The Variable-Order Fractional Diffusion Equation And Inverse Problems

In this dissertation we discuss three kinds of forward problems and inversion problems for the variable fractional orders in the time variable fractional order dispersion equation,the variable-order fractional advection-diffusion equation with a nonlinear source on a finite domain and the two-dimensional variable-order fractional percolation equation,respectively. Finite difference scheme is established for each forward problem, and the stability and convergence are proved. Based on the numerical solution, the homotopy regularization algorithm is applied to determine the variable fractional orders under different conditions. The main contents are given as follows:Chapter 1, introduces the background and motivation of this thesis as well as the research trends, and the main works.Chapter 2, gives some concepts of the variable fractional derivatives, including four kinds of definitions, and their relations.Chapter 3, an implicit finite difference scheme is introduced to solve the variable-order time-fractional diffusion equation, and an inverse problem of determining the variable fractional order is considered using the additional measurements at one interior point. The homotopy regularization algorithm is applied to solve the inverse problem, and numerical examples are presented.In chapter 4, we consider a variable-order space fractional diffusion equation. Explicit approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, also the homotopy regularization algorithm is utilized to solve the inversion problem to determine the fractional orders, and numerical examples are presented.Chapter 5 deals with the 2-D variable-order space fractional diffusion equation, and Euler alternative difference scheme is established with stability and consistency.Furthermore, numerical inversions are performed using the inversion algorithm to determine the variable fractional order.Chapter 6, a summary is given, and some related problems which could be considered in the future are proposed.