On Galerkin spectral element method for solving Riesz fractional diffusion equation based on Legendre polynomials

Document Type : Research Article

Authors

1 LaR2A, FS, Abdelmalek Essaadi University, Tetouan, Morocco

2 LaR2A, FS, Abdelmalek Essaad University, Tetouan, Morocco

Abstract

This paper presents a Galerkin spectral element method for solving a fractional diffusion equation, considering initial and boundary conditions. We construct a discrete scheme for time, employing the Crank-Nicolson method to approximate the Caputo fractional derivative on a uniform mesh. Then we introduce a Galerkin variational formulation to establish the unconditional stability of the scheme. Moreover, we apply the spectral element method based on Legendre polynomials in the space direction and obtain the fully discrete scheme. The error analysis of the fully discrete scheme is treated in $L_2$ sense. we present a computational analysis to deal with the Galerkin spectral element method, to compute the corresponding bilinear form, on the implementation process. Finally, we prove the effectiveness of the method through numerical experiments and some simulations using MATLAB software.

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Journal of Mathematical Modeling
Volume 13, Issue 2
May 2025
Pages 281-301

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