TWO REGULARIZATION METHODS FOR IDENTIFYING THE UNKNOWN SOURCE IN A MULTITERM TIME-FRACTIONAL DIFFUSION EQUATION

We study an inverse source problem for a multiterm time-fractional diffusion equation from a noisy final data in a general bounded domain. This problem is ill-posed. Uniqueness and a conditional stability for the inverse problem are derived based on an expression of the solution and some properties of the multinomial Mittag-Leffler function. Further we introduce the modified quasiboundary regularization method and the Landweber iterative regularization method to solve the inverse source problem. Convergence estimates between the regularization solution and the exact solution are given under the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule, respectively. Finally, we use the finite difference method to solve the direct problem and the inverse source problem in the one-dimensional case, and apply the finite element method to solve them in the two-dimensional case. Numerical examples are provided to show the effectiveness of the proposed method in the one- and two-dimensional cases.