Simultaneous inversion for a fractional order and a time source term in a time-fractional diffusion-wave equation

In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg-Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm.

Automated summary

In this article, the inverse problem of determining a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation is considered using a nonlocal condition. The uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem are proven. The Levenberg-Marquardt method is employed to recover the fractional order and the time source term simultaneously, and a finite-dimensional approximation algorithm is established to find a regularized numerical solution. Furthermore, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Numerical results in both one and multi-dimensional spaces are presented to demonstrate the robustness of the proposed algorithm.