A fast finite difference method for 2D time variable fractional mobile/immobile equation

In this paper, we establish a fast Crank–Nicolson L1 finite difference scheme for two-dimensional time variable fractional mobile/immobile diffusion equations. First, we discretize the time fractional derivative by the Crank–Nicolson formula on uniform meshes, and discretize the spatial derivative by the central difference quotient formula on uniform meshes to obtain a numerical scheme. Then, the Von-Neumann stability analysis method is used to analyze the stability and the optimal error estimate. On the other hand, we optimize the numerical format based on the exponential-sum-approximation technique, effectively reducing the amount of computation and storage. Finally, numerical examples validate the effectiveness of the algorithm.