Reconstruction of a Space-dependent Source Term for a Time Fractional Diffusion Equation by a Modified Quasi-boundary Value Regularization Method

Abstract

In this paper, a modified quasi-boundary value regularization method is proposed to solve an inverse source problem for time fractional diffusion equation. First, the regularization problem is proposed and the existence and the uniqueness of the regularized solution are proven. Then based on some source condition for the source term and the selection strategies for the regularization parameter, the a prior convergence rate and the posteriori convergence rate of regularization solution are derived by using the eigenfunction expansion method. Next, an all-in-one matrix form with block-Toeplitz structure is obtained by using the finite difference discretization for the regularized problem, and an efficient preconditioning technique is adopted to approximately solve the regularization solution. Finally, several numerical examples are presented to evaluate the effectiveness of the inversion method.