ 1 + j ! j 2 º ˙ 2 F¹ u º¹ ! ºk L 2 ¹ R º ,where F¹ u º representstheFourier transformof u .Subsequently,wedenoteby H ˙ ¹ º = u 2 L 2 ¹ ºj9 ~ u 2 H ˙ ¹ R º s : t : ~ u j = u , ˙ 0 ,thefractionalSobolevspaceonany˝niteclosedinterval,e.g. = ¹ a ; b º ,withnorm k u k H ˙ ¹ º = inf ~ u 2 H ˙ R ; ~ u j = u k ~ u k H ˙ ¹ R º .Wede˝nethefollowingusefulnormsas: kk l H ˙ ¹ º = k a D ˙ x ¹ºk 2 L 2 ¹ º + kk 2 L 2 ¹ º 1 2 ; kk r H ˙ ¹ º = k x D ˙ b ¹ºk 2 L 2 ¹ º + kk 2 L 2 ¹ º 1 2 ; kk c H ˙ ¹ º = k x D ˙ b ¹ºk 2 L 2 ¹ º + k a D ˙ x ¹ºk 2 L 2 ¹ º + kk 2 L 2 ¹ º 1 2 ; wheretheequivalenceof kk l H ˙ ¹ º and kk r H ˙ ¹ º areshownin[53,99,100]. Lemma7.3.1. Let ˙ 0 and ˙ , n 1 2 .Then,thenorms kk l H ˙ ¹ º and kk r H ˙ ¹ º areequivalent to kk c H ˙ ¹ º . Wealsode˝ne C 1 0 ¹ º asthespaceofsmoothfunctionswithcompactsupportin ¹ a ; b º .Wedenote by l H ˙ 0 ¹ º , r H ˙ 0 ¹ º ,and c H ˙ 0 ¹ º astheclosureof C 1 0 ¹ º withrespecttothenorms kk l H ˙ ¹ º , kk r H ˙ ¹ º ,and kk c H ˙ ¹ º .Itisshownin[53,100]thattheseSobolevspacesareequalandtheir seminormsarealsoequivalentto jj H ˙ ¹ º = a D ˙ x ¹º ; x D ˙ b ¹º 1 2 .Therefore,wecanprovethat ¹ a D ˙ x u ; x D ˙ b v º j u j l H ˙ ¹ º j v j r H ˙ ¹ º and ¹ x D ˙ b u ; a D ˙ x v º j u j r H ˙ ¹ º j v j l H ˙ ¹ º ,in which isapositiveconstant. Moreover,byletting 0 C 1 ¹ I º and C 1 0 ¹ I º bethespaceofsmoothfunctionswithcompactsupport in ¹ 0 ; T ¼ and » 0 ; T º ,respectively,wede˝ne l H s ¹ I º and r H s ¹ I º astheclosureof 0 C 1 ¹ I º and C 1 0 ¹ I º withrespecttothenorms kk l H s ¹ I º and kk r H s ¹ I º .Otherequivalentusefulsemi-normsassociated 172 with H s ¹ I º arealsointroducedin[53,99],as jj l H s ¹ I º = k 0 D s t ¹ºk L 2 ¹ I º , jj r H s ¹ I º = k t D s T ¹ºk L 2 ¹ I º , jj H s ¹ I º = 0 D s t ¹º ; t D s T ¹º I 1 2 ,where jj H s ¹ I º jj 1 2 l H s ¹ I º jj 1 2 r H s ¹ I º . Borrowingde˝nitionsfrom[145],wede˝nethefollowingspaces,whichweuselaterin constructionofcorrespondingsolutionandtestspacesofourproblem.Thus,byletting 1 = ¹ a 1 ; b 1 º , j = ¹ a j ; b j º j 1 for j = 2 ; ; d ,wede˝ne X 1 = H 1 2 0 ¹ 1 º ,whichisassociatedwith thenorm kk c H 1 2 ¹ 1 º ,andaccordingly, X j ; j = 2 ; ; d as X 2 = H 2 2 0 ¹ a 2 ; b 2 º ; L 2 ¹ 1 º \ L 2 ¹¹ a 2 ; b 2 º ; X 1 º ; (7.20) : : : X d = H d 2 0 ¹ a d ; b d º ; L 2 ¹ d 1 º \ L 2 ¹¹ a d ; b d º ; X d 1 º ; (7.21) associatedwithnorms kk X j = ˆ kk 2 H j 2 0 ¹ a j ; b j º ; L 2 ¹ j 1 º + kk 2 L 2 ¹ a j ; b j º ; X j 1 ˙ 1 2 ; for j = 2 ; 3 ; ; d . Lemma7.3.2. Let j 0 and j , n 1 2 .Then,for j = 1 ; 2 ; ; d kk X j ˆ j Õ i = 1 k x i D i š 2 b i ¹ºk 2 L 2 ¹ j º + k a i D i š 2 x i ¹ºk 2 L 2 ¹ j º + kk 2 L 2 ¹ j º ˙ 1 2 : SolutionandTestSpaces Wede˝nethespace" U andtspace" V ,respectively,as U = l 0 H 2 I ; L 2 ¹ d º \ L 2 ¹ I ; X d º ; V = r 0 H 2 I ; L 2 ¹ d º \ L 2 ¹ I ; X d º ; (7.22) 173 endowedwithnorms k u k U = n k u k 2 l H 2 ¹ I ; L 2 ¹ d ºº + k u k 2 L 2 ¹ I ; X d º o 1 2 ; k v k V = n k v k 2 r H 2 ¹ I ; L 2 ¹ d ºº + k v k 2 L 2 ¹ I ; X d º o 1 2 ; (7.23) where I = » 0 ; T ¼ ,and l 0 H 2 I ; L 2 ¹ d º = n u k u ¹ t ; ºk L 2 ¹ d º 2 H 2 ¹ I º ; u j t = 0 = u j x = a j = u j x = b j = 0 o ; r 0 H 2 I ; L 2 ¹ d º = n v k v ¹ t ; ºk L 2 ¹ d º 2 H 2 ¹ I º ; v j t = T = v j x = a j = v j x = b j = 0 o ; equippedwithnorms k u k l H 2 ¹ I ; L 2 ¹ d ºº and k u k r H 2 ¹ I ; L 2 ¹ d ºº ,respectively.Wecanshowthat thesenormstakethefollowingforms k u k l H 2 ¹ I ; L 2 ¹ d ºº = k u ¹ t ; ºk L 2 ¹ d º l H 2 ¹ I º = k 0 D 2 t ¹ u ºk 2 L 2 ¹ º + k u k 2 L 2 ¹ º 1 2 ; k u k r H 2 ¹ I ; L 2 ¹ d ºº = k u ¹ t ; ºk L 2 ¹ d º r H 2 ¹ I º = k t D 2 T ¹ u ºk 2 L 2 ¹ º + k u k 2 L 2 ¹ º 1 2 : (7.24) Also,usingLemma7.3.2,wecanshowthat k u k L 2 ¹ I ; X d º = k u ¹ t ;: ºk X d L 2 ¹ I º (7.25) = n k u k 2 L 2 ¹ º + d Õ j = 1 k x j D j 2 b j ¹ u ºk 2 L 2 ¹ º + k a j D j 2 x j ¹ u ºk 2 L 2 ¹ º o 1 2 : 174 Therefore,Eqn.(7.23)canbewrittenas k u k U = n k u k 2 L 2 ¹ º + k 0 D 2 t ¹ u ºk 2 L 2 ¹ º + d Õ j = 1 k x j D j 2 b j ¹ u ºk 2 L 2 ¹ º + k a j D j 2 x j ¹ u ºk 2 L 2 ¹ º o 1 2 ; (7.26) k v k V = n k v k 2 L 2 ¹ º + k t D 2 T ¹ v ºk 2 L 2 ¹ º + d Õ j = 1 k x j D j 2 b j ¹ v ºk 2 L 2 ¹ º + k a j D j 2 x j ¹ v ºk 2 L 2 ¹ º o 1 2 : (7.27) 7.3.2WeakFormulation Thefollowinglemmashelpusobtaintheweakformulationofdeterministicprobleminthephysical domainandconstructthenumericalscheme. Lemma7.3.3. [99]:Forall 2¹ 0 ; 1 º ,if u 2 H 1 ¹» 0 ; T ¼º suchthat u ¹ 0 º = 0 ,and v 2 H š 2 ¹» 0 ; T ¼º , then ¹ 0 D t u ; v º = ¹ 0 D š 2 t u ; t D š 2 T v º ,where ¹ ; º representsthestandardinnerproductin = » 0 ; T ¼ . Lemma7.3.4. [87]:Let 1 << 2 , a and b bearbitrary˝niteorin˝niterealnumbers.Assume u 2 H ¹ a ; b º suchthat u ¹ a º = 0 ,also x D š 2 b v isintegrablein ¹ a ; b º suchthat v ¹ b º = 0 .Then, ¹ a D x u ; v º = ¹ a D š 2 x u ; x D š 2 b v º . Lemma7.3.5. Let 1 < j < 2 for j = 1 ; 2 ; ; d ,and u ; v 2X d .Then, a j D j x j u ; v d = a j D j 2 x j u ; x j D j 2 b j v d ; x j D j b j u ; v d = x j D j 2 b j u ; a j D j 2 x j v d : ForanyrealizationofEqn.(7.13),weobtaintheweaksystem,i.e.thevariationalformof thedeterministiccounterpartoftheproblem,subjecttothegiveninitial/boundaryconditions,by multiplyingtheequationwithpropertestfunctionsandintegrateoverthewholecomputational 175 domain D .UsingLemmas7.3.3-7.3.5,thebilinearformcanbewrittenas a ¹ u ; v º = ¹ 0 D 2 t u ; t D 2 T v º D (7.28) d Õ j = 1 k j h ¹ a j D j 2 x j u ; x j D j 2 b j v º D + ¹ x j D j 2 b j u ; a j D j 2 x j v º D i ; andthus,byletting U and V bethepropersolution/testspaces,theproblemreadsas:˝nd u 2 U suchthat a ¹ u ; v º = ¹ f ; v º D ; 8 v 2 V ; (7.29) wheref = h ¹ t ; x º + f ¹ t º . 7.3.3Petrov-GalerkinSpectralMethod Wede˝nethefollowing˝nitedimensionalsolutionandtestspaces.WeemployLegendrepolyno- mials ˚ m j ¹ ˘ º ; j = 1 ; 2 ; ; d ,andJacobipoly-fractonomialof˝rstkind ˝ n ¹ º [183,186],asthe spatialandtemporalbases,respectively,givenintheircorrespondingstandarddomainas ˚ m j ¹ ˘ º = ˙ m j P m j + 1 ¹ ˘ º P m j 1 ¹ ˘ º ; (7.30) ˝ n ¹ º = ˙ n ¹ 1 º P ˝ n ¹ º = ˙ n ¹ 1 + º ˝ P ˝;˝ n 1 ¹ º ; (7.31) inwhich ˘ 2 1 ; 1 ¼ , m j = 1 ; 2 ; , ˙ m j = 2 + 1 º m j , 2 1 ; 1 ¼ , n = 1 ; 2 ; ,and ˙ n = 2 + 1 º n . Therefore,byperforminga˚nemappings = 2 t T 1 and ˘ = 2 x a j b j a j 1 fromthecomputational domaintothestandarddomain,weconstructthesolutionspace U N as U N = span n ˝ n ¹ t º d Ö j = 1 ˚ m j ˘ ¹ x j º (7.32) : n = 1 ; 2 ; ; N ; m j = 1 ; 2 ; ; M j o : Wenotethatthechoiceoftemporalandspatialbasisfunctionsnaturallysatisfytheinitialand boundaryconditions,respectively.Theparameter ˝ inthetemporalbasisfunctionsplaysaroleof ˝netunningparameter,whichcanbechosenproperlytocapturethesingularityofexactsolution. 176 Moreover,weemployLegendrepolynomials r j ¹ ˘ º ; j = 1 ; 2 ; ; d ,andJacobipoly-fractonomial ofsecondkind ˝ k ¹ º ,asthespatialandtemporaltestfunctions,respectively,givenintheircorre- spondingstandarddomainas r j ¹ ˘ º = e ˙ r j P r j + 1 ¹ ˘ º P r j 1 ¹ ˘ º ; (7.33) ˝ k ¹ º = e ˙ k ¹ 2 º P ˝ k ¹ º = e ˙ k ¹ 1 º ˝ P ˝; ˝ k 1 ¹ º ; (7.34) where ˘ 2 1 ; 1 ¼ , r j = 1 ; 2 ; , e ˙ r j = 2 1 º r j + 1 , 2 1 ; 1 ¼ , k = 1 ; 2 ; ,and e ˙ k = 2 1 º k + 1 . Therefore,bysimilara˚nemappingweconstructthetestspace V N as V N = span n ˝ k ¹ t º d Ö j = 1 r j ˘ j ¹ x j º (7.35) : k = 1 ; 2 ; ; N ; r j = 1 ; 2 ; ; M j o : Thus,since U N ˆ U and V N ˆ V ,theproblemsEqn.(7.29)readas:˝nd u N 2 U N suchthat a h ¹ u N ; v N º = l ¹ v N º ; 8 v N 2 V N ; (7.36) where l ¹ v N º = ¹ f ; v N º .Thediscretebilinearform a h ¹ u N ; v N º canbewrittenas a h ¹ u N ; v N º = ¹ 0 D 2 t u N ; t D 2 T v N º D (7.37) d Õ j = 1 k j h ¹ a j D j 2 x j u N ; x j D j 2 b j v N º D + ¹ x j D j 2 b j u N ; a j D j 2 x j v N º D i : Weexpandtheapproximatesolution u N 2 U N ,satisfyingthediscretebilinearformEqn.(7.37),in thefollowingform u N ¹ t ; x º = (7.38) N Õ n = 1 M 1 Õ m 1 = 1 M d Õ m d = 1 ^ u n ; m 1 ; ; m d h ˝ n ¹ t º d Ö j = 1 ˚ m j ˘ ¹ x j º i ; 177 andobtainthecorrespondingLyapunovsystembysubstitutingEqn.(7.38)intoEqn.(7.37)by choosing v N ¹ t ; x º = ˝ k ¹ t º d Ö j = 1 r j ˘ j ¹ x j º ; k = 1 ; 2 ;:::; N ; r j = 1 ; 2 ;:::; M j : Therefore, h S T M 1 M 2 M d (7.39) + d Õ j = 1 M T M 1 M j 1 S j M j + 1 M d i U = F ; inwhich representstheKroneckerproduct, F denotesthemulti-dimensionalloadmatrixwhose entriesaregivenas F k ; r 1 ; ; r d = ¹ D f ¹ t ; x º ˝ k ¹ t º d Ö j = 1 r j ˘ j ¹ x j º d D ; (7.40) and U isthematrixofunknowncoe˚cients.Thematrices S T and M T denotethetemporalsti˙ness andmassmatrices,respectively;andthematrices S j and M j denotethespatialsti˙nessandmass matrices,respectively.Weobtaintheentriesofspatialmassmatrix M j analyticallyandemploy properquadraturerulestoaccuratelycomputetheentriesofothermatrices S T , M T and S j . Wenotethatthechoicesofbasis/testfunctions,employedindevelopingthePGschemeleadsto symmetricmassandsti˙nessmatrices,providingusefulpropertiestofurtherdevelopafastsolver. ThefollowingTheorem7.3.6providesauni˝edfastsolver,developedintermsofthegeneralized eigensolutionsinordertoobtainaclosed-formsolutiontotheLyapunovsystemEqn.(7.39). Theorem7.3.6 (Uni˝edFastFPDESolver[145]) . Let f ® e m j ; m j g M j m j = 1 bethesetofgeneraleigen- solutionsofthespatialsti˙nessmatrix S j withrespecttothemassmatrix M j .Moreover,let f ® e ˝ n ; ˝ n g N n = 1 bethesetofgeneraleigen-solutionsofthetemporalmassmatrix M T withrespectto thesti˙nessmatrix S T .Then,thematrixofunknowncoe˚cients U isexplicitlyobtainedas U = N Õ n = 1 M 1 Õ m 1 = 1 M d Õ m d = 1 n ; m 1 ; ; m d ® e ˝ n ® e m 1 ® e m d ; (7.41) 178 where n ; m 1 ; ; m d isgivenby n ; m 1 ; ; m d = ¹ ® e ˝ n ® e m 1 ® e m d º F h ¹ ® e ˝ T n S T ® e ˝ n º Î d j = 1 ¹ ® e T m j M j ® e m j º i n ; m 1 ; ; m d ; (7.42) inwhichthenumeratorrepresentsthestandardmulti-dimensionalinnerproduct,and n ; m 1 ; ; m d isobtainedintermsoftheeigenvaluesofallmassmatricesas n ; m 1 ; ; m d = h 1 + ˝ n Í d j = 1 ¹ m j º i : 7.3.4StabilityAnalysis Weshowthewell-posednessofdeterministicproblemandprovethestabilityofproposedPG scheme. Lemma7.3.7. Let 2¹ 0 ; 1 º , = I d ,and u 2 l 0 H š 2 ¹ I ; L 2 ¹ d ºº .Then, 0 D š 2 t u ; t D š 2 T v k u k l H š 2 ¹ I ; L 2 ¹ d ºº k v k r H š 2 ¹ I ; L 2 ¹ d ºº ; 8 v 2 r 0 H š 2 ¹ I ; L 2 ¹ d ºº : Moreover, j a d D d š 2 x d u ; x d D d š 2 b d v d j (7.43) j u j c H d š 2 ¹ a d ; b d º ; L 2 ¹ d 1 º j v j c H d š 2 ¹ a d ; b d º ; L 2 ¹ d 1 º ; and j x d D d š 2 b d u ; a d D d š 2 x d v d j (7.44) j u j c H d š 2 ¹ a d ; b d º ; L 2 ¹ d 1 º j v j c H d š 2 ¹ a d ; b d º ; L 2 ¹ d 1 º : Lemma7.3.8 (Continuity) . ThebilinearformEqn. (7.28) iscontinuous,i.e., 8 u 2 U ; 9 > 0 ; s.t. j a ¹ u ; v ºj k u k U k v k V ; 8 v 2 V : (7.45) Proof. TheproofdirectlyconcludesfromEqn.(7.43)andLemma7.3.7. 179 Theorem7.3.9 (Stability) . Thefollowinginf-supconditionholdsforthebilinearformEqn. (7.28) , i.e., inf u , 0 2 U sup v , 0 2 V j a ¹ u ; v ºj k v k V k u k U > 0 ; (7.46) where = I d and sup u 2 U j a ¹ u ; v ºj > 0 . Theorem7.3.10 (well-posedness) . Forall 0 << 1 , , 1 ,and 1 < j < 2 ,and j = 1 ; ; d , thereexistsauniquesolutiontoEqn. (7.29) ,continuouslydependenton f ,where f belongstothe dualspaceof U . Proof. Lemmas7.3.8(continuity)and7.3.9(stability)yieldthewell-posednessofweakform Eqn.(7.29)in(1+d)-dimensionduetothegeneralizedBabu²ka-Lax-Milgramtheorem. Sincethede˝nedbasisandtestspacesareHilbertspaces,and U N ˆ U and V N ˆ V ,wecan provethatthedevelopedPetrov-Gelerkinspectralmethodisstableandthefollowingcondition holds inf u N , 0 2 U N sup v , 0 2 V N j a ¹ u N ; v N ºj k v N k V k u N k U > 0 ; (7.47) with > 0 andindependentof N ,where sup u N 2 U N j a ¹ u N ; v N ºj > 0 . 7.4NumericalResults Weinvestigatetheperformanceofdevelopednumericalmethodsbyconsideringcoupleof numericalsimulations.WecompareMCSandPCMinrandomspacediscretizationwhileusing PGmethodinphysicaldomain.Wenotethatbyseveralnumericalexamples,wemakesurethat thedevelopedPGmethodisstableandaccurateinsolvingeachdeterministicproblem;theresults arenotprovidedhere. 7.4.1Low-DimensionalRandomInputs Asthe˝rstcase,weconsiderastochasticfractionalinitialvalueproblem(IVP)withrandom fractionalindexbylettingthedi˙usioncoe˚cienttobezero,andalsoignoringtheadditional 180 Figure7.2: L 2 -normconvergencerateofMCMandPCMforstochasticfractionalIVPEqn.(7.48). randominputandonlytaking h ¹ t º astheexternalforcingterm.Therefore,weobtain 0 D ¹ ˘ º t u ¹ t ; ˘ º = h ¹ t º ; (7.48) subjecttozeroinitialcondition,where u ¹ t ;˘ º : ¹ 0 ; T ¼ ! R .Welet u ext ¹ t º = 2 t 3 + 2 , h ¹ t º = 0 D ¹ ˘ º t u ext ¹ t º foreachrealizationof .Inthiscase,bychoosingthetunningparameter ˝ inthetemporalbasisfunctiontobe 2 ,wecane˚cientlyemployPGnumericalschemeandalso obtaintheexactexpectationbyrenderingtheexactsolutiontoberandomwithsimilardistribution astherandomfractionalindex.Fig.7.2showsthe L 2 -normconvergencerateofMCSandPCMin comparisonofsolutionexpectationwith E ext » u ¼ = E » u ext ¼ .Theresultscon˝rmsconvergesrateof 0 : 5 forMCS,whileinPCM,thestatisticsofsolutionconvergesaccuratelyveryfast,usingonlyfew numbersofrealizations.Inthisexample,byignoringtheadditionalrandominputtothesystem, wetaketheadvantageofhavingtheexactrandomsolutiontobeavailable. Asanotherexample,wealsoconsiderEqn.(7.48)withadditionalrandominput,expandedby KLexpansionwith M = 4 ,as: 0 D ¹ ˘ º t u ¹ t ; ˘ º = h ¹ t º + M Õ k = 1 a k sin 2 k ˇ t T ˘ k ; (7.49) withtwocases h ¹ t º = t 2 and h ¹ t º = sin ¹ ˇ t º .Fig.7.3showsthemeanvalueandvarianceofsolution for 10 4 samplingofMCScomparedto 625 realizationsinPCM. Moreover,weconsider(1+1)-Done-sidedSFPDEgiveninEqn.(7.13),where d = 1 andthe di˙usioncoe˚cientis k l .Weignoretheadditionalrandominputandconsider h ¹ t ; x º astheonly 181 Figure7.3:ExpectationofsolutiontoEqn.(7.49)withuncertainty(standarddeviation)bounds, employingMCSandPCMfor(left) h ¹ t º = t 2 and(right) h ¹ t º = sin ¹ ˇ t º . Figure7.4: L 2 -normconvergencerateofMCMandPCMforSFPDEEqn.(7.50). externalforcingterm.Therefore,weobtain 0 D ¹ ˘ 1 º t u ¹ t ; x ; ˘ º k l 1 D ¹ ˘ 2 º x u ¹ t ; x ; ˘ º = h ¹ t ; x º ; (7.50) subjecttozeroinitial/boundaryconditions,where u ¹ t ; x ; ˘ º : ¹ 0 ; T ¼ 1 ; 1 º ! R ,andtheonly randomvariablesarethefractionalindices and .Welet u ext ¹ t ; x º = t 3 + ˝ ¹ 1 + x º 3 + 1 2 ¹ 1 + x º 4 + , andchoose ˝ = š 2 and = š 2 .Foreachrealizationof and ,weobtaintheforcefunction h ¹ t ; x º bysubstitutingthecorresponding u ext toEqn.(7.50).De˝ning E ext » u ¼ = E » u ext ¼ ,Fig.7.4 showsthe L 2 -normconvergenceofsolutionexpectationascomparedtotheexactexpectation.We observethatPCMconvergesaccuratelywithonlyfewnumberofrealizations. Consideringadditionalrandominput,expandedbyKLexpansionwith M = 4 ,theproblemcan 182 Figure7.5:ExpectationofsolutiontoEqn.(7.51),employingMCSandPCMat t = 0 : 125 ; 0 : 625 ; 1 . berecastas 0 D ¹ ˘ º t u ¹ t ; x ; ˘ º k l 1 D ¹ ˘ º x u ¹ t ; x ; ˘ º (7.51) = h ¹ t ; x º + M Õ k = 1 a k sin 2 k ˇ t T ˘ k subjecttozeroinitial/boundaryconditions.Fig.7.5showsthemeanvalueandvarianceofsolution forMCSandPCMatdi˙erenttimes. Remark7.4.1. Wenotethatgenerallyuseofthesparsegridoperatorsinobtainingsolution statisticsismoree˙ectivewhendimensionoftherandomspaceishigherthan6.Thus,inthe numericalexamplesforlow-dimensionalrandominputs,weemploytheeasy-to-implementtensor productnodalsets. 7.4.2Moderate-toHigh-DimensionalRandomInputs WerendertheproblemwithhighernumberoftermsinKLexpansionofrandominputsinEqn.(7.51) bychoosing M = 10 and M = 20 .Thisyieldsthedimensionofrandomspace N = 12 and N = 22 , respectively.AsmentionedinRemark7.4.1,inthecaseofhigh-dimensionalrandomspace constructinggridbasedontensorproductruleresultsinveryexpensivecomputationofsolution statisticsduetoexhaustiveincreaseofforwardsolverinstruction.Table7.1showsthecomparison betweendi˙erentlevelofSmolyakalgorithmandtensorproductrule.Therefore,toobtainthe 183 solutionstatistics,weemploytheSmolyaksparsegridgeneratorinthedevelopedPCM.Foreach casesofKLexpansion,wegeneratethesparsegridontwolevels w = 1 and w = 2 ,i.e. A ¹ 1 ; 12 º , A ¹ 2 ; 12 º , A ¹ 1 ; 22 º ,and A ¹ 2 ; 22 º ,whereweletthehigherresolutioncasebeabenchmarkvalueto thesolutionstatistics,basedonwhichwecomputeandnormalizetheerror.Weobservethatfor bothcases N = 12 and N = 22 ,thenormalizederrorincomputingtheexpectationandstandard deviationofsolutionareoforders O¹ 10 7 º and O¹ 10 3 º ,respectively. 184 CHAPTER8 NONLINEARVIBRATIONOFFRACTIONALVISCOELASTICCANTILEVERBEAM: APPLICATIONTOSTRUCTURALHEALTHMONITORING 8.1Background Weinvestigatethenonlinearvibrationofaviscoelasticcantileverbeamwithfractionalconstitu- tiverelation,subjecttobaseexcitation.Weconsiderthegeneralformofdistributed-orderfractional di˙erentialequationanduseextendedHamilton'sprincipletoderivethegoverningequationsof motionforfractionalKelvin-Voigtviscoelasticmodel,whichisthensolvedviaaspectraldecompo- sitioninspace.BydirectnumericalintegrationofresultingtemporalfractionalODE,weobserve ananomalouspower-lawdecayrateofamplitudeinthelinearizedmodel.Thenonlinearequation issolvedbyperturbationanalysis,wherewereplacetheexpensivenumericaltimeintegrationwith acubicalgebraicequationtosolveforfrequencyresponseofthesystem.Wereportthesuper sensitivityofresponseamplitudetothefractionalelementparametersatfreevibration,andbifur- cationinsteady-stateamplitudeatprimaryresonance.Wefurtherusetheobservedvibration-based featuresofsystemresponsefordi˙erentvaluesoffractionalderivativeordertodevelopaparameter estimationframework,whichcanbeusedtoassesstheheathofconsideredbeambyassuminga thresholdinthemodelparameters. 8.2MathematicalFormulation Weformulatethemathematicalmodeloftheconsideredphysicalsystem.Wediscussthe mainassumptionsandtheorems,usedtoderivetheequationofmotion.Weemployspectral decompositiontodiscretizetheproblemandfurtherusetheperturbationmethodtosolvethe resultingnonlinearequations. 185 8.2.1NonlinearIn-PlaneVibrationofaVisco-ElasticCantileverBeam Weconsiderthenonlinearresponseofaslenderisotropicvisco-elasticcantileverbeamwithlumped mass M atthetip,subjecttoharmonictransversebaseexcitation, V b .WeusethenonlinearEuler- Bernoullibeamtheorytoobtainthegoverningequations,wherethegeometricnonlinearitiesina cantileverbeamwithsymmetriccrosssectionisincludedintheequationsofmotion.Weassume thatthebeamisidealizedasaninextensionalone,i.e.,stretchingoftheneutralaxisisinsigni˝cant, andthee˙ectsofwarpingandsheardeformationareignored.Wealsoassumethattheconsidered slenderbeamwithsymmetricalcrosssectionundergoespurelyplanar˛exuralvibration.Therefore, weconsiderthein-planetransversevibrationofthebeamandreducetheproblemto1-dimension. Fig.8.1showsthelateraldeformationofthecantileverbeamwithcrosssectionarea A andmass perunitlength m ,wheretheaxialdisplacementalonglengthofbeamandthelateraldisplacement aredenotedby u ¹ s ; t º and v ¹ s ; t º ,respectively.Asthebeamdeforms,welettheinertialcoordinate system ¹ x ; y ; z º rotatesaboutthe z axisbytherotationangle ¹ s ; t º tothecoordinatesystem ¹ ˘;; º , where 2 6 6 6 6 6 6 6 6 4 e ˘ e e 3 7 7 7 7 7 7 7 7 5 = © « cos ¹ º sin ¹ º 0 sin ¹ º cos ¹ º 0 001 ª ® ® ® ® ® ¬ 2 6 6 6 6 6 6 6 6 4 e x e y e z 3 7 7 7 7 7 7 7 7 5 ; and e i istheunitvectorof i coordinate.Thus,angularvelocityandcurvatureofthebeamatany pointalongthelengthofthebeam s andanytime t canbewrittenas ! ¹ s ; t º = Û e z ; ˆ ¹ s ; t º = 0 e z ; (8.1) whereoverdotandprimedenotethederivativewithrespecttotimeandspace,respectively. Thetotaldisplacementandvelocityofanarbitrarypointalongthe y axistakestheform: r = ¹ u sin ¹ ºº e x + ¹ v + V b + cos ¹ ºº e y ; (8.2) Û r = ¹ Û u Û cos ¹ ºº e x + ¹ Û v + Û V b Û sin ¹ ºº e y : (8.3) Wealsoletanarbitraryelement CD ofthebeam'sneutralaxis,whichisoflength ds andlocated atadistance s fromtheorigin O ,movetotheelement C D ,seeFig.8.3.Thedisplacement 186 Figure8.1:In-planelateraldeformationofaslenderisotropiccantileverbeam. u ¹ s ; t º and v ¹ s ; t º aretheaxialandlateraldisplacements,and ¹ s ; t º istherotationangleabout z axis. Figure8.2:Detailedin-planelateraldeformationofaslenderisotropiccantileverbeam.The˝gure showstotaldeformationofanarbitrarypoint(theredpoint)asthebeamundergoesdeformation. Thisdeformationiscomprisedoftheaxialdisplacementofthebeam u ,thelateraldisplacementof beaminadditiontothebasemotion v + V b ,andthedisplacementduetorotation . componentsofpoints C and D aredenotedby ¹ u ; v º and ¹ u + du ; v + d v º ,respectively.Thestrain e ¹ s ; t º atthearbitrarypoint C isthengivenby e = ds ds ds = p ¹ ds + du º 2 + d v 2 ds ds = q ¹ 1 + u 0 º 2 + v 0 2 1 : (8.4) Theinextensionalityconstraint,i.e. e = 0 ,becomes 1 + u 0 = ¹ 1 v 0 2 º 1 š 2 : (8.5) 187 Figure8.3:Deformationofanarbitraryelementofthebeam. CD extends,traverses,androtatesto C D . Moreover,basedontheassumptionofnotransversesheardeformationandusing(8.5),wehave = tan 1 v 0 1 + u 0 = tan 1 v 0 ¹ 1 v 0 2 º 1 š 2 : (8.6) Usingtheexpansion tan 1 ¹ x º = x 1 3 x 3 + ,thecurvaturecanbeapproximateduptocubicterm as = v 0 ¹ 1 v 0 2 º 1 š 2 1 3 v 0 3 ¹ 1 v 0 2 º 3 š 2 + (8.7) ' v 0 ¹ 1 + 1 2 v 0 2 º 1 3 v 0 3 ' v 0 + 1 6 v 0 3 Therefore,theangularvelocityandcurvatureofthebeam,i.e. Û and 0 ,respectively,canbe approximatedas: Û ' Û v 0 + 1 2 Û v 0 v 0 2 ' Û v 0 ¹ 1 + 1 2 v 0 2 º ; (8.8) 0 ' v 00 + 1 2 v 00 v 0 2 ' v 00 ¹ 1 + 1 2 v 0 2 º : (8.9) BytheEuler-Bernoullibeamassumptionsaslender,no-transverse-shearwithnostrainsintheplane ofcrosssectionalplane,thestrain-curvaturerelationtakestheform " ¹ s ; t º = 0 ¹ s ; t º (8.10) 8.2.2Viscoelasticity:BoltzmannSuperpositionPrinciple Manyexperimentalobservationsintheliteratureshowviscoelasticbehaviorofmaterialindi˙erent environmental/boundaryconditions,meaningthattheydonotbehavepurelyelasticandthereexists 188 someinternaldissipationmechanism.Insuchcases,theresultingstresshasamemorydepending onthevelocityofallearlierdeformations,whichcanbedescribedbytheBoltzmannsuperposition principle.Whenthespecimenisunderloading,thematerialinstantaneouslyreactselastically andthen,immediatelystartstorelax,wheredissipationtakesplace.Thus,asastepincreasein elongation(fromthestretch = 1 tosome )isimposed,thedevelopedstressinthematerialwill beafunctionoftimeandthestretch: K ¹ ; t º = G ¹ t º ˙ ¹ e º ¹ º ; (8.11) where G ¹ t º isthereducedrelaxationfunctionand ˙ ¹ e º istheelasticresponse(inabsenceof anyviscosity). ˙ ¹ e º canalsobeinterpretedastensilestressresponseinasu˚cientlyhighrate loadingexperiment.TheBoltzmannsuperpositionprinciplestatesthatthestressesfromdi˙erent smalldeformationsareadditive,meaningthatthetotaltensilestressofthespecimenattime t is obtainedfromthesuperpositionofin˝nitesimalchangesinstretchatsomepriortime ˝ j ,givenas G ¹ t ˝ j º @˙ ¹ e º » ¹ ˝ j º¼ @ ¹ ˝ j º .Therefore, ˙ ¹ t º = Õ ˝ j < t G ¹ t ˝ j º @˙ ¹ e º » ¹ ˝ j º¼ @ ¹ ˝ j º ˝ j ˝ j ; (8.12) whereinthelimitingcase ˝ j ! 0 givestheintegralformoftheequationas ˙ ¹ t º = ¹ t G ¹ t ˝ º @˙ ¹ e º » ¹ ˝ º¼ @ @ @˝ d ˝ = ¹ t G ¹ t ˝ º Û ˙ ¹ e º d ˝: (8.13) ExponentialRelaxation,ClassicalModels: Therelaxationfunction G ¹ t º istraditionallyanalyzed intothesummationofexponentialfunctionswithdi˙erentexponentsandconstantsas G ¹ t º = Í C i e t š ˝ i Í C i : (8.14) Forthesimplecaseofasingleexponentialterm(Maxwellmodel),wehave G ¹ t º = e t š ˝ .Thus,in thecaseofzeroinitialstrainwehave ˙ ¹ t º = ¹ t 0 e t ~ t ºš ˝ E Û " d ~ t ; (8.15) 189 Figure8.4:Classicalvisco-elasticmodelsasacombinationofspring(purelyelastic)anddash-pot (purelyviscous)elements.Kelvin-Voigt(top)andMaxwell(bottom)rheologicalmodels. whichsolvestheinteger-orderdi˙erentialequation Û " = 1 E Û ˙ + 1 ˙ ,wheretherelaxationtimeconstant ˝ = š E ,isobtainedfromexperimentalobservations.TheMaxwellmodelisinfactacombination ofpurelyelasticandviscouselementsinseries,seeFig.8.4.Otherdi˙erentcombinationsof purelyelasticandviscouselementsinseriesandparallelgiverisetovariousrheologicalmodels withdistinctiveproperties,eachofwhichcanbeusedtomodeldi˙erenttypesofmaterial.The keyissueisthattheyrequirecomplicatedcombinationsofelasticandviscouselementsinorderto modelthecomplexhereditarybehaviorofmaterial,yettheycannotfullycaptureitasthebuilding blocksdonotre˛ectanymemorydependenceinthematerialresponse.Moreover,theyintroduce arelativelylargenumberofmodelparameters,whichadversetheconditionofill-posedinverse problemofmodel˝tting. Power-LawRelaxation,FractionalModels: Themechanicalstressappearedatthedeformation ofviscoelasticmaterialsdecreasesaspower-lawfunctionsintime,suggestingthatrelaxationof stressobeysapowerlawbehaviorandtherelaxationtimecannotbedescribedwithsingletime scaleanymore[116].Therefore,bylettingthekernelin(8.13)haveapower-lawform,thetensile stresstakestheformof ˙ ¹ t º = ¹ t g ¹ º ¹ t ˝ º E Û " d ˝ = E g ¹ º ¹ t Û " ¹ t ˝ º d ˝; (8.16) wheretheelasticresponse ˙ ¹ e º = E " .Ifwechoose g ¹ º = 1 ¹ 1 º ,thentheintegro-di˙erential operator(8.16)givestheLiouville-Weylfractionalderivative.Underthehypothesisofcausal histories,statingthattheviscoelasticbodyisquiescentforalltimepriortosomestartingpoint 190 t = 0 ,theequation(8.16)canbewrittenas ˙ ¹ t º = " ¹ 0 + º E g ¹ º t + E g ¹ º ¹ t 0 Û " ¹ t ˝ º d ˝; (8.17) = " ¹ 0 + º E g ¹ º t + E C 0 D t "; = RL 0 D t "; where C 0 D t and RL 0 D t aretheCaputoandRiemann-Liouvillefractionalderivatives.Theconsti- tutiveequation(8.17)introducestheScottBlairelement[115,116,138,160],whichcanbethough ofasaninterpolationbetweenapureelastic(spring)andapureviscous(dashpot)elements. Inamoregeneralsense,wherethematerialcontainaspectrumofpower-lawrelaxation,the singleorderfractionalconstitutivemodelcanbeextendedtothedistributed-orderone.Thus,we lettherelaxationfunction G ¹ t º in(8.13)notbeonlyasingleorderpower-lawasin(8.16),butrather bedistributedoverarange.Thisleadstoadistributedformofconstitutiveequationsexpressedas ¹ max min ¹ º 0 D t ˙ ¹ t º d = ¹ max min ¹ º 0 D t " ¹ t º d ; (8.18) where ¹ º and ¹ º aredistributionfunctionsthatcancon˝nethetheoreticalterminals min , max , min ,and max accordingtothephysicalrealizationofproblem.Bychoosingdi˙erent distributionfunctions ¹ º and ¹ º ,onecandesigndistinctiverheologicalmodelstogetdi˙erent typesofbehavior.Wenotethatifweletthedistributionfunctionsbedeltafunctions,thedistributed ordermodelbecomesthefollowingmulti-termmodel: 1 + p ˙ Õ k = 1 a k 0 D k t ! ˙ ¹ t º = c + p " Õ k = 1 b k 0 D k t ! " ¹ t º : (8.19) Here,welet ¹ º = ¹ º and ¹ º = E 1 ¹ º + E ¹ 0 º ,andthus,recoverthefractional Kelvin-Voigtmodelas ˙ ¹ t º = E 1 " ¹ t º + E RL 0 D t " ¹ t º ; (8.20) where 2¹ 0 ; 1 º .Sinceweonlyhaveonesinglederivativeorder ,wedropthesubscriptzerofor thesakeofsimpli˝cation. 191 8.2.3ExtendedHamilton'sPrinciple WederivetheequationsofmotionbyemployingtheextendedHamilton'sprinciple ¹ t 2 t 1 ¹ T W º dt = 0 ; where T and W arethevariationsofkineticenergyandtotalwork[119].Theonlysourceof externalinputtotheoursystemofinterestisthebaseexcitation,whichsuperposesbasevelocityto thebeamvelocity,andthuscontributestothekineticenergy.Hence,thetotalworkonlyincludes theinternalworkdonebytheinducedstressesanditsvariationcanbeexpressedinthegeneral formas[32] W = ¹ V ˙" d v ; (8.21) wheretheintegralistakenoverthewholesystemvolume V .Thevolumetricstress ˙ includes boththeconservativepart, ˙ c ,duetoelasticandthenon-conservativepart, ˙ nc ,duetoviscous deformation,wheretheformerconstitutesthepotentialenergyofthesystem.Therehasbeen someattemptsintheliteraturetoseparatetheconservativeandnon-conservativepartsoffractional constitutiveequationstode˝nethefreeenergyofthesystem[105].Wenotethatasthisseparation isnottrivialforsophisticatedfractionalconstitutiveequations,andaswedonotdealwithfree energyofoursystem,wewouldratherleavethetotalworknotseparatedandthusdonotcompute thepotentialenergyandworkdonebynon-conservativeforcesseparately.Intheconsidered cantileverbeamwithsymmetricconstantcrosssections,werecasttheintegral(8.21)as W = ¯ L 0 ¯ A ˙" dAds .Weobtainthevariationofstrainas " = 0 ,using(8.10).Therefore,by assumingtheconstitutiveequation(8.20),thevariationoftotalworkisexpressedas w = ¹ L 0 ¹ A E 1 0 E RL 0 D t 0 0 º dAds (8.22) = ¹ L 0 E 1 ¹ A 2 dA 0 + E ¹ A 2 dA RL 0 D t 0 0 ds = ¹ L 0 E 1 I 0 + E I RL 0 D t 0 0 ds 192 where I = ¯ A 2 dA .Byapproximation(8.9),wewritethevariationofcurvatureas 0 = ¹ 1 + 1 2 v 0 2 º v 00 + v 00 v 0 v 0 : (8.23) Therefore,thevariationoftotalenergybecomes w = ¹ L 0 E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º v 00 ds (8.24) + ¹ L 0 E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º v 00 v 0 v 0 ds Byexpandingthetermsandintegratingbyparts,wehave w = ¹ L 0 E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º 00 v ds (8.25) ¹ L 0 E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º v 00 v 0 0 v ds + E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º v 0 L 0 E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º 0 v L 0 + E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º v 00 v 0 v L 0 Theprescribedgeometryboundaryconditionsatthebaseofthebeam, s = 0 ,allowthevariation ofde˛ectionandits˝rstderivativetobezeroat s = 0 ,i.e. v ¹ 0 ; t º = v 0 ¹ 0 ; t º = 0 .Therefore, w = ¹ L 0 E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º 00 v ds (8.26) ¹ L 0 E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º v 00 v 0 0 v ds + E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º s = L v 0 ¹ L ; t º E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º 0 s = L v ¹ L ; t º + E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º v 00 v 0 s = L v ¹ L ; t º 193 Let % bemassperunitvolumeofthebeam, M and J bethemassandrotatoryinertiaofthelumped massatthetipofbeam.Byconsideringthedisplacementandvelocityofthebeamgivenin(8.2) and(8.3),respectively,thekineticenergyisobtainedas T = 1 2 ¹ L 0 ¹ A % Û r 2 dAds + 1 2 M Û u 2 + ¹ Û v + Û V b º 2 s = L + 1 2 J Û 2 s = L ; (8.27) = 1 2 ¹ L 0 ¹ A % n ¹ Û u Û cos ¹ ºº 2 + ¹ Û v + Û V b Û sin ¹ ºº 2 o dAds + 1 2 M Û u 2 + ¹ Û v + Û V b º 2 s = L + 1 2 J Û 2 s = L ; = 1 2 ¹ L 0 ¹ A % n Û u 2 2 Û u Û cos ¹ º + 2 Û 2 cos 2 ¹ º + Û v 2 + Û V 2 b + 2 Û v Û V b 2 ¹ Û v 2 + Û V b º Û sin ¹ º + 2 Û 2 sin 2 ¹ º o dAds + 1 2 M Û u 2 + ¹ Û v + Û V b º 2 s = L + 1 2 J Û 2 s = L ; = 1 2 ¹ L 0 ¹ A % n Û u 2 + Û v 2 + Û V 2 b + 2 Û v Û V b 2 Û u Û cos ¹ º + 2 Û 2 2 ¹ Û v 2 + Û V b º Û sin ¹ º o dAds + 1 2 M Û u 2 + ¹ Û v + Û V b º 2 s = L + 1 2 J Û 2 s = L : Let ˆ = ¹ A % dA ; J 1 = ¹ A % dA ; J 2 = ¹ A % 2 dA : ˆ isthemassperunitlengthofthebeam, J 1 isthe˝rstmomentofinertiaandiszerobecausethe referencepointofcoordinatesystemattachedtothecrosssectioncoincideswiththemasscentroid, and J 2 isthesecondmomentofinertia,whichisverysmallforslenderbeamandcanbeignored [66].Assumingthatthevelocityalongthelengthofthebeam, Û u ,isrelativelysmallcomparedto thelateralvelocity Û v + Û V b ,thekineticenergyofthebeamcanbereducedto T = 1 2 ˆ ¹ L 0 ¹ Û v + Û V b º 2 ds + 1 2 M ¹ Û v + Û V b º 2 s = L + 1 2 J Û 2 s = L ; (8.28) whereitsvariationcanbetakenas T = ˆ ¹ L 0 ¹ Û v + Û V b º Û v ds + M ¹ Û v + Û V b º Û v s = L + J Û Û s = L ; (8.29) 194 inwhich Û isgivenin(8.8)and Û canbeobtainedas Û '¹ 1 + 1 2 v 0 2 º Û v 0 + v 0 Û v 0 v 0 .Therefore, T = ˆ ¹ L 0 ¹ Û v + Û V b º Û v ds + M ¹ Û v + Û V b º Û v s = L + J Û v 0 ¹ 1 + v 0 2 º Û v 0 + v 0 Û v 0 2 v 0 s = L : (8.30) Thetimeintegrationof T takesthefollowingformthroughintegrationbyparts ¹ t 2 t 1 Tdt (8.31) = ¹ t 2 t 1 ( ˆ ¹ L 0 ¹ Û v + Û V b º Û v ds + M ¹ Û v + Û V b º Û v s = L + J Û v 0 ¹ 1 + v 0 2 º Û v 0 + v 0 Û v 0 2 v 0 s = L ) dt = ¹ t 2 t 1 ˆ ¹ L 0 ¹ Û v + Û V b º Û v dsdt + M ¹ t 2 t 1 ¹ Û v + Û V b º Û v s = L dt + J ¹ t 2 t 1 Û v 0 ¹ 1 + v 0 2 º Û v 0 + v 0 Û v 0 2 v 0 s = L dt = ˆ ¹ L 0 ¹ t 2 t 1 ¹ Û v + Û V b º Û v dtds + M ¹ t 2 t 1 ¹ Û v + Û V b º Û v dt s = L + J ¹ t 2 t 1 Û v 0 ¹ 1 + v 0 2 º Û v 0 + v 0 Û v 0 2 v 0 dt s = L = ˆ ¹ L 0 " ¹ Û v + Û V b º v t 2 t 1 ¹ t 2 t 1 ¹ Ü v + Ü V b º v dt # ds + M ¹ Û v + Û V b º v s = L t 2 t 1 M ¹ t 2 t 1 ¹ Ü v + Ü V b º v dt s = L + J Û v 0 ¹ 1 + v 0 2 º v 0 s = L t 2 t 1 J ¹ t 2 t 1 Ü v 0 ¹ 1 + v 0 2 º + v 0 Û v 0 2 v 0 dt s = L = ¹ t 2 t 1 ( ˆ ¹ L 0 ¹ Ü v + Ü V b º v ds + M ¹ Ü v + Ü V b º v s = L + J Ü v 0 ¹ 1 + v 0 2 º + v 0 Û v 0 2 v 0 s = L ) dt ; 195 whereweconsiderthat v = v 0 = 0 at t = t 1 and t = t 2 .Therefore,theextendedHamilton's principletakestheform ¹ t 2 t 1 ( (8.32) ¹ L 0 " ˆ ¹ Ü v + Ü V b º E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º 00 + E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º v 00 v 0 0 # v ds M ¹ Ü v + Ü V b º s = L v ¹ L ; t º J Ü v 0 ¹ 1 + v 0 2 º + v 0 Û v 0 2 s = L v 0 ¹ L ; t º E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º s = L v 0 ¹ L ; t º + E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º ¹ 1 + 1 2 v 0 2 º 0 s = L v ¹ L ; t º E 1 I v 00 ¹ 1 + 1 2 v 0 2 º + E I RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º v 00 v 0 s = L v ¹ L ; t º ) dt = 0 : Invokingthearbitrarinessofvirtualdisplacement v ,weobtainthestrongformoftheequationof motionas: ˆ Ü v + E 1 I v 00 ¹ 1 + 1 2 v 0 2 º 2 00 + E I ¹ 1 + 1 2 v 0 2 º RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º 00 (8.33) E 1 I v 0 v 00 2 ¹ 1 + 1 2 v 0 2 º 0 E I v 0 v 00 RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º 0 = ˆ Ü V b ; whichissubjecttothefollowingnaturalboundaryconditions: J Ü v 0 ¹ 1 + v 0 2 º + v 0 Û v 0 2 + E 1 I v 00 ¹ 1 + 1 2 v 0 2 º 2 (8.34) + E I ¹ 1 + 1 2 v 0 2 º RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º s = L = 0 ; M ¹ Ü v + Ü V b º E 1 I v 00 ¹ 1 + 1 2 v 0 2 º 2 + E I ¹ 1 + 1 2 v 0 2 º RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º 0 + E 1 I v 0 v 00 2 ¹ 1 + 1 2 v 0 2 º + E I v 0 v 00 RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º s = L = 0 : Followingasimilarapproachasin(8.9)inderivingthebeamcurvature,weobtaintheapproxi- mationsbelow,whereweonlyconsideruptothirdordertermsandremovethehigherorderterms 196 (HOTs). v 00 ¹ 1 + 1 2 v 0 2 º 2 = v 00 + v 00 v 0 2 + HOTs ¹ 1 + 1 2 v 0 2 º RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º = RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 v 0 2 RL 0 D t v 00 + HOTs v 0 v 00 2 ¹ 1 + 1 2 v 0 2 º = v 0 v 00 2 + HOTs v 0 v 00 RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º = v 0 v 00 RL 0 D t v 00 + HOTs Therefore,thestrongformcanbeapproximateduptothethirdorderandtheproblemthenreads as:˝nd v 2 V suchthat m Ü v + v 00 + v 00 v 0 2 00 v 0 v 00 2 0 + E r RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 v 0 2 RL 0 D t v 00 00 (8.35) E r v 0 v 00 RL 0 D t v 00 0 = m Ü V b ; m Ü v + v 00 + v 00 v 0 2 + E r RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 E r v 0 2 RL 0 D t v 00 00 (8.36) v 0 v 00 2 + E r v 0 v 00 RL 0 D t v 00 0 = m Ü V b ; subjecttothefollowingboundaryconditions: v s = 0 = v 0 s = 0 = 0 ; (8.37) Jm ˆ Ü v 0 ¹ 1 + v 0 2 º + v 0 Û v 0 2 + v 00 + v 00 v 0 2 + E r RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 E r v 0 2 RL 0 D t v 00 s = L = 0 ; Mm ˆ ¹ Ü v + Ü V b º v 00 + v 00 v 0 2 + E r RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 E r v 0 2 RL 0 D t v 00 0 + v 0 v 00 2 + E r v 0 v 00 RL 0 D t v 00 s = L = 0 ; where m = ˆ E 1 I and E r = E E 1 . 197 8.2.4Nondimensionalization Letthedimensionlessvariables s = s L ; v = v L ; t = t 1 mL 4 1 š 2 ; E r = E r 1 mL 4 š 2 ; (8.38) J = J ˆ L 3 ; M = M ˆ L ; V b = V b L : Weobtainthefollowingdimensionlessequationbysubstitutingtheabovedimensionlessvariables. m L mL 4 @ 2 v @ t 2 (8.39) + 1 L 2 @ 2 @ s 2 " L L 2 @ 2 v @ s 2 + L L 2 @ 2 v @ s 2 ¹ L L @ v @ s º 2 + E r ¹ mL 4 º š 2 2 1 ¹ mL 4 º š 2 L L 2 ¹ L L º 2 RL 0 D t @ 2 v @ s 2 ¹ @ v @ s º 2 + E r ¹ mL 4 º š 2 1 ¹ mL 4 º š 2 L L 2 RL 0 D t @ 2 v @ s 2 + 1 2 E r ¹ mL 4 º š 2 ¹ L L @ v @ s º 2 1 ¹ mL 4 º š 2 L L 2 RL 0 D t @ 2 v @ s 2 # 1 L @ @ s " L L @ v @ s ¹ L L 2 @ 2 v @ s 2 º 2 + E r ¹ mL 4 º š 2 L L @ v @ s L L 2 @ 2 v @ s 2 1 ¹ mL 4 º š 2 L L 2 RL 0 D t @ 2 v @ s 2 # = m L mL 4 @ 2 V b @ t 2 ; whichcanbesimpli˝edto @ 2 v @ t 2 + @ 2 @ s 2 " @ 2 v @ s 2 + @ 2 v @ s 2 ¹ @ v @ s º 2 + E r 2 RL 0 D t @ 2 v @ s 2 ¹ @ v @ s º 2 + E r RL 0 D t @ 2 v @ s 2 (8.40) + 1 2 E r ¹ @ v @ s º 2 RL 0 D t @ 2 v @ s 2 # @ @ s " @ v @ s ¹ @ 2 v @ s 2 º 2 + E r @ v @ s @ 2 v @ s 2 RL 0 D t @ 2 v @ s 2 # = @ 2 V b @ t 2 ; 198 Thedimensionlessboundaryconditionsarealsoobtainedbysubstitutingdimensionlessvariables in(8.37).Wecanshowsimilarlythattheypreservetheirstructureas: v s = 0 = @ v @ s s = 0 = 0 ; J ˆ L 3 m ˆ 1 mL 4 " @ 3 v @ t @ 2 s 1 + @ v @ s 2 ! + @ v @ s @ 2 v @ t @ s ! 2 # + 1 L " @ 2 v @ s 2 + @ 2 v @ s 2 ¹ @ v @ s º 2 + E r 2 RL 0 D t @ 2 v @ s 2 ¹ @ v @ s º 2 + E r RL 0 D t @ 2 v @ s 2 + 1 2 E r ¹ @ v @ s º 2 RL 0 D t @ 2 v @ s 2 # s = 1 = 0 ; M ˆ Lm ˆ L mL 4 @ 2 v @ 2 t + @ 2 V b @ 2 t ! 1 L 2 @ v @ s " @ 2 v @ s 2 + @ 2 v @ s 2 ¹ @ v @ s º 2 + E r 2 RL 0 D t @ 2 v @ s 2 ¹ @ v @ s º 2 + E r RL 0 D t @ 2 v @ s 2 + 1 2 E r ¹ @ v @ s º 2 RL 0 D t @ 2 v @ s 2 # + 1 L 2 " @ v @ s ¹ @ 2 v @ s 2 º 2 + E r @ v @ s @ 2 v @ s 2 RL 0 D t @ 2 v @ s 2 # s = 1 = 0 ; Therefore,thedimensionlessequationofmotionbecomes(afterdropping forthesakeofsimplic- ity) Ü v + v 00 + v 00 v 0 2 + E r RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 E r v 0 2 RL 0 D t v 00 00 (8.41) v 0 v 00 2 + E r v 0 v 00 RL 0 D t v 00 0 = Ü V b ; whichissubjecttothefollowingdimensionlessboundaryconditions v s = 0 = v 0 s = 0 = 0 ; (8.42) J Ü v 0 ¹ 1 + v 0 2 º + v 0 Û v 0 2 + v 00 + v 00 v 0 2 + E r RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 E r v 0 2 RL 0 D t v 00 s = 1 = 0 ; M ¹ Ü v + Ü V b º v 00 + v 00 v 0 2 + E r RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 E r v 0 2 RL 0 D t v 00 0 + v 0 v 00 2 + E r v 0 v 00 RL 0 D t v 00 s = 1 = 0 ; 199 8.2.5WeakFormulation Weobtaintheweakformoftheproblembymultiplyingtheequationwithpropertestfunctions ~ v ¹ s º2 ~ V andintegratingoverthedimensionlessspatialcomputationaldomain s = » 0 ; 1 ¼ .Thetest functionsatis˝esthegeometricboundaryconditions,i.e. ~ v ¹ 0 º = ~ v 0 ¹ 0 º = 0 .Therefore,bychanging theorderofintegralandtemporalderivatives,andthroughintegrationbyparts,theweakformof problemcanbewrittenas ¹ 1 0 Ü v ~ v ds + ¹ 1 0 v 00 + v 00 v 0 2 + E r RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 E r v 0 2 RL 0 D t v 00 00 ~ v ds (8.43) ¹ 1 0 v 0 v 00 2 + E r v 0 v 00 RL 0 D t v 00 0 ~ v ds = ¹ 1 0 Ü V b ~ v ds ; wherewetransferthespatialderivativeloadtothetestfunctionthroughintegrationbypartsas @ 2 @ t 2 ¹ 1 0 v ~ v ds + ¹ 1 0 v 00 + v 00 v 0 2 + E r RL 0 D t v 00 ¹ 1 + 1 2 v 0 2 º + 1 2 E r v 0 2 RL 0 D t v 00 ~ v 00 ds (8.44) + ¹ 1 0 v 0 v 00 2 + E r v 0 v 00 RL 0 D t v 00 ~ v 0 ds + M ¹ Ü v + Ü V b º ~ v s = 1 + J Ü v 0 ¹ 1 + v 0 2 º + v 0 Û v 0 2 ~ v 0 s = 1 = Ü V b ¹ L 0 ~ v ds : Byrearrangingtheterms,weget @ 2 @ t 2 ¹ 1 0 v ~ v ds + M v ~ v s = 1 + J v 0 ~ v 0 s = 1 ! + J Ü v 0 v 0 2 + v 0 Û v 0 2 ~ v 0 s = 1 (8.45) + ¹ 1 0 v 00 ~ v 00 ds + E r ¹ 1 0 RL 0 D t v 00 ~ v 00 ds + ¹ 1 0 v 00 v 0 2 ~ v 00 ds + ¹ 1 0 v 0 v 00 2 ~ v 0 ds + E r 2 ¹ 1 0 RL 0 D t v 00 v 0 2 ~ v 00 ds + E r 2 ¹ 1 0 v 0 2 RL 0 D t v 00 ~ v 00 ds + E r ¹ 1 0 v 0 v 00 RL 0 D t v 00 ~ v 0 ds = Ü V b ¹ 1 0 ~ v ds + M ~ v s = 1 ! : 200 8.2.6AssumedMode:ASpectralGalerkinApproximationInSpace Weemploythefollowingmodaldiscretizationtoobtainareduced-ordermodelofthebeam. Therefore, v ¹ s ; t º' v N ¹ s ; t º = N Õ n = 1 q n ¹ t º ˚ n ¹ s º ; (8.46) wherethespatialfunctions ˚ n ¹ s º ; n = 1 ; 2 ; ; N areassumed apriori andthetemporalfunctions q n ¹ t º ; n = 1 ; 2 ; ; N aretheunknownmodalcoordinates.Theassumedmodes ˚ n ¹ s º indiscretiza- tion(8.46)areobtainedinsection8.5bysolvingthecorrespondingeigenvalueproblemoflinear counterpartofourmodel.Hence,weconstructtheproper˝nitedimensionalspacesofbasis/test functionsas: V N = ~ V N = span n ˚ n ¹ x º : n = 1 ; 2 ; ; N o : (8.47) Thus,since V N = ~ V N ˆ V = ~ V ,problem(8.45)readas:˝nd v N 2 V N suchthat @ 2 @ t 2 ¹ 1 0 v N ~ v N ds + M v N ~ v N s = 1 + J v 0 N ~ v 0 N s = 1 ! (8.48) + J Ü v 0 N v 0 N 2 + v 0 N Û v 0 N 2 ~ v 0 N s = 1 + ¹ 1 0 v 00 N ~ v 00 N ds + E r ¹ 1 0 RL 0 D t v 00 N ~ v 00 N ds + ¹ 1 0 v 00 N v 0 N 2 ~ v 00 N ds + ¹ 1 0 v 0 N v 00 N 2 ~ v 0 N ds + E r 2 ¹ 1 0 RL 0 D t v 00 N v 0 N 2 ~ v 00 N ds + E r 2 ¹ 1 0 v 0 N 2 RL 0 D t v 00 N ~ v 00 N ds + E r ¹ 1 0 v 0 N v 00 N RL 0 D t v 00 N ~ v 0 N ds = Ü V b ¹ 1 0 ~ v N ds + M ~ v N s = 1 ! ; forall ~ v N 2 ~ V N . 8.2.7SingleModeApproximation Weassumethattheonlyactivemodeofvibrationistheprimaryone,whichencapsulatesmostofthe fundamentaldynamicsofourcomplexsystem.Therefore,westartwithaunimodaldiscretization 201 v N = q ¹ t º ˚ ¹ s º ,wherewelet N = 1 in(8.46)anddropsubscript 1 forsimplicity.Uponsubstituting, weobtain M Ü q + J¹ Ü qq 2 + q Û q 2 º + K l q + E r C l RL 0 D t q + 2 K nl q 3 (8.49) + E r C nl 2 RL 0 D t q 3 + 3 q 2 RL 0 D t q = b Ü V b ; inwhich M = ¹ 1 0 ˚ 2 ds + M ˚ 2 ¹ 1 º + J ˚ 0 2 ¹ 1 º ; J = J ˚ 0 4 ¹ 1 º ; (8.50) K l = C l = ¹ 1 0 ˚ 00 2 ds ; K nl = C nl = ¹ 1 0 ˚ 0 2 ˚ 00 2 ds ; M b = ¹ 1 0 ˚ ds + M ˚ ¹ 1 º : 8.3LinearizedEquation:DirectNumericalTimeIntegration Welinearizetheobtainedequation,governingthetimeevolutionofthe˝rstvibrationmode byremovingthenonlinearterms.Therefore,intheabsenceofbaseexcitation,(8.49)takesthe followingform Ü q + E r c l RL 0 D t q + k l q = 0 (8.51) inwhichthecoe˚cients c l = C l M and k l = K l M aregivenin(8.50).Thelinearizedequation(8.51) canbethoughtofasafractionallydampedoscillator,shownschematicallyinFig.8.5(right).This settingdescribesthevibrationofalumpedfractionalKelvin-Voigtmodel.Byletting E r = 1 ,the dimensionlessparameters c l = k l = 1 : 24 withaunitmassatthetip,i.e. M = 1 .We˝ndthetime responseofthelinearizedmodel(8.51)usingadirect˝nitedi˙erencetimeintegrationscheme, whichemploys L 1 scheme[96,104]andNewmarkmethodtoapproximatethefractionalderivative andtheinertialterm,respectively.TheNewmarkmethodisofsecondorderaccuracyandthusthe overallaccuracyofthedevelopedschemeisgovernedbytheerrorlevelof L 1 scheme,whichisof order 2 . 202 Fig.8.5(left)showsthetimeresponseoffreevibrationofafractionallydampedoscillator.The absolutevalueof q ¹ t º versustimefordi˙erentvaluesof isplottedinLog-Logscale.Weobserve thatinthelongtime,theamplitudeofoscillationdecayswithapower-law,whoserateisgoverned bytheorderoffractionalderivative andincreasesbyincreasing (seebluelinesinthe˝gure). Byreplacingthefractionaldamperwithaclassicalinteger-orderone,weseethattheamplitude decaysexponentiallyandnotanymorebyapower-law(seethedottedredlineinthe˝gure).These resultsareinperfectagreementwiththepower-lawandexponentialrelaxationkernel,described inSec.8.2.2.Wenotethatsincethefractionalelementisinherentlyaviscoelasticelementthat interpolatesbetweenthetwospringanddash-potelements(seeSec.8.2.2formorediscussionand references),itcontributesbothinthesti˙nessanddampingratioofthesystem.As increases, thefractionalelementconvergestopurelyviscouselement,andthusthesystembecomessofter (lesssti˙),resultinginfrequencyreduction.Thisfrequencyshiftcanbeseenfromthedriftof consecutiveamplitudepeakstotherightas isincreased.Thefractionallinearoscillatorsarealso consideredin[161]asacaseofsystemswithmemory,wheretheirinteractionwitha˛uctuating environmentcausesthetimeevolutionofthesystemtobeintermittent.Theauthorsin[161]apply theKoopmanoperatortheorytothecorrespondingintegerordersystemandthenmakeaL evy transformationintimetorecoverlong-termmemorye˙ects;theyobserveapower-lawbehavior intheamplitudedecayofthesystem'sresponse.Suchananomalousdecayratehasalsobeen investigatedin[150]foranextendedtheoryofdecayofclassicalvibrationalmodelsbroughtinto nonlinearresonances.Theauthorsreportaxponential"decayinvariablesdescribingthe dynamicsofthesysteminthepresenceofdissipationandalsoasharpchangeinthedecayrate closetoresonance. Remark8.3.1. Thechangeinfractionalderivativeorder, ,isanotionofsti˙ening/softeningof aviscoelasticmaterialmodeledviafractionalconstitutiveequations.AsshowninFig.8.5(left), thevalueof directlya˙ectsthedecayrateoffreevibration.Thisstrongrelationcanbeusedto developapredictionframework,whichtakestimeseriesoffreevibrationsasaninput,andreturns anestimationofthelevelofmaterialsti˙nessasare˛ectionofthehealthofthesystemofinterest. 203 Figure8.5:Power-LawDecay:Timeresponseoflinearfractionallydampedoscillatorusing Newmarkand L 1 scheme.Thefractionaldamperhastwoconstants E r c l and asthecoe˚cient andderivativeorderoffractionaloperator. 8.4PerturbationAnalysisofNonlinearEquation Nonlineartermsinequation(8.49)giverisetoexpensivetimeintegrationschemes.Weuse perturbationanalysistoinvestigatethebehaviorofanonlinearsystem,wherewereducethe nonlinearfractionaldi˙erentialequationtoanalgebraicequationtosolveforthesteadystate amplitudeandphaseofvibration. 8.4.1MethodofMultipleScales Toinvestigatethedynamicsofthesystemdescribedby(8.49),weusethemethodofmultiplescales [128,141].Thenewindependenttimescalesandtheinteger-orderderivativewithrespecttothem arede˝nedas T m = m t ; D m = @ @ T m ; m = 0 ; 1 ; 2 ; : (8.52) Itisalsoconvenienttoutilizeanotherrepresentationofthefractionalderivativeasinequation (5.82)in[147],whichaccordingtotheRiemann-Liouvillefractionalderivative,isequivalenttothe 204 fractionalpoweroftheoperatorofconventionaltime-derivative,i.e. RL 0 D t = ¹ d dt º .Therefore, d dt = D 0 + D 1 + ; (8.53) d 2 dt 2 = D 2 0 + 2 D 0 D 1 + ; RL 0 D t = ¹ d dt º = D 0 + D 1 0 D 1 + ; Thesolution q ¹ t º canthenberepresentedintermsofseries q ¹ T 0 ; T 1 ; º = q 0 ¹ T 0 ; T 1 ; º + q 1 ¹ T 0 ; T 1 ; º + 2 q 2 ¹ T 0 ; T 1 ; º + (8.54) Weassumethatthecoe˚cientsintheequationofmotionhasthefollowingscaling J M = m nl ; K l M = k l = ! 2 0 ; C l M = c l ; K nl M = k nl ; C nl M = c nl ; (8.55) andthebaseexcitation M b M Ü V b isaharmonicfunctionofform F cos ¹ t º .Thus,(8.49)canbe expandedas ¹ D 2 0 + 2 D 0 D 1 + º¹ q 0 + q 1 + º (8.56) + m nl ¹ D 2 0 + 2 D 0 D 1 + º¹ q 0 + q 1 + º¹ q 0 + q 1 + º 2 + m nl ¹ q 0 + q 1 + º ¹ ¹ D 0 + D 1 + º¹ q 0 + q 1 + º º 2 + ! 2 0 ¹ q 0 + q 1 + º + E r c l ¹ D 0 + D 1 0 D 1 + º¹ q 0 + q 1 + º + 2 k nl ¹ q 0 + q 1 + º 3 + 1 2 E r c nl ¹ D 0 + D 1 0 D 1 + º¹ q 0 + q 1 + º 3 + 3 2 E r c nl ¹ q 0 + q 1 + º 2 h ¹ D 0 + D 1 0 D 1 + º¹ q 0 + q 1 + º i = F cos ¹ T 0 º : 205 Bycollectingsimilarcoe˚cientsofzero-thand˝rstordersof ,weobtainthefollowingequations O¹ 0 º : D 2 0 q 0 + ! 2 0 q 0 = 0 ; (8.57) O¹ 1 º : D 2 0 q 1 + ! 2 0 q 1 = 2 D 0 D 1 q 0 m nl q 2 0 D 2 0 q 0 + q 0 ¹ D 0 q 0 º 2 E r c l D 0 q 0 2 k nl q 3 0 1 2 E r c nl D 0 q 3 0 3 2 E r c nl q 2 0 D 0 q 0 + F cos ¹ T 0 º : (8.58) Thesolutionto(8.57)isoftheform q 0 ¹ T 0 ; T 1 º = A ¹ T 1 º e i ! 0 T 0 + c : c (8.59) wheredenotesthecomplexconjugate.Bysubstituting(8.59)intotheright-hand-sideof(8.58), weobservethatdi˙erentresonancecasesarepossible.Ineachcase,weobtainthecorresponding solvabilityconditionsbyremovingthesecularterms,i.e.thetermsthatgrowintimeunbounded. Then,wewrite A inthepolarform A = 1 2 ae i ' ,wheretherealvaluedfunctions a and ' arethe amplitudeandphaselagoftimeresponse,respectively.Thus,thesolution q ¹ t º becomes q ¹ t º = a ¹ t º cos ¹ ! 0 t + ' ¹ t ºº + O¹ º ; (8.60) wherethegoverningequationsof a and ' areobtainedbyseparatingtherealandimaginaryparts. 8.4.1.1Case1:NoLumpedMassAtTheTip Inthiscase, M = J = 0 ,andthus,giventhefunctions ' 1 ¹ x º insection8.5,thecoe˚cientsare computedas M = 1 , K l = C l = 12 : 3624 , M b = 0 : 782992 ,and K nl = C nl = 20 : 2203 .We considerthefollowingcases: FreeVibration, F = 0 :SuperSensitivityto Inthiscase,thebeamisnotexternallyexcitedandthus, F = 0 .Byremovingthesecularterms thatarethecoe˚cientsof e i ! 0 T 0 inthesolvabilitycondition,we˝ndthegoverningequationsof 206 solutionamplitudeandphaseas da dT 1 = E r ! 1 0 sin ¹ ˇ 2 º 1 2 c l a + 3 8 c nl a 3 ; (8.61) d ' dT 1 = 1 2 c l E r ! 1 0 cos ˇ 2 + 3 4 c nl E r ! 1 0 cos ˇ 2 a 2 + 3 4 ! 1 0 k nl a 2 : (8.62) Wecanseefromthe˝rstequation(8.61)thattheamplitudeoffreevibrationdecaysout,wherethe decayrate ˝ d = c l E r ! 1 0 sin ¹ ˇ 2 º directlydependsonvaluesofthefractionalderivative and thecoe˚cients E r (seeFig.8.6).Weintroducethesensitivityindex S ˝ d ; asthepartialderivative Figure8.6:Freevibrationoftheviscoelasticcantileverbeamwithnolumpedmassatthetip.The rateofdecayofamplitudestronglydependsonthefractionalderivativeorder andthecoe˚cient E r .Theleft˝gure(log-linearscale)showstherapidincreaseinamplitudedecayingas is increasedand E r = 0 : 1 .Theright˝gure(linearscale)showsthephaselag ' ¹ t º ,whereitsincrease ratedecreasesas isincreased. ofdecayratewithrespectto ,i.e. S ˝ d ; = d ˝ d d = ˇ 2 c l E r ! 1 0 cos ¹ ˇ 2 º + c l E r ! 1 0 sin ¹ ˇ 2 º log ¹ ! 0 º : (8.63) ThesensitivityindexiscomputedandplottedinFig.8.7forthesamesetofparametersasinFig. 8.6.Thereexistsacriticalvalue cr = 2 ˇ tan 1 ˇ 2log ¹ ! 0 º ; (8.64) where S ˝ d ; = 0 .WeobserveinFig.8.7thatbyincreasing when < cr ,i.e.introducingmore viscositytothesystem,thedissipationrate,andthusdecayrate,increases;thiscanbethoughtof asasoftening(sti˙ness-decreasing)region.Furtherincreasing when > cr ,willreversely 207 resultsindecreaseofdecayrate;thiscanbethoughtofasahardening(moresti˙ening)region.We alsonotethat cr solelydependsonvalueof ! 0 ,givenin(8.55),andeventhoughthevalueof E r a˙ectsdecayrate,itdoesnotchangethevalueof cr . Figure8.7:Freevibrationoftheviscoelasticcantileverbeamwithnolumpedmassatthetip. Thisgraphshowssensitivityofthedecayrate ˝ d withrespecttochangeof .Increasing when < cr leadstohigherdissipationanddecayrate.Thereversee˙ectisobservedwhen > cr .By softeningandhardeningwere˛ecttotheregionswhereincreasing (introducingextraviscosity) leadstohigherandlowerdecayrate,respectively. PrimaryResonanceCase, ˇ ! 0 Inthecaseofprimaryresonance,theexcitationfrequencyisclosetothenaturalfrequencyofthe system.Welet = ! 0 + ,where iscalledthedetuningparameterandthus,writetheforce functionas 1 2 Fe i T 1 e i ! 0 T 0 + c : c .Inthiscase,theforcefunctionalsocontributestothesecular terms.Therefore,we˝ndthegoverningequationsofsolutionamplitudeandphaseas da dT 1 = E r ! 1 0 sin ¹ ˇ 2 º 1 2 c l a + 3 8 c nl a 3 + 1 2 f ! 1 0 sin ¹ T 1 ' º ; (8.65) a d ' dT 1 = 1 2 c l E r ! 1 0 cos ¹ ˇ 2 º a + 3 4 c nl E r ! 1 0 cos ¹ ˇ 2 º a 3 + 3 4 ! 1 0 k nl a 3 (8.66) 1 2 f ! 1 0 cos ¹ T 1 ' º ; inwhichthefourparameters f ; E r ; f ; g mainlychangethefrequencyresponseofthesystem.The equations(8.65)and(8.66)canbetransformedintoanautonomoussystem,wherethe T 1 doesnot appearexplicitly,byletting = T 1 ': 208 Thesteadystatesolutionoccurwhen da dT 1 = d ' dT 1 = 0 ,thatgives E r ! 1 0 sin ¹ ˇ 2 º c l 2 a + 3 c nl 8 a 3 = f 2 ! 0 sin ¹ º ; (8.67) c l 2 E r ! 1 0 cos ¹ ˇ 2 º a 3 4 c nl E r ! 1 0 cos ¹ ˇ 2 º + ! 1 0 k nl a 3 = f 2 ! 0 cos ¹ º ; (8.68) andthus,bysquaringandaddingthesetwoequations,weget c l 2 E r ! 1 0 sin ¹ ˇ 2 º a + 3 c nl 8 E r ! 1 0 sin ¹ ˇ 2 º a 3 2 (8.69) + c l 2 E r ! 1 0 cos ¹ ˇ 2 º a 3 4 c nl E r ! 1 0 cos ¹ ˇ 2 º + ! 1 0 k nl a 3 2 = f 2 4 ! 2 0 : Thiscanbewritteninasimplerwayas h A 1 a + A 2 a 3 i 2 + h B 1 a + B 2 a 3 i 2 = C ; (8.70) where A 1 = c l 2 E r ! 1 0 sin ¹ ˇ 2 º ; A 2 = 3 c nl 8 E r ! 1 0 sin ¹ ˇ 2 º ; C = f 2 4 ! 2 0 ; B 1 = c l 2 E r ! 1 0 cos ¹ ˇ 2 º ; B 2 = 3 4 c nl E r ! 1 0 cos ¹ ˇ 2 º + ! 1 0 k nl : Hence,thesteadystateresponseamplitudeistheadmissiblerootof ¹ A 2 2 + B 2 2 º a 6 + ¹ 2 A 1 A 2 + 2 B 1 B 2 º a 4 + ¹ A 2 1 + B 2 1 º a 2 C = 0 ; (8.71) whichisacubicequationin a 2 .Thediscriminantofacubicequationoftheform ax 3 + bx 2 + cx + d = 0 isgivenas # = 18 abcd 4 b 3 d + b 2 c 2 4 ac 3 27 a 2 d 2 .Thecubicequation(8.71)hasonereal rootwhen #< 0 andthreedistinctrealrootswhen #> 0 .Themainfourparameters f ; E r ; f ; g dictatethevalueofcoe˚cients f A 1 ; A 2 ; B 1 ; B 2 ; C g ,thevalueofdiscriminant # ,andthusthenumber ofadmissiblesteadystateamplitudes.Weseethatfor˝xedvaluesof f ; E r ; f g ,bysweepingthe detuningparameter fromlowertohigherexcitationfrequency,thestablesteadystateamplitude bifurcatesintotwostablebranchesandoneunstablebranch,wheretheyconvergebacktoastable amplitudebyfurtherincreasing .Fig.8.8(left)showsthebifurcationdiagrambysweepingthe 209 detuningparameter andfordi˙erentvaluesof when E r = 0 : 3 and f = 1 .Thesolidand dashedblacklinesarethestableandunstableamplitudes,respectively.Thebluelinesconnectthe bifurcationpoints(reddots)foreachvalueof .Weseethatthebifurcationpointsarestrongly relatedtothevalueof ,meaningthatbyintroducingextraviscositytothesystem,i.e.increasing thevalueof ,theamplitudesbifurcateandthenconvergebackfaster.TherightpanelofFig.8.8 showsthefrequencyresponseofthesystem,i.e.themagnitudeofsteadystateamplitudesversus excitationfrequency.Astheexcitationfrequencyisswepttotheright,thesteadystateamplitude increases,reachesapeakvalue,andthenjumpsdown(seee.g.reddashedlinefor = 0 : 4 ).The peakamplitudeandthejumpmagnitudedecreasesas isincreased. Figure8.8:Primaryresonanceoftheviscoelasticcantileverbeamwithnolumpedmassatthetip. Steadystateamplitude(right)anditsbifurcationdiagram(left)bychangingthedetuningparameter fordi˙erentvaluesof and E r = 0 : 3 ; f = 1 . Thecoe˚cient E r = E 1 E istheproportionalcontributionoffractionalandpureelasticelement. Atacertainvaluewhileincreasingthisparameter,weseethatthebifurcationdisappearsandthe frequencyresponseofsystemslightlychanges.Fig.8.9showsthefrequencyresponseofthesystem fordi˙erentvaluesof f ; E r g when f = 0 : 5 .Ineachsub-˝gure,welet be˝xedandthenplotthe frequencyresponsefor E r = f 0 : 1 ; 0 : 2 ; ; 1 g ;theamplitudepeakmovesdownas E r isincreased. Forhighervaluesof E r ,weseethatas isincreased,theamplitudepeaksdriftbacktotheleft, showingasofteningbehaviorinthesystemresponse. 210 Figure8.9:Frequency-Responsecurveforthecaseofprimaryresonanceintheviscoelasticcan- tileverbeamwithnolumpedmassatthetip.Eachsub-˝gurecorrespondstoa˝xedvalueof and f when E r = f 0 : 1 ; 0 : 2 ; ; 1 g .Ase˙ectoffractionalelementbecomesmorepronounced,i.e. and E r increase,thecurvemovesdownanddrifttoleft. 8.4.1.2Case2:LumpedMassAtTheTip Inthiscase, M = J = 1 ,andthus,giventhefunctions ˚ 1 ¹ x º insection8.5,thecoe˚cients arecomputedas M = 1 + 70 : 769 J + 7 : 2734 M , J = 5008 : 25 , K l = C l = 98 : 1058 , M b = 0 : 648623 2 : 69692 M ,and K nl = C nl = 2979 : 66 .SimilartoCase1,weconsiderthefollowing cases: FreeVibration, F = 0 FollowingthesamestepsasinCase1,weseethattheequationgoverningamplitudepreserveits structure,butthegoverningequationofphasecontainsanextratermaccommodatingthe m nl . da dT 1 = E r ! 1 0 sin ¹ ˇ 2 º 1 2 c l a + 3 8 c nl a 3 ; (8.72) d ' dT 1 = 1 2 c l E r ! 1 0 cos ˇ 2 + 3 4 c nl E r ! 1 0 cos ˇ 2 a 2 (8.73) + 3 4 ! 1 0 k nl a 2 1 4 m nl ! 0 a 2 : Thisextratermdoesnotsigni˝cantlyalterthebehaviorofphaseandthewholesystem. 211 PrimaryResonanceCase, ˇ ! 0 Similartothefreevibration,weseethattheequationgoverningamplitudepreservesitsstructure whilethegoverningequationofphasecontainsanextratermaccommodatingthe m nl da dT 1 = E r ! 1 0 sin ¹ ˇ 2 º 1 2 c l a + 3 8 c nl a 3 + 1 2 f ! 1 0 sin ¹ T 1 ' º ; (8.74) a d ' dT 1 = 1 2 c l E r ! 1 0 cos ¹ ˇ 2 º a + 3 4 c nl E r ! 1 0 cos ¹ ˇ 2 º a 3 + 3 4 ! 1 0 k nl a 3 (8.75) 1 2 f ! 1 0 cos ¹ T 1 ' º 1 4 m nl ! 0 a 3 : Transformingtheequationsintoanautonomoussystembyletting = T 1 ' ,weobtainthe governingequationofsteadystatesolutionas c l 2 E r ! 1 0 sin ¹ ˇ 2 º a + 3 c nl 8 E r ! 1 0 sin ¹ ˇ 2 º a 3 2 (8.76) c l 2 E r ! 1 0 cos ¹ ˇ 2 º a 3 4 c nl E r ! 1 0 cos ¹ ˇ 2 º + ! 1 0 k nl + 1 3 m nl ! 0 a 3 2 = f 2 4 ! 2 0 ; which,similartoCase1,canbewrittenas ¹ A 2 2 + B 2 2 º a 6 + ¹ 2 A 1 A 2 + 2 B 1 B 2 º a 4 + ¹ A 2 1 + B 2 1 º a 2 C = 0 ; whereallthe A 1 , A 2 , B 1 ,and C arethesameasinCase1,but B 2 = 3 4 c nl E r ! 1 0 cos ¹ ˇ 2 º + ! 1 0 k nl + 1 3 m nl ! 0 : Thecorrespondingcubicequationcanbesolvedtoobtainthebifurcationdiagramandalsothe frequencyresponseofthesystem.However,inadditiontoCase1,wehaveanextraparameter m nl whicha˙ectstheresponseofthesystem. Remark8.4.1. Themodelparametersinfactdescribethepropertiesofthesystemofinterest.As thesystemundergoescyclicloading,thedevelopmentofdamageduetoinitialimperfectionsand defectsinsidethematerialchangesthesystemproperties,andthus,it'sresponse.Thiscanbeused todevelopamachinelearningtoolthattakestheobservedexperimentaldataasinputandthen, predictsthesystemhealthbyinferringmodelparameters.Thedevelopedframeworkhereprovides aforwardsimulationtoconstructmanycasesoftrainingsetsforthelearningtool. 212 8.5EigenvalueProblemofLinearModel Theassumedmodes ˚ i ¹ s º indiscretization(8.46)areobtainedbysolvingthecorresponding eigenvalueproblemoffreevibrationofundampedlinearcounterpartstoourmodel.Thus,the dimensionlesslinearizedundampedequationofmotiontakestheform @ 2 @ t 2 v ¹ s ; t º + @ 4 @ s 4 v ¹ s ; t º = 0 : (8.77) subjecttolinearizedboundaryconditions: v ¹ 0 ; t º = 0 ; v 00 ¹ 1 ; t º = J Ü v 0 ¹ 1 ; t º ; (8.78) v 0 ¹ 0 ; t º = 0 ; v 000 ¹ 1 ; t º = M Ü v ¹ 1 ; t º ; where Û ¹º = d dt and ¹º 0 = d ds .Wederivethecorrespondingeigenvalueproblembyapplyingthe separationofvariables,i.e. v ¹ x ; t º = X ¹ s º T ¹ t º to(8.77).Therefore, Ü T ¹ t º X ¹ s º + T ¹ t º X 0000 ¹ s º = 0 ; (8.79) Ü T ¹ t º T ¹ t º + X 0000 ¹ s º X ¹ s º = 0 ; Ü T ¹ t º T ¹ t º = X 0000 ¹ s º X ¹ s º = ; whichgivesthefollowingequations Ü T ¹ t º + ! 2 T ¹ t º = 0 ; (8.80) X 0000 ¹ s º 4 X ¹ s º = 0 ; (8.81) where 4 = ! 2 andtheboundaryconditionsare X ¹ 0 º = 0 ; X 00 ¹ 1 º = J ! 2 X 0 ¹ 1 º ; X 0 ¹ 0 º = 0 ; X 000 ¹ 1 º = M ! 2 X ¹ 1 º : thesolutionto(8.81)isoftheform X ¹ s º = A sin ¹ s º + B cos ¹ s º + C sinh ¹ s º + D cosh ¹ s º ,where C = A and D = B ,usingtheboundaryconditionsat s = 0 .Therefore, X ¹ s º = A ¹ sin ¹ s º sinh ¹ s º º + B ¹ cos ¹ s º cosh ¹ s º º : 213 Applyingthe˝rstbondaryconditionat s = 1 ,i.e. X 00 ¹ 1 º = J ! 2 X 0 ¹ 1 º gives B = sin ¹ º + sinh ¹ º + J 3 ¹ cos ¹ º cosh ¹ ºº cos ¹ º + cosh ¹ º J 3 ¹ sin ¹ º sinh ¹ ºº A ; thatresultsin X ¹ s º = A " ¹ sin ¹ s º sinh ¹ s º º sin ¹ º + sinh ¹ º + J 3 ¹ cos ¹ º cosh ¹ ºº cos ¹ º + cosh ¹ º J 3 ¹ sin ¹ º sinh ¹ ºº ¹ cos ¹ s º cosh ¹ s º º # : Finally,usingthesecondboundaryconditionat s = 1 givesthefollowingtranscendentalequation tosolvefor 'sforthecasethat M = J = 1 : 1 + 4 + cos ¹ º cosh ¹ º + ¹ sin ¹ º cosh ¹ º cos ¹ º sinh ¹ º º (8.82) + 3 ¹ sin ¹ º cosh ¹ º sinh ¹ º cosh ¹ º º + 4 ¹ sin ¹ º sinh ¹ º + cos ¹ º cosh ¹ º º = 0 : (8.83) The˝rsteigenvalueiscomputedas 2 1 = ! 1 = 1 : 38569 ,whichresultstothefollowing˝rst normalizedeigenfunction,giveninFig.8.10. X 1 ¹ s º = 5 : 50054sin ¹ 1 s º 0 : 215842cos ¹ 1 s º 5 : 50054sinh ¹ 1 s º + 0 : 215842cosh ¹ 1 s º ; Figure8.10:The˝rsteigenfunctions, X 1 ¹ s º ,oftheundampedlinearcounterpartofourmodel.It isusedasthespatialfunctionsinthesinglemodeapproximation. Wenotethat(8.82)reducesto 1 + cos ¹ º cosh ¹ º = 0 forthecasethatthereisnolumpedmass atthetipofbeam;thisinfactgivesthenaturalfrequenciesofalinearcantileverbeam.Inthis case,the˝rsteigenvalueiscomputedas 2 1 = ! 1 = 3 : 51602 ,whichresultstothefollowing˝rst normalizedeigenfunction,giveninFig.8.11. 214 X 1 ¹ s º = 0 : 734096sin ¹ 1 s º cos ¹ 1 s º 0 : 734096sinh ¹ 1 s º + cosh ¹ 1 s º : Figure8.11:The˝rsteigenfunctions, X 1 ¹ s º ,oftheundampedlinearcounterpartofourmodelwith nolumpedmassatthetip.Itisusedasthespatialfunctionsinthesinglemodeapproximation. 215 CHAPTER9 SUMMARYANDFUTUREWORKS FractionalPDEsarethepropermathematicalmodelstodescribetheanomalousbehaviorinawide rangeofphysicalphenomenon.Theorderoffractionalderivativesintheseequationsareconsid- eredasanadditionalsetofmodelparameter,whosevaluesarestronglytiedtotheexperimental observations.Estimationofparametersinfractionalmodelsisnotatrivialproceduresincethe mainparameters,i.e.fractionalderivativesappearastheorderofderivatives.Thisimposesan extrachallengeindevelopingpropermathematicalframeworks,whicharecomputationallye˚cient tohandlesuchinverseproblems.Moreimportantly,theinherentrandomnessofexperimentalob- servationsintroduceuncertaintyintheorderofderivativesandthusmodeloutput;suchuncertainty demandsanout-of-boxthinkingtobeassessed.Theseschallengesevenbecomemoreimportant toresolveastheinherentbottleneckofnon-localityinfractionalPDEsleadstoexpensivecom- putationswithexcessivecomputer-memorystoragerequirementsandinsu˚cientcomputational accuracy.Utilizationoflocalnumericalmethods,suchas˝nitedi˙erencemethodcantakelong timeonordinarycomputersevenforonedimensionalproblemswithasinglederivativeorders. Theseleadtomoreseriousissueinhigherdimensionalnonlinearproblemswithdistributedorder derivatives,wheretheproblemosde˝nedoveracomplexgeometry. Toovercomethesechallenges,we˝rstdevelopaPetrov-Galerkinspectralelementmethod, whichcansolvefractionalboundaryvalueproblemsineachtimestepofnumericaltimeintegration techniques.Inchapter2,wedevelopedanew C 0 -continuousPetrov-Galerkinspectralelement methodfortheone-sidedspace-fractionalHelmholtzequation 0 D x u ¹ x º u ¹ x º = f ¹ x º , 2¹ 1 ; 2 º , subjecttohomogeneousboundaryconditions.Weobtainedaweakform,inwhichtheentire fractionalderivativeloadwastransferredontothetestfunctions,allowingustoe˚cientlyemploy thestandardmodalspectralelementbaseswhileincorporatingJacobipoly-fractonomialsasthetest functions.Weseamlesslyextendedthestandardprocedureofassemblingto non-localassembling inordertoconstructthegloballinearsystemfromlocal(elemental)mass/sti˙nessmatricesand 216 non-localhistorymatrices.Thekeytothee˚ciencyofthedevelopedPGmethodistwofold:i) ourformulationallowstheconstructionofelementalmassandsti˙nessmatricesinthestandard domain » 1 ; 1 ¼ once,andii)wee˚cientlyobtainthenon-local(history)sti˙nessmatrices,in whichthenon-localityispresented analytically .Wealsoinvestigatedlocalbasis/testfunctions inadditiontolocalbasiswithglobaltestfunctions.Wedemonstratedthattheformerchoice leadstoabetter-conditionedsystemandapproximabilityinthespectralelementformulationwhen higherpolynomialordersareneeded.Moreover,weshowedtheexponentialrateofconvergence consideringsmoothsolutionsaswellassingularsolutionswithinteriorsingularity;also,the spectral(algebraic)rateofconvergenceinsingularsolutionswithsingularitiesatboundaries.We alsopresentedtheretrievalprocessofhistorymatricesonuniformgrids,whichresultsinfasterand moree˚cientconstructionandsolutionofthelinearsystemcomparedtotheon-linecomputation. Inaddition,weconstructedtwonon-uniformgridsoverthecomputationaldomain(namely,kernel- drivenandgeometricallyprogressivegrids),anddemonstratedthee˙ectivenessofthenon-uniform gridsinaccuratelycapturingsingularsolutions,usingfewernumberofelementsandhigherorder polynomials.We˝nallyperformedasystematicnumericalstudyofnon-locale˙ectsviabothfull andpartial(history)fadinginordertobetterenhancethecomputationale˚ciencyofthescheme. Inchapter3,wedevelopedtwospectrally-accurateschemes,namelythePetrov-Galerkinspectral methodandthefractionalspectralcollocationmethodfordistributedorderfractionaldi˙erential equations.Thetwoschemeswereconstructedbasedontherecentlydevelopedspectraltheory forfractionalSturm-Liouvilleproblems(FSLPs).InthePetrov-Galerkinmethod,weemployed theJacobipoly-fractonomialsasthebases,whicharetheeigenfunctionsofFSLP-I,andthepoly- fractonomialeigenfunctionsofFSLP-IIasthetestfunctions.Wecarriedoutthediscretestability analysisoftheproposedschemeemployingsomeequivalent/bilinear-inducednormsbasedon thede˝neddistributedSobolevspacesandtheirassociatednorms.Inaddition,weperformeda convergencestudyoftheproposedscheme.Inthecollocationmethod,weemployedfractional LagrangeinterpolantssatisfyingtheKroneckerdeltapropertyatthecollocationpoints,andthen weobtainedthecorrespondingdistributeddi˙erentiationmatricestodiscretizethestrongproblem. 217 Theexistingschemesintheliteraturearemostlyemploying˝nitedi˙erencemethods.The mainchallengeinthesemethods,incomparisontospectralmethods,isthehistorycalculation aswellasextensivememoryallocationwhiletheydeliver˝xedalgebraicaccuracies.Therecent spectraltheoryonfractionalSturm-Liouvilleproblems(FSLPs)in[186]naturallymotivatesthe useofPetrov-Galerkinspectralmethods,wherethearisingbilinearformsarecomprisedofleft-and right-sidedfractionalderivatives.Theeigen-functionsofFSLPscanbeemployednaturallyasthe basesandtestspaces,wheretheirleft-andright-sidedderivativesareobtainedanalytically.These functionsconsistofapolynomialpartandafractionalpart,wheretheformerleavesthefractional order, ,asafreeparametertocapturesolutionsingularities,hence,totuneuptheaccuracyof theschemefrombeingalgebraicallyconvergenttoexponentialconvergent.Infact,theCase-IIIof numericalexamplesdemonstratedhowaproperchoiceoffractionalpartofthebasesprovidesthe exactsolutionwithonlyonetermexpansion.Furthermore,weprovedthatthedistributedbilinear formcanbeapproximatedwithaspectral/exponentialaccuracyusingaproperquadraturerule. ThePGspectralmethodtreatsthenonlocale˙ectse˚cientlythroughaglobalspectralmethod andprovidesanicemathematicalframeworkforperformingtheoreticalstudies,however,treating nonlinearproblemsremainsachallenge.Tothisend,weconstructedaspectrallyaccuratefractional spectralcollocationmethodemployingfractionalLagrangeinterpolants,whereforlinearproblems thetwodevelopedschemesbecomeequivalentintermsoftherateofconvergence. Thedistributionfunction, ˚ ¹ º ,de˝nedthedistributionofthedi˙erentiationfractional-order, ,anditcouldarbitrarilycon˝nethedomainoverwhichthefractionaldi˙erentiationistaken.If ˚ wasintegrableinacompactsupportin » min ; max ¼ ,then H min ¹ R º ˚ H¹ R º H max ¹ R º . Hence, ˚ couldplayacrucialruleinde˝ningtheunderlyingsolutionspaceproperly.Inanomalous physicalprocesses,thedistributionfunctioncanbeobtainedfromexperimentaldata,wherethe inherentdatauncertaintycanbeincorporatedthroughthe ˚ obtainedfromtheobserveddata,hence, leadingtoarobustdata-drivensimulationframeworkformulti-physicsproblems. Inchapter4,wecombinedthetwomodalandnodalexpansions,anddevelopedapseudo-spectral accurateschemebasedwhereweemployedtwotypesoffractionalLagrangeinterpolantsasthe 218 nodalbasisandtestfunctionsintheweaksenseofproblemandobtainedthecorrespondingweak distributeddi˙erentiationmatrix.Wefurtherinvestigatedtheconditionnumberoftheresulting linearsystemfordi˙erentchoicesofdistributionfunctionandinterpolationpoints.Weshowedthat amongtheconsideredchoices,therootsofJacobipolynomialsleadstobetterconditionnumber. Moreover,weintroducedasetofdistributedpre-conditionersbasedonthedistributionfunctionin theDODEsandJacobipoly-fractonomialsofsecondkind.Weshowedthatapplyingthedesigned pre-conditionerscanfurtherimprovetheconditionnumberofthelinearsystem.Theconstructed basisfunctionsarecomprisedofapolynomialpartandafractionalpart,wheretheformerleaves thefractionalorder, ,asafreeparametertocapturesolutionsingularities.Weshowedinthe exampleofinitialvalueproblemsthatbytunningtheinterpolationparameter,wecanachievethe highestrateofconvergencewithminimaldegreesoffreedom. ThedevelopedPetrov-Galerkin(PG)spectralmethodinchapter3hasalsothebene˝tofspectral accuracyinsolvingDODEs.However,theremainingchallengewastotreatnonlinearproblems.We showedthroughseveralexamplesof ¹ 1 + 1 º -Dand ¹ 1 + 2 º -Dtimedependentspacedistributed-order nonlinearproblems,thattheproposedpseudo-spectralschemecane˚cientlytreatnonlinearity, whilekeepingthesamerateofconvergence.Wecomputedtheassociatednonlinearvectorswith lesscomplexity,usingtheKroneckerdeltapropertyofthebasisandtestfunctions.Moreover,in comparisontofractionalcollocationmethods,weshowedthattheproposedschemeleadstoabetter conditioning,yetstillrequiresperformingadditionalquadratureintegrationinspatialdomain.The currentschemealsobene˝tsfromthewell-establishedmathematicalframeworkofBabu s ka-Lax- Milgramtheorem,whichcanbeusedalongwiththede˝nedunderlyingdistributedsobolevspace andthesharpestimates,providedbytheequivalent/bilinear-inducedassociatednorms,toperform theanalysisofscheme.Weintenttocarryoutandreporttheseanalysisinourfutureworks. Inchapter5,weextendedthederivationtofractionalPDEsanddevelopedaPetrov-Galerkin spectralmethodforhighdimensionaltemporally-distributedfractionalpartialdi˙erentialequations withtwo-sidedderivativesinaspace-timehypercube.WeemployedJacobipoly-fractonomialsand Legendrepolynomialsasthetemporalandspatialbasis/testfunctions,respectively.Tosolvethe 219 correspondingLyapunovlinearsystem,wefurtherformulatedafastlinearsolverandperformed thecorrespondingdiscretestabilityanderroranalysis.Wealsocarriedoutseveralnumerical simulationstoexaminetheperformanceofthemethod. Inordertoformulateasensitivityframework,inchapter6,wedevelopedafractionalsensitivity equationmethod(FSEM)inordertoanalyzethesensitivityoffractionalmodels(FIVPs,FBVPs, andFPDEs)withrespecttotheirparameters.Wederivedtheadjointgoverningdynamicsof sensitivitycoe˚cients,i.e.fractionalsensitivityequations(FSEs),bytakingthepartialderivative ofFDEwithrespecttothemodelparameters,andshowedthattheypreservethestructureof originalFDE.Wealsointroducedanewfractionaloperator,associatedwithlogarithmic-power lawkernel,forthe˝rsttimeinthecontextofFSEM.Weextendedtheexistingproperunderlying functionspacestorespecttheextraregularitiesimposedbyFSEsandprovedthewell-posedness ofproblem.Moreover,wedevelopedaPetrov-Galerkin(PG)spectralmethodbyemployingJacobi polyfractonomialsandLegendrepolynomialsasbasis/testfunctions,andproveditsstability.We furtherusedthedevelopedFSEMtoformulateanoptimizationprobleminordertoconstructthe fractionalmodelbyestimatingthemodelparameters.Wede˝nedtwotypesofmodelerroras objectivefunctionsandproposedatwo-stagessearchalgorithmtominimizethem.Wepresented thestepsofiterativealgorithminapseudocode.Finally,weexaminedtheperformanceofproposed numericalschemeinsolvingcoupledFPDEandFSEs,wherewenumericallystudytheconvergence rateoferror.Wealsoinvestigatedthee˚ciencyofdevelopediterativealgorithminestimatingthe derivativeorderfordi˙erentcasesoffractionalmodels. Inchapter7,wedevelopedamathematicalframeworktonumericallyquantifythesolution uncertaintyofastochasticFPDE,associatedwiththerandomnessofmodelparameters.The stochasticFPDEisreformulatedbyrenderingtheproblemwithrandomfractionalindices,subject toadditionalrandomnoise.WeusedthetruncatedKarhunen-Loéveexpansiontoparametrize theadditivenoise.Then,byemployinganon-intrusiveprobabilisticcollocationmethod(PCM), wepropagatedtheassociatedrandomnesstothesystemresponse,byusingSmolyaksparsegrid generatortoconstructthesetofsamplepointintherandomspace.Wealsoformulatedaforward 220 solvertosimulatethedeterministiccounterpartofthestochasticproblemforeachrealizationof randomvariables.Weshowedthatthedeterministicproblemismathematicallywell-posedina weaksense.Furthermore,byemployingJacobipoly-fractonomialsandLegendrepolynomialsas thetemporalandspatialbasis/testfunctions,respectively,wedevelopedaPetrove-Galerkinspectral methodtosolvethedeterministicprobleminthephysicaldomain.Wealsoprovedthattheinf-sup conditionholdsfortheproposednumericalscheme,andthus,itisstable.Byconsideringseveral numericalexampleswithlow-tohigh-dimensionalrandomspaces,weexaminedtheperformance ofourstochasticdiscretization.Weshowedthatineachcase,PCMconvergesveryfasttoavery highlevelofaccuracywithveryfewnumberofsampling. Finally,inchapter8,wefurtherapplythedevelopedmathematicaltoolstoinvestigatethe nonlinearvibrationofaviscoelasticcantileverbeam.Intheabsenceofexternalexcitation,the responseamplitudeoffreevibrationrevealsasuper-sensitivitywithrespecttothefractionalorder. Primaryresonanceofthebeamsubjecttobaseexcitationalsodisclosesasofteningbehaviorinthe frequencyresponseofthebeam.Theseuniquefeaturescanbeusedfurthertobuildavibration-based healthmonitoringplatform. 9.0.1FutureWorks Manyopenissuesremaininthis˝eldtobeaddressedinourfuturework.Here,welistsomeof themasfollows: PGspectralelementmethodfortwosidedderivatives: Theone-sidedHelmholtzequationin chapter2canbeextendedtoitstwo-sidedversion.However,inthatcaseduetothepresenceofthe left-andright-sidedderivatives,thecorrespondingintegration-by-partsrequirethetestfunctions tovanishatbothboundaries.Therefore,theintroducedchoicesoftestfunctionsinthisworkwould notformapropertestspaceforthetwo-sidedversion.Thisrequiresfurtherinvestigationasfuture work. Fractionaloperatorwithlogarithmicpower-lawkernel: Thesensitivityanalysisoffractional di˙erentialequationsintroducednewclassofintegro-di˙erentialoperatorswithweakersingular 221 kerneloflogarithmicpower-lawtype.Properdevelopmentofcalculusfortheseoperatorsisneeded toe˚cientlydealwithcomputationofsensitivity˝elds.Followingthesamederivationasinchapter 6,wecantakethederivativeoffractionalStrum-Liouvilleeigenvalueproblemtoobtainanewclass ofeigenvalueproblemsfortheseoperators.Thisisstillanopenproblemtobeinvestigatedfurther. Applicationtobio-tissuemechanics: Manyapplicationofviscoelasticmodelingareinbio- engineeringandbio-tissuemechanics.Theyalsoincludeproblems,wherehumanbodyundergoes certaindynamicalloadingduetoenvironmental/workconditions.Theexcessivebodymotioncan adverselyinduceundesiredvibrationtovitalorganssuchashumanbrain,leadingtoirrecoverable damages.Fractionalmodelsprovideatooltomodelandstudytheviscoelasticbehaviorofhuman organs.Thedevelopedframeworkinthisthesiscanbefurtheremployedto˝ndaccurateparam- etersofthesemodel,andthuscalculatingsafeoperatingregions,withinwhich,theorganwould experiencelessdamage.Suchapplicationrequirestheextensionofdevelopedparameterestimation anduncertaintyframeworktothecasewithlimitedrealdatasetsastheyarenotlargelyavailable forhumanbodies. Modalanalysisoffractionalviscoelasticity: Theeigenfunctionsoffractionaltimederivatives arenotexponentialfunctionsanymore.Forlinearfractionaloscillator,theMittagLe˜erfunctions canbeusedastheyincorporatememorydependence.But,inmorecomplexviscoelasticmodeling, whichleadtononlinearityinfractionaldi˙erentialequation,itisnotveryclearhowecanuse thesefunctions.Theextensionofmodalanalysisforviscoelasticmaterialsaswellasinteractive systemswithviscoelasticbehavior(suchaselasticsolidimmersedinviscous˛uid)canbefurther investigatedinfutureworks. Variableorderfractionalmodel: Theorderoffractionalmodelsdirectlyrelatestothedistinctive characteristicsofunderlyingphysicalphenomena.Overacourseoftime,thesepropertieschanges, andthusthefractionalmodelsshouldbere-calibrated.Instead,avariableordermodelcanbe developed,wheretheevolutionoffractionalordersaresuchthattheycomplywiththetime evolutionofphysicalproperties.Thiswillrequiresadditionaldynamics,whichmodels/describe thechangeofderivativeordersintime. 222 Applicationofrealdatainthedevelopedfractionalmodelconstructionframework: In general,theinverseproblemofparameterestimationisanill-posedproblem.Eventhoughwe showedcomputationallythattheintroducedmodelerrorsastheobjectivefunctiontominimize hassolelyoneminimum,weshouldextenttheanalysistomathematicallyprovetheexistence ofminimum(s).Moreimportantly,themethodshouldbeexaminedbyrealdatasets,astheir incompletenessimposesextrachallengeinminimizingtheobjectivefunction. 223 BIBLIOGRAPHY 224 BIBLIOGRAPHY [1] Acar,GizemD&BrianFFeeny.2017.Bend-bend-twistvibrationsofawindturbineblade. 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