This dissertation covers several topics in Applied Mathematics,including nonlinear Poisson equation(NLPE) with application in electrostatics and solvation analysis for biological system,partial differential equation(PDE) transform for hyperbolic conservation laws,high order fractional PDE transform for molecular surface construction and Poisson-Kohn-Sham equation for modeling geometric, thermal and tunneling effects on nano-transistors.Electrostatic interactions are ubiquitous in nature and fundamental for chemical, biological and material sciences. The Poisson equation is a widely accepted model for electrostatic analysis. However, the Poisson equation is derived based on electric polarizations in a linear, isotropic and homogeneous dielectric medium. We introduce a nonlinear Poisson equation to take into consideration of hyperpolarization effects due to intensive charges and possible nonlinear, anisotropic and heterogeneous media. Variational principle is utilized to derive the nonlinear Poisson model from an electrostatic energy functional. To apply the proposed nonlinear Poisson equation for the solvation analysis, we also construct a nonpolar solvation energy functional based on the nonlinear Poisson equation by using the geometric measure theory. The proposed nonlinear Poisson theory is extensively validated by the electrostatic analysis of the Kirkwood model, 17 small molecules and 20 proteins at a fixed temperature as well as 21 compounds at different temperatures. A good agreement between our results and experimental data as well as theoretical results suggests that the proposed nonlinear Poisson model is a potentially useful model for electrostatic analysis involving hyperpolarization effects.In our next work, we introduce the use of the PDE transform, paired with the Fourier pseudospectral method (FPM),as a new approach for hyperbolic conservation law problems, which remains an interesting and challenging task due to the diversity of physical origins and complexity of the physical situations.The PDE transform, based on the use of arbitrarily high order evolution PDEs, is a new algorithm for splitting signals, surfaces and data to functional mode functions, such as trend, edge, noise etc.A fast PDE transform implemented by the fast Fourier Transform (FFT) is introduced to avoid stability constraint of integrating high order evolution PDEs. An adaptive measure of total variations is utilized to automatically switch on and off the PDE transform during the time integration of conservation law equations. A variety of standard benchmark test problems of hyperbolic conservation laws is employed to systematically validate the performance of the present PDE transform based FPM. The impact of two PDE transform parameters, i.e., the highest order and the propagation time, is carefully studied to deliver the best effect of suppressing Gibbs' oscillations. The PDE orders of 2-6 are used for hyperbolic conservation laws of low oscillatory solutions, while the PDE orders of 8-12 are often required for problems involving highly oscillatory solutions, such as shock-entropy wave interactions. The present results are compared with those in the literature. It is found that the present approach not only works well for problems that favor low order shock capturing schemes, but also exhibits superb behavior for problems that require the use of high order shock capturing methods.Furthermore, we study the high-order factional PDE transform based on fractional derivative with application in molecular surface generation. Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present.Our work introduces arbitrarily high-order PDEs to describe fractional hyper-diffusions. The fractional PDEs are constructed via fractional variational principle.Furthermore, we construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDE transform are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvationfree energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.The last part of my work is in the filed of nano-scale electronic transistors. The miniaturization of nano-scale electronic transistors, such as metal oxide semiconductor field effect transistors (MOSFETs), has given rise to a pressing demand in the new theoretical understanding and practical tactic for dealing with quantum mechanical effects in integrated circuits. We study the effects of geometry of semiconductor-insulator interfaces, phonon-electron interactions, and quantum tunneling of nano-transistors. Mathematical models of these factors are based on a unified two-scale energy functional that describes free energy of electrons and their interactions with external environments. Related numerical tools and algorithms are introduced to perform simulations on 3D nano four-gate MOSFETs with different geometries of silicon/silicon dioxide interfaces. Phonon-electron interactions are modeled in fashion of density functional theory and integrated in the general free energy formulation. Quantum tunneling effects are defined as electron tunneling ratios and calculated for each type of nano-MOSFETs. Performances of nano-transistors are explored in terms of current-voltage (I-V) curves and quantized transport energy profiles in a wide range of device parameters.
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