Simultaneous inversion of time-dependent source term and fractional order for a time-fractional diffusion equation

Abstract

In this article, we consider a simultaneous identification of fractional order and time dependent source term in a time fractional diffusion equation. Firstly, we establish the uniqueness of the simultaneous identification problem by Laplace transformation and analytical continuation. Then, using the Tikhonov regularization, we convert the inverse problem into a Tikhonov functional optimization problem. The existence of minimizer to the Tikhonov functional is obtained. Meanwhile, we adopt an alternating minimization algorithm to solve the regularized optimization problem. Based on the analysis of Lipschitz continuity for the forward operator, the convergence of the inversion algorithm is proven. The performance of the inversion algorithm is tested by several numerical examples.