A graded mesh technique for numerical approximation of a multi-term Caputo time-fractional Fokker-Planck equation in 2D space

Abstract

This paper focuses on the design of an efficient numerical approach for solving a two-dimensional multi-term Caputo time fractional Fokker-Planck (TFFP) model. The solution of such problem, in general, shows a weak singularity at the time origin. A numerical technique based on a graded time mesh is proposed to handle the singular behavior of the solution. The multi-term Caputo time fractional derivatives in the TFFP model are discretized by means of the L1 scheme on the nonuniform mesh, while a high-order compact alternating direction implicit finite difference scheme is designed to approximate the spatial derivatives. Convergence and stability analysis of the suggested method is analyzed. Two numerical examples subjected to smooth and nonsmooth exact solutions are presented to demonstrate the applicability and accuracy of the method. The results obtained by the proposed graded mesh technique are compared with the results obtained by the uniform mesh technique.