Identification of the spatial dependent source and the fractional order in a time-fractional convection diffusion-wave equation

In this paper, we dedicate our efforts to investigating an inverse problem of simultaneously recovering a spatial-dependent source and the fractional order by using some internal measurement data for a time-fractional convection diffusion-wave equation. We prove the uniqueness of the inverse problem and the Lipschitz continuity of the forward operator. Then, the inverse problem is formulated as a variational problem using a Tikhonov-type regularization. Based on the continuity of the forward operator, we demonstrate that the minimizer of the Tikhonov-type functional exists and converges to the exact solution under an a priori choice of the regularization parameter. The steepest descent method combined with the Nesterov acceleration is adopted to solve the variational problem. Four numerical examples are presented to show the effectiveness of our proposed method, including examples in one-dimensional and two-dimensional cases.