Inverse Problems For Time-Space Fractional Diffusion Equation

This thesis deals with numerical solution to the time-space Caputo-Riesz fractional diffusion equation and inversion for the fractional orders and the source term. A finite difference scheme to solve the forward problem is set forth based on Caputo’s discretization to the time fractional derivative and Grünward-Letnikov’s discretization to the two-sided space fractional derivative, and the unconditional stability and convergence for the difference scheme are proved by fine estimation to the spectral radius of the coefficient matrix of the scheme. The derivatives of the solution operator on the fractional orders are analyzed at the same time by which the homotopy regularization algorithm is introduced. Numerical inversions for determining the fractional orders and the source coefficient with random noisy data using the final observations or the measurements at one inner point are presented. The numerical solutions give good approximations to the exact solution demonstrating the effectiveness of the proposed algorithm.The main contents are given as follows:In Chapter 1, background and significance of this dissertation are introduced, and the main researching works and structure of this dissertation are given.In Chapter 2, numerical methods are studied for solving the time-space Caputo-Riesz fractional diffusion equation. For the forward problem with general continuous and initial conditions, an implicit finite difference scheme is set forth and its stability and convergence are proved in a simple way, and numerical tests are presented.In Chapter 3, the homotopy regularization algorithm is introduced based on the analysis to the derivative of the fractional orders of the analytical solution. Numerical inversion for the two fractional orders is performed using homotopy regularization algorithm in the case of using noisy data.In Chapter 4, the homotopy regularization algorithm is applied to solve the inverse source problem for the inhomogeneous time-space fractional equation, and several factors which have influences on the inversion algorithm are discussed.In Chapter 5, a summary of the thesis is given, and some related problems which could be considered for the future work are discussed.