A Jacobi Gauss–Lobatto and Gauss–Radau collocation algorithm for solving fractional Fokker–Planck equations

Abstract

In this article, we construct a new numerical approach for solving the time-fractional Fokker–Planck equation. The shifted Jacobi polynomials are used as basis functions, and the fractional derivative is described in the sense of Caputo. The proposed approach is a combination of shifted Jacobi Gauss–Lobatto scheme for the spatial discretization and the shifted Jacobi Gauss–Radau scheme for temporal approximation. The problem is then reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. In addition, our numerical algorithm is also applied for solving the space-fractional Fokker–Planck equation and the time–space-fractional Fokker–Planck equation. Numerical results are consistent with the theoretical analysis, indicating the high accuracy and effectiveness of the proposed algorithm.