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.

在本文中，我们考虑了具有时间分数导数的扩散过程的后向问题的正则化。由于考虑中的方程涉及时间分数导数，因此我们将与时间分数导数有关的新扰动引入原始方程。这导致了分数阶准可逆性方法。从理论上讲，我们给出了正则化解的规则性，并且在适当的正则化参数选择规则下也证明了相应的收敛速度。在数值上，进行了一些数值实验以说明我们方法的有效性，并与现有的准可逆性方法进行了数值比较。理论和数值结果均表明了该方法的优势。