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SEMI-ANALYTICAL SOLUTIONS FOR THE BRUSSELATOR REACTION–DIFFUSION MODEL

Part of: Functional-differential and differential-difference equations Qualitative theory Parabolic equations and systems

Published online by Cambridge University Press:  23 October 2017

H. Y. ALFIFI Open the ORCID record for H. Y. ALFIFI [Opens in a new window]
H. Y. ALFIFI*
Affiliation:
Department of Basic Sciences, College of Education, Imam Abdulrahman Bin Faisal University, Dammam, Saudi Arabia email hyalfifi@iau.edu.sa
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Abstract

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Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

Keywords

semi-analytical solutionsreaction–diffusionBrusselator modelHopf bifurcationslimit cyclesystem of ordinary differential equations

MSC classification

Primary: 35K57: Reaction-diffusion equations
Secondary: 34C05: Location of integral curves, singular points, limit cycles 34K18: Bifurcation theory

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 59 , Issue 2 , October 2017 , pp. 167 - 182
Copyright
© 2017 Australian Mathematical Society 

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