Fractional derivatives are integro-differential convolution type operators with power law kernels, which seamlessly generalize the notion of standard integer order differentiation to their fractional counter parts. In such operators, the order of differentiation is a non-integer number, which in the limiting cases of integer numbers recovers the standard derivatives. The fractional differential equations (FDEs) particularly, have been shown in the literature to provide a rigorous mathematical tool that can be used to describe the anomalous behavior in a wide range of physical phenomenon. They introduce however, the order of fractional derivatives as an additional set of model parameter, whose values are essentially obtained from experimental observations. The inherent randomness in measurements, incomplete sets of data, significant approximations and assumptions upon which the model is built, and the random nature of quantities being modeled pervade uncertainty in the corresponding mathematical formulations. This renders the model parameters, including the fractional orders, random and thus, the fractional model stochastic. We develop proper data-driven mathematical frameworks to efficiently infuse experimental observations/data into the corresponding mathematical models in the context of stochastic fractional partial differential equations.We extend the fractional order derivatives to the distributed order ones, where the differential orders are distributed over a range of values rather than being just a fixed integer/fraction as it is in standard/fractional ODEs/PDEs. Such distributed operators can also be considered as expectation of fractional derivative with random orders in the context of stochastic modeling. We develop two spectrally-accurate schemes, namely a Petrov-Galerkin spectral method and a spectral collocation method for distributed order fractional differential equations. In both methods, we employ the fractional (non-polynomial) functions, called Jacobi poly-fractonomials, which are the analytical eigenfunctions of the fractional Strum-Liouville eigenvalue problem of first and second kind. We also define the underlying distributed Sobolev space and the associated norms, where we carry out the corresponding discrete stability and error analyses of the proposed scheme. We develop a fractional sensitivity equation method, where we obtain the new set of adjoint fractional sensitivity equations, in which we introduce another fractional integor-differential operator, associated with logarithmic-power law kernel, for the first time in the context of fractional sensitivity analysis. We show that the developed sensitivity analysis provides a machine learning tool, which build a bridge between experiments and mathematical models to gear observable data via proper optimization techniques. We also develop an operator-based uncertainty quantification framework in the context of stochastic fractional partial differential equations, in which we characterize different sources of uncertainties and further propagate the associated randomness to the fractional model output quantity of interest.We further apply the developed mathematical tools to investigate the nonlinear vibration of a viscoelastic cantilever beam. In the absence of external excitation, the response amplitude of free vibration reveals a super-sensitivity with respect to the fractional order. Primary resonance of the beam subject to base excitation also discloses a softening behavior in the frequency response of the beam. These unique features can be used further to build a vibration-based health monitoring platform.
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