High Order Numerical Scheme for Generalized Fractional Diffusion Equations

TL;DR

This paper introduces a higher order finite difference scheme for solving generalized fractional diffusion equations, incorporating various fractional derivatives and analyzing its stability, convergence, and numerical performance.

Contribution

It presents a novel higher order finite difference scheme for GFDEs that accounts for different fractional derivatives via scale and weight functions, with stability and convergence analysis.

Findings

The scheme accurately approximates solutions for various fractional derivatives.

Numerical tests confirm the stability and convergence of the proposed method.

The method effectively models physical systems with generalized fractional derivatives.

Abstract

In this paper, a higher order finite difference scheme is proposed for Generalized Fractional Diffusion Equations (GFDEs). The fractional diffusion equation is considered in terms of the generalized fractional derivatives (GFDs) which uses the scale and weight functions in the definition. The GFD reduces to the Riemann-Liouville, Caputo derivatives and other fractional derivatives in a particular case. Due to importance of the scale and the weight functions in describing behaviour of real-life physical systems, we present the solutions of the GFDEs by considering various scale and weight functions. The convergence and stability analysis are also discussed for finite difference scheme (FDS) to validate the proposed method. We consider test examples for numerical simulation of FDS to justify the proposed numerical method.