A Fractional-order Quasi-reversibility Method to a Backward Problem for the Multi-term Time-fractional Diffusion Equation

In the present paper, we devote our efforts to a backward problem for an anomalous diffusion model with multi-term time fractional derivatives. Such a problem is ill-posed. For this purpose, we introduce a fractional-order quasi-reversibility regu-larization method that is a new perturbation related to the time fractional derivative into the original equation. Based on some properties of the multinomial Mittag-Leffler function as well as the Fourier method, we theoretically give some regularity results of the regularized solution, and prove the corresponding convergence rate under the a-priori regularization parameter choice rule in the general dimensional case. Finally, several numerical examples are given to demonstrate the effectiveness of the proposed method. The numerical results are well in line with our expectations.