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

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 $\mathcal{O}(\ddot{I}{„}^{\hat{I}}½)$ $(\hat{I}½=1,2,3,4)$. 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 $\mathcal{O}(\ddot{I}{„}^2+h^2)$ are theoretically proved and numerical verified.

At first sight, fractional derivatives defined using non-singular kernels may appear very attractive. Thus, it is unsurprising that these simpler operators have become quite popular since their appearance about five years ago. But these operators with non-singular kernels have serious shortcomings that strongly discourage their use, see [Fract. Calc. Appl. Anal., 23, 610-634, 2020]