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.