On the effective method for the space-fractional advection-diffusion equation by the Galerkin method

Haifa Bin Jebreen; Hongzhou Wang; Haifa Bin Jebreen; Hongzhou Wang

doi:10.3934/math.20241173

AIMS Mathematics

2024, Volume 9, Issue 9: 24143-24162. doi: 10.3934/math.20241173

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Research article

On the effective method for the space-fractional advection-diffusion equation by the Galerkin method

Haifa Bin Jebreen ^{1
,
,},
Hongzhou Wang ²

1.
Department of mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Beijing Key Lab on Mathematical Characterization, Analysis, and Applications of Complex Information, MIIT Key Laboratory of Mathematical Theory and Computation in Information Security, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, China

Received: 22 June 2024 Revised: 19 July 2024 Accepted: 22 July 2024 Published: 14 August 2024
MSC : 26A33, 54A25, 65M70

The present work is dedicated to a study that focuses on solving space-fractional advection-diffusion equations (SFADEs) using the Galerkin method. Through our analysis, we demonstrate the effectiveness of this approach in solving the considered equations. After introducing the Chebyshev cardinal functions (CCFs), the Caputo fractional derivative (CFD) was represented based on these bases as an operational matrix. Applying the Galerkin method reduces the desired equation to a system of algebraic equations. We have proved that the method converges analytically. By solving some numerical examples, we have demonstrated that the proposed method is effective and yields superior outcomes compared to existing methods for addressing this problem.
- Chebyshev cardinal functions,
- space-fractional advection-diffusion equations,
- Galerkin method,
- convergence analysis
Citation: Haifa Bin Jebreen, Hongzhou Wang. On the effective method for the space-fractional advection-diffusion equation by the Galerkin method[J]. AIMS Mathematics, 2024, 9(9): 24143-24162. doi: 10.3934/math.20241173

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Abstract

The present work is dedicated to a study that focuses on solving space-fractional advection-diffusion equations (SFADEs) using the Galerkin method. Through our analysis, we demonstrate the effectiveness of this approach in solving the considered equations. After introducing the Chebyshev cardinal functions (CCFs), the Caputo fractional derivative (CFD) was represented based on these bases as an operational matrix. Applying the Galerkin method reduces the desired equation to a system of algebraic equations. We have proved that the method converges analytically. By solving some numerical examples, we have demonstrated that the proposed method is effective and yields superior outcomes compared to existing methods for addressing this problem.

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