High order algorithms for Fokker-Planck equation with Caputo-Fabrizio fractional derivative

TL;DR

This paper develops high-order numerical algorithms for the Fokker-Planck equation involving Caputo-Fabrizio fractional derivatives, enabling accurate modeling of complex physical phenomena with proven stability and convergence.

Contribution

It introduces novel discretization schemes for Caputo-Fabrizio derivatives with high-order accuracy and analyzes their stability and convergence for solving related fractional diffusion equations.

Findings

Discretization schemes with error order $ au^ u$ for $ u=1,2,3,4

Unconditional stability of the proposed schemes

Convergence with error $ au^2+h^2$

Abstract

Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures with different scales. Extending the discretizations for fractional substantial calculus [Chen and Deng, \emph{ ESAIM: M2AN.} \textbf{49}, (2015), 373--394], we first provide the numerical discretizations of the Caputo-Fabrizio fractional derivative with the global truncation error $\mathcal{O}(\tau^\nu)$ $ (\nu=1,2,3,4)$. Then we use the derived schemes to solve the Caputo-Fabrizio fractional diffusion equation. By analysing the positive definiteness of the stiffness matrices of the discretized Caputo-Fabrizio operator, the unconditional stability and the convergence with the global truncation error $\mathcal{O}(\tau^2+h^2)$ are theoretically proved and numerical verified.