Theoretical And Numerical Investigation Of Some Fractional Differential Equations

Fractional PDEs appear in modeling transport dynamics in complex systems specially having the anomalous diffusion.It has been found that some phenomena in various fields of science,mathematics,engineering,and economics can be beter described by using fractional derivatives.Due to the non-local nature of the fractional derivatives,local methods such as finite difference and finite element loss the advantage that they show for usual partial differential equations.This means that the main disadvantage of global methods such as spectral methods is no longer an issue for fractional PDEs.In contrast,the spectral method to fractional PDEs enjoys great advantage when the solution is smooth enough.The main focus of this thesis is to develop some efficient methods to solve a class of FPDEs.The outline of the thesis is as follows:In Chapter 1,we give a brief review about the recent progress on fractional differential equation,present the motivations and main contents of the thesis,and list some relevant preliminaries.In Chapter 2,we investigate the well-posedness of some basic FPDEs,discuss the choice of boundary conditions.In particular,we consider the existence and uniqueness of continuous solution to a FPDE with mixed Riemann-Liouville and Caputo fractional derivatives.The obtained results will be found useful for the discussion in the next chapters.In Chapter 3,a family of tensor-based methods,called Proper Generalized Decom-position(PGD)methods is proposed and analyzed for numerical solutions of a two-dimensional fractional differential equation with variable coefficients.The PGD method has originally been developed for traditional differential equations,and has demonstrated the capability of representing the solution with a significant reduction of the calculation and storage cost.The purpose of this chapter is to show that the PGD method can also be successfully used for fractional equations with variable coefficients.We will illustrate the efficiency by a series of numerical experiments.In Chapter 4,we study the well-posedness of a fractional Stokes equation and its numerical solution.We first establish the well-posedness of the weak problem by suitably define the.fractional Laplacian operator and associated functional spaces.The existence and uniqueness of the weak solution is proved by using the classical saddle-point theory.Then,based on the proposed variational framework,we construct an efficient spectral method for numerical approximations of the weak solution.The main contribution of this part is threefold:1)a theoretical framework for the variational solutions of the fractional Stokes equation;2)an efficient spectral method for solving the weak problem,together with a detailed numerical analysis providing useful error estimates for the approximative solution;3)a fast implementation technique for the proposed method and investigation of the discrete system.Finally,some numerical experiments are carried out to confirm the theoretical results.