In magnetic resonance imaging of the human brain, the diffusion process of tissue water is considered in the complex tissue environment of cells, membranes and connective tissue. Models based on fractional order Bloch–Torrey equations are known to provide insights into tissue structures and the microenvironment.

In this paper, we consider new two-dimensional multi-term time and space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains, which involve the Caputo time fractional derivative and the Riemann–Liouville space fractional derivative. An unstructured-mesh Galerkin finite element method is used to discretize the spatial fractional derivative, while for each time fractional derivative we use the $L1$ scheme on a temporal graded mesh. The stability and convergence of the fully discrete scheme are proved. Numerical examples are given to verify the efficiency of our method.

在人脑的磁共振成像中，在细胞，膜和结缔组织的复杂组织环境中考虑了组织水的扩散过程。已知基于分数阶Bloch-Torrey方程的模型可提供有关组织结构和微环境的见解。

在本文中，我们考虑了在不规则凸域上具有可变系数的新的二维多维时空分数Bloch-Torrey方程，其中涉及Caputo时间分数导数和Riemann-Liouville空间分数导数。使用非结构网格Galerkin有限元方法离散化空间分数导数，而对于每个时间，分数导数都使用$\mathrm{大号}\mathrm{1个}$时间渐变网格上的一种方案。证明了完全离散方案的稳定性和收敛性。数值例子验证了我们方法的有效性。