Modeling, Analysis, Approximation, and Application of Viscoelastic Structures and Anomalous Transport

(Variable-order) fractional partial differential equations are emerging as a competitive means to integer-order PDEs in characterizing the memory and hereditary properties of physical processes, e.g., anomalously diffusive transport, viscoelastic mechanics and financial mathematics, and thus have attracted widespread attention. In particular, optimal control problems governed by fractional partial differential equations are attracting increasing attentions since they are shown to provide competitive descriptions of challenging physical phenomena. Nevertheless, variable-order fractional models exhibit salient features compared with their constant-order analogues and introduce mathematical difficulties that are not typical encountered in the context of integer-order and constant-order fractional partial differential equations. This dissertation intends to employ viscoelastic materials as an illustrative example to investigate the modeling, mathematical analysis and numerical approximations to variable-order fractional derivative problems, including viscoelastic Euler Bernoulli beams as well as the corresponding inverse problem, viscoelastic Timoshenko beams, viscoelastic Kirchhoff plates as well as the optimal control problem of the two-time-scale time fractional advection-diffusion-reaction equation. From analysis point of view, two techniques are developed to accommodate the impact of the variable fractional order. One method is to employ the spectral decomposition method to reduce the problem to the component ordinary differential equations (ODEs), in which the impact of the variable order could be handled such that the analysis can be processed. Another method is to consider the variable-order fractional term as a perturbation of the first-order time derivative and then apply the auxiliary equation method and Fredholm alternative for compact operators to analyze the well-posedness of the variable-order FPDEs. The proposed mathematical and numerical methods provide potential tools to analyze and compute the variable-order fractional problems.