In this paper, we consider a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine the initial data from a noisy final data. Based on a series expression of the solution, a conditional stability for the initial data is given. Further, we propose a modified quasi-boundary value regularization method to deal with the backward problem and obtain two kinds of convergence rates by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed methods.

In this paper, we consider a fully discrete finite element method (FEM) to solve the two-dimensional nonlinear Fisher' equation with Riesz fractional derivatives in space. This method is chiefly performed by using Crank–Nicolson discretization in conjunction with a linearized approach in time and FEM in space. The existence, uniqueness of the weak solution, and the numerical stability of the scheme are proved in great detail. The optimal error estimate computed by $L^2$-norm showed both in time and space is derived by introducing a fractional orthogonal projection. Moreover, several numerical examples are conducted on unstructured triangular meshes by a properly designed algorithm.