Numerical solution of the simultaneous identification problem for a time-fractional diffusion equation

Abstract

It is a key problem to determine the initial diffusion time and initial value in a time fractional diffusion model when using the fractional diffusion model to describe the emerging anomalous diffusion phenomenons in real world. This paper considers a simultaneous identification of the initial diffusion time, fractional order and the initial value for a time fractional diffusion equation. First, the uniqueness results are proven by employing the Laplace transformation method in conjunction with the method of eigenfunction expansions, and by utilizing certain types of additional information regarding the solution. Then the Tikhonov regularization method is adopted to approximate the identification problem, and the existence of optimal solutions to the Tikhonov functional is proven. Next an alternating minimization algorithm is adopted to solve the regularization optimization problem, and the convergence of the alternating minimization algorithm is analyzed. At last, the efficiency and stability of the reconstruction algorithm are tested by several one- and two-dimensional examples.