Solving the reaction-diffusion Brusselator system using Generalized Finite Difference Method

Ángel García; Francisco Ureña; Antonio M. Vargas; Ángel García; Francisco Ureña; Antonio M. Vargas

doi:10.3934/math.2024644

AIMS Mathematics

2024, Volume 9, Issue 5: 13211-13223. doi: 10.3934/math.2024644

Previous Article Next Article

Research article Special Issues

Solving the reaction-diffusion Brusselator system using Generalized Finite Difference Method

1.
Departamento de Construcción y Fabricación, UNED, Madrid, Spain
2.
Departamento de Matemáticas Fundamentales, UNED, Madrid, Spain

Received: 14 December 2023 Revised: 25 March 2024 Accepted: 29 March 2024 Published: 09 April 2024
MSC : 35K57, 65N06

In this paper, we investigate the numerical solution of the Brusselator system using a meshless method. A numerical scheme is derived employing the formulas of the Generalized Finite Difference Method, and the convergence of the approximate solution to the exact solution is examined. In order to demonstrate the applicability and accuracy of the method, several examples are proposed.
- Brusselator,
- Generalized Finite Differences,
- meshless method,
- convergence
Citation: Ángel García, Francisco Ureña, Antonio M. Vargas. Solving the reaction-diffusion Brusselator system using Generalized Finite Difference Method[J]. AIMS Mathematics, 2024, 9(5): 13211-13223. doi: 10.3934/math.2024644

Related Papers:

Abstract

In this paper, we investigate the numerical solution of the Brusselator system using a meshless method. A numerical scheme is derived employing the formulas of the Generalized Finite Difference Method, and the convergence of the approximate solution to the exact solution is examined. In order to demonstrate the applicability and accuracy of the method, several examples are proposed.

References

[1]	A. Whye-Teong, The two-dimensional reaction-diffusion Bruselator system: A dual-reciprocity boundary element solution, Eng. Anal. Bound. Elem., 27 (2003), 897–903. https://doi.org/10.1016/S0955-7997(03)00059-6 doi: 10.1016/S0955-7997(03)00059-6
[2]	M. Mohammadi, R. Mokhtart, R. Schaback, A Meshless Method for Solving the 2D Brusselator Reaction-Diffusion System, CMES, 101 (2014), 113–138.
[3]	Z. Zafar, K. Rehan, M. Mushtaq, M. Rafiq, Numerical treatment for nonlinear Brusselator chemical model, J. Differ. Equ. Appl., 23 (2017), 521–538, http://doi.org/10.1080/10236198.2016.1257005 doi: 10.1080/10236198.2016.1257005
[4]	R. K. Saeed, S. A. Manaa, F. H. Easif, Numerical Solution of Brusselator Model by Expansion Methods, Aust. J. Basic Appl. Sci., 4 (2010), 3389–3403.
[5]	R. C. Mittal, R. Jiwari, Numerical solution of two-dimensional reaction-diffusion Brusselator system, Appl. Math. Comput., 217 (2011), 5404–5415. https://doi.org/10.1016/j.amc.2010.12.010 doi: 10.1016/j.amc.2010.12.010
[6]	W. Sun, W. Qu, Y. Gu, P. W. Li, An arbitrary order numerical framework for transient heat conduction problems, Int. J. Heat Mass Tran., 218 (2024), 124798. https://doi.org/10.1016/j.ijheatmasstransfer.2023.124798 doi: 10.1016/j.ijheatmasstransfer.2023.124798
[7]	W. Sun, H. Ma, W. Qu, A hybrid numerical method for non-linear transient heat conduction problems with temperature-dependent thermal conductivity, Appl. Math. Lett., 146 (2024), 108868. https://doi.org/10.1016/j.aml.2023.108868 doi: 10.1016/j.aml.2023.108868
[8]	J. Benito, F. Ureña, L. Gavete, B. Alonso, Application of the Generalized Finite Difference Method to improve the approximated solution of PDEs, CMES, 38 (2009), 39–58.
[9]	J. Benito, F. Ureña, L. Gavete, Influence of several factors in the generalized finite difference method, Appl. Math. Modell., 25 (2001), 1039–1053. https://doi.org/10.1016/S0307-904X(01)00029-4 doi: 10.1016/S0307-904X(01)00029-4
[10]	L. Gavete, F. Ureña, J. Benito, A. Garcia, M. Ureña, E. Salete, Solving second order non-linear elliptic partial differential equations using generalized finite difference method, J. Comput. Appl. Math., 318 (2017), 378–387. https://doi.org/10.1016/j.cam.2016.07.025 doi: 10.1016/j.cam.2016.07.025
[11]	F. Ureña, L. Gavete, A. Garcia, J. Benito, A. M. Vargas, Solving second order non-linear parabolic PDEs using generalized finite difference method (GFDM), J. Comput. Appl. Math., 354 (2019), 221–241. https://doi.org/10.1016/j.cam.2018.02.016 doi: 10.1016/j.cam.2018.02.016

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)