The fractional Landweber method for identifying the space source term problem for time-space fractional diffusion equation

Abstract

This paper is devoted to solve an inverse problem for identifying the source term of a time-fractional nonhomogeneous diffusion equation with a fractional Laplacian in a non-local boundary. Based on the expression of the solution for the direct problem, the inverse problem for searching the space source term is converted into solving the first kind of Fredholm integral equation. The conditional stability for the inverse source problem is investigated. The fractional Landweber method is used to deal with this inverse problem and the regularized solution is also obtained. Furthermore, the convergence rates for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Several numerical examples are given to show the proposed method is efficient and stable.