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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 two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme.

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