In this paper, we shall present the alternating direction implicit (ADI) Galerkin finite element method (FEM) for solving the distributed-order time-fractional mobile–immobile equation in two dimensions. In the time direction, the backward Euler method is used to deal with the temporal first-order derivative, and the weighted and shifted Grünwald formula is employed to discretize the distributed-order time-fractional derivative. Galerkin FEM is used for discretization of the spatial direction, and then an ADI Galerkin finite element scheme is constructed. The stability and $L^2$-norm convergence are proved. Several numerical experiments are provided to verify our theoretical analysis.

在本文中，我们将提出交替方向隐式（ADI）Galerkin有限元方法（FEM），以二维求解分布式时分分数阶动-不动方程。在时间方向上，使用后向Euler方法处理时间一阶导数，并使用经过加权和移位的Grünwald公式离散化时间阶导数。将Galerkin有限元用于空间方向的离散化，然后构建ADI Galerkin有限元方案。稳定性和${\mathrm{大号}}^2$-范数收敛。提供了一些数值实验来验证我们的理论分析。