Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T12:44:05.134Z Has data issue: false hasContentIssue false
  • English
  • Français

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 [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

Abramowitz, M. and Stegun, I. M., Handbook of mathematical functions (Dover, New York, 1965).Google Scholar
Adomian, G., Solving frontier problem of physics: the decomposition method (Kluwer Academic, New York, 1994).CrossRefGoogle Scholar
Adomian, G., “The diffusion-Brusselator equation”, Comput. Math. Appl. 29 (1995) 1–3; doi:10.1016/0898-1221(94)00244-F.CrossRefGoogle Scholar
Al Noufaey, K. S. and Marchant, T. R., “Semi-analytical solutions for the reversible Selkov model with feedback delay”, Appl. Math. Comput. 232 (2014) 49–59; doi:10.1016/j.amc.2014.01.059.Google Scholar
Alfifi, H. Y., Marchant, T. R. and Nelson, M. I., “Generalised diffusive delay logistic equations: semi-analytical solutions”, Dyn. Contin. Discrete Impuls. Syst. Ser. B 19 (2012) 579–596; http://online.watsci.org/contents2012/v19n4-5b.html.Google Scholar
Alfifi, H. Y., Marchant, T. R. and Nelson, M. I., “Semi-analytical solutions for the 1- and 2-D diffusive Nicholson’s blowflies equation”, IMA J. Appl. Math. 79 (2014) 175–199; doi:10.1093/imamat/hxs060.Google Scholar
Alfifi, H. Y., Marchant, T. R. and Nelson, M. I., “Non-smooth feedback control for Belousov–Zhabotinskii reaction–diffusion equations: semi-analytical solutions”, J. Math. Chem. 57 (2016) 157–178; doi:10.1007/s10910-016-0641-8.Google Scholar
Alharthi, M. R., Marchant, T. R. and Nelson, M. I., “Mixed quadratic–cubic autocatalytic reaction–diffusion equations: semi-analytical solutions”, Appl. Math. Model. 38 (2014) 5160–5173; doi:10.1016/j.apm.2014.04.027.CrossRefGoogle Scholar
Ang, W. T., “The two-dimensional reaction–diffusion Brusselator system: a dual-reciprocity boundary element solution”, Eng. Anal. Bound. Elem. 27 (2003) 897–903; doi:10.1016/S0955-7997(03)00059-6.CrossRefGoogle Scholar
Biazar, J. and Ayati, Z., “An approximation to the solution of the Brusselator system by Adomian decomposition method and comparing the results with Runge–Kutta method”, Int. J. Contemp. Math. Sci. 2 (2007) 983–989; doi:10.1.1.558.4425.CrossRefGoogle Scholar
Brown, K. J. and Davidson, F. A., “Global bifurcation in the Brusselator system”, Nonlinear Anal. Theory Methods Appl. 24 (1995) 1713–1725; doi:10.1016/0362-546X(94)00218-7.CrossRefGoogle Scholar
Erneux, T., Applied delay differential equations (Springer, New York, 2009).Google Scholar
Fletcher, C. A., Computational Galerkin methods (Springer, New York, 1984).CrossRefGoogle Scholar
Guo, G., Wu, J. and Ren, X., “Hopf bifurcation in general Brusselator system with diffusion”, Appl. Math. Mech. (English Ed.) 32 (2011) 1177–1186; doi:10.1007/s10483-011-1491-6.CrossRefGoogle Scholar
Hale, J. K., Theory of functional differential equations (Springer, New York, 1977).CrossRefGoogle Scholar
Jiwari, R. and Yuan, J., “A computational modeling of two dimensional reaction–diffusion Brusselator system arising in chemical processes”, J. Math. Chem. 52 (2014) 1535–1551; doi:10.1007/s10910-014-0333-1.CrossRefGoogle Scholar
Kumar, S., Khan, Y. and Yildirim, A., “A mathematical modeling arising in the chemical systems and its approximate numerical solution”, Asia Pac. J. Chem. Eng. 7 (2012) 835–840; doi:10.1002/apj.647.CrossRefGoogle Scholar
Li, B. and Wang, M. X., “Diffusion-driven instability and Hopf bifurcation in Brusselator system”, Appl. Math. Mech. (English Ed.) 29 (2008) 825–832; doi:10.1007/s10483-008-0614-y.CrossRefGoogle Scholar
Liu, B. and Marchant, T. R., “The microwave heating of two-dimensional slabs with small Arrhenius absorptivity”, IMA J. Appl. Math. 26 (1999) 137–166; doi:10.1093/imamat/62.2.137.CrossRefGoogle Scholar
Ma, M. and Hu, J., “Bifurcation and stability analysis of steady states to a Brusselator model”, Appl. Math. Comput. 236 (2014) 580–592; doi:10.1016/j.amc.2014.02.075.Google Scholar
Marchant, T. R., “Cubic autocatalytic reaction diffusion equations: semi-analytical solutions”, Proc. R. Soc. Lond. A 458 (2002) 873–888; doi:10.1098/rspa.2001.0899.CrossRefGoogle Scholar
Marchant, T. R. and Liu, B., “The steady state microwave heating of slabs with small Arrhenius absorptivity”, J. Engrg. Math. 33 (1998) 219–236; doi:10.1023/A:1004227904688.CrossRefGoogle Scholar
Marchant, T. R. and Nelson, M. I., “Semi-analytical solution for one- and two-dimensional pellet problems”, Proc. R. Soc. Lond. A 460 (2004) 2381–2394; doi:10.1098/rspa.2004.1286.CrossRefGoogle Scholar
Mittal, R. C. and Jiwari, R., “Numerical study of two-dimensional reaction–diffusion Brusselator system”, Appl. Math. Comput. 217 (2011) 5404–5415; doi:10.1080/15502287.2010.540300.Google Scholar
Peng, R. and Wang, M. X., “Pattern formation in the Brusselator system”, J. Math. Anal. Appl. 309 (2005) 151–166; doi:10.1016/j.jmaa.2004.12.026.CrossRefGoogle Scholar
Prigogine, I. and Lefever, R., “Symmetry breaking instabilities in dissipative systems. II”, J. Chem. Phys. 48 (1968) 1665–1700; doi:10.1063/1.1668896.Google Scholar
Smith, G. D., Numerical solution of partial differential equations: finite difference methods (Clarendon Press, Oxford, 1985).Google Scholar
Twizell, E. H., Gumel, A. B. and Cao, Q., “A second-order scheme for the Brusselator reaction–diffusion system”, J. Math. Chem. 26 (1999) 297–316; doi:10.1023/A:1019158500612.CrossRefGoogle Scholar
Verwer, J. G., Hundsdorfer, W. H. and Sommeijer, B. P., “Convergence properties of the Runge–Kutta–Chebyshev method”, Numer. Math. 57 (1990) 157–178; doi:10.1007/BF01386405.CrossRefGoogle Scholar
Wazwaz, A. M., “The decomposition method applied to systems of partial differential equations and to the reaction–diffusion Brusselator model”, Appl. Math. Comput. 110 (2000) 251–264; doi:10.1016/S0096-3003(99)00131-9.Google Scholar
Xu, L., Zhao, L. J., Chang, Z. X., Feng, J. T. and Zhang, G., “Turing instability and pattern formation in a semi-discrete Brusselator model”, Modern Phys. Lett. B 27 (2013) 1–9; doi:10.1142/S0217984913500061.Google Scholar
You, Y., “Global dynamics of the Brusselator equations”, Dyn. Partial Differ. Equ. 4 (2007) 167–196; doi:10.4310/DPDE.2007.v4.n2.a4.CrossRefGoogle Scholar