An alternating direction implicit Galerkin finite element method for the distributed-order time-fractional mobile–immobile equation in two dimensions

Abstract 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 Grunwald 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.