A fractional-order quasi-reversibility method to a backward problem for the time fractional diffusion equation

In this paper, we consider the regularization of the backward problem of diffusion process with time-fractional derivative. Since the equation under consideration involves the time-fractional derivative, we introduce a new perturbation which is related to the time-fractional derivative into the original equation. This leads to a fractional-order quasi-reversibility method. In theory, we give the regularity of the regularized solution and the corresponding convergence rate is also proved under the appropriate regularization parameter choice rule. In numerics, some numerical experiments are presented to illustrate the effectiveness of our method and some numerical comparison with the existing quasi-reversibility method is conducted. Both theoretical and numerical results show the advantage of the new method.