On an explicit finite difference method for fractional diffusion equations

TL;DR

This paper develops an explicit finite difference method combining FTCS and Grunwald-Letnikov derivatives to numerically solve fractional diffusion equations, enabling stable and accurate simulations of anomalous transport phenomena.

Contribution

It introduces a novel explicit fractional FTCS scheme with stability analysis, extending traditional methods to fractional diffusion equations.

Findings

The method is stable within derived bounds.

Numerical results agree with analytical solutions.

Applicable to various fractional dynamics equations.

Abstract

A numerical method to solve the fractional diffusion equation, which could also be easily extended to many other fractional dynamics equations, is considered. These fractional equations have been proposed in order to describe anomalous transport characterized by non-Markovian kinetics and the breakdown of Fick's law. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald-Letnikov definition of the fractional derivative operator to obtain an explicit fractional FTCS scheme for solving the fractional diffusion equation. The resulting method is amenable to a stability analysis a la von Neumann. We show that the analytical stability bounds are in excellent agreement with numerical tests. Comparison between exact analytical solutions and numerical predictions are made.