In this paper, we consider the numerical methods for both the forward and backward problems of a time-space fractional diffusion equation. For the two-dimensional forward problem, we propose a finite difference method. The stability of the scheme and the corresponding Fast Preconditioned Conjugated Gradient algorithm are given. For the backward problem, since it is ill-posed, we use a quasi-boundary-value method to deal with it. Based on the Fourier transform, we obtain two kinds of order optimal convergence rates by using an a-priori and an a-posteriori regularization parameter choice rules. Numerical examples for both forward and backward problems show that the proposed numerical methods work well.

在本文中，我们考虑时空分数扩散方程的前向和后向问题的数值方法。对于二维前向问题，我们提出了有限差分法。给出了该方案的稳定性以及相应的快速预条件共轭梯度算法。对于后向问题，由于它是不适定的，因此我们使用拟边值方法来处理它。基于傅里叶变换，我们利用先验和后验正则化参数选择规则获得了两种阶最优收敛速度。前向和后向问题的数值例子表明，所提出的数值方法效果良好。