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Computational Study of Traveling Wave Solutions of Isothermal Chemical Systems

Part of: Chemistry Qualitative theory

Published online by Cambridge University Press:  17 May 2016

Yuanwei Qi  and
Yi Zhu
Yuanwei Qi*
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
Yi Zhu*
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA
*
*Corresponding author. Email addresses:[email protected] (Y. Qi), [email protected] (Y. Zhu)
*Corresponding author. Email addresses:[email protected] (Y. Qi), [email protected] (Y. Zhu)
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Abstract

This article studies propagating traveling waves in a class of reaction-diffusion systems which model isothermal autocatalytic chemical reactions as well as microbial growth and competition in a flow reactor. In the context of isothermal autocatalytic systems, two different cases will be studied. The first is autocatalytic chemical reaction of order m without decay. The second is chemical reaction of order m with a decay of order n, where m and n are positive integers and m>n≥1. A typical system in autocatalysis is A+2B→3B and B→C involving two chemical species, a reactant A and an auto-catalyst B and C an inert chemical species.

The numerical computation gives more accurate estimates on minimum speed of traveling waves for autocatalytic reaction without decay, providing useful insight in the study of stability of traveling waves.

For autocatalytic reaction of order m = 2 with linear decay n = 1, which has a particular important role in chemical waves, it is shown numerically that there exist multiple traveling waves with 1, 2 and 3 peaks with certain choices of parameters.

Keywords

Isothermal chemical systemsmicrobial growth in a flow reactortraveling wavemultiple peak solutionsminimum speedexistencenon-existence

MSC classification

Secondary: 34C20: Transformation and reduction of equations and systems, normal forms 34C25: Periodic solutions 92E20: Classical flows, reactions, etc.
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 5 , May 2016 , pp. 1461 - 1472
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Ai, S. and Huang, W., Traveling waves for a reaction-diffusion system in population dynamics and epidemiology, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), pp. 663–675.Google Scholar
[2]Ballyk, M., Le, D., Jones, D.A., and Smith, H.L., Effects of random motility on microbial growth and compitition in a flow reactor, SIAM J. Appl. Math., 59 (1999), pp. 573–596.Google Scholar
[3]Bricmont, J., Kupiainen, A., and Xin, J., Global large time self-similarity of a thermal-diffusive combustion system with critical nonlinearity, J. Differentail Equations, 130 (1996), pp. 9–35.Google Scholar
[4]Chen, X.F. and Qi, Y., Sharp estimates on minimum travelling wave speed of reaction diffusion systems modelling autocatalysis, SIAM J. Math. Anal., 39 (2007), pp. 437–448.Google Scholar
[5]Chen, X.F. and Qi, Y., Propagation of local disturbances in reaction diffusion systems modeling quadratic autocatalysis, SIAM J. Appl. Math., 69 (2008), pp.273–282.Google Scholar
[6]Chen, X.F. and Qi, Y., Travelling waves of auto-catalytic chemical reaction of general order-an elliptic approach, J. Differential Equations, 246 (2009), pp.3038–3057.Google Scholar
[7]Chen, X.F., Qi, Y., and Zhang, Y.J., Existence of traveling waves of auto-catalytic systems with decay. Submitted to Comm Partial Differential Equations.Google Scholar
[8]Chen, X.F., Lai, X., Qi, Y., Qin, C., and Zhang, Y.J., Multiple traveling waves of an autocatalysis system with linear decay, preprint.Google Scholar
[9]Gray, P., Instabilities and oscillations in chemical reactions in closed and open systems, Proc. Roy. Soc. A, 415 (1988), pp. 1–34.Google Scholar
[10]Guo, J.S. and Tsai, J.C., Traveling waves of two-component reaction-diffusion systems arising from higher order autocatalytic models, Quarterly App. Math., 67 (2009), pp. 559–578.Google Scholar
[11]Hosono, Y. and Ilyas, B., Existence of travelling waves with any positive speed for a diffusive epidemic model, Nonlin. World, 1 (1994), pp. 277–290.Google Scholar
[12]Hosono, Y. and Ilyas, B., Travelling waves for a simple diffusive epidemic model, Math. Models Meth. Appl. Sci., 5 (1995), pp. 935–966.Google Scholar
[13]Hosono, Y., Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), pp. 115–125.Google Scholar
[14]Huang, W., Travelling waves for a biological reaction-diffusion model, J. Dynam. Diff. Eqns., 16 (2004), pp. 745–765.Google Scholar
[15]Kermack, W.O. and Mckendric, A.G., Contribution to the mathematical theory of epidemic, Proc. Roy. Soc. A, 115 (1927), pp. 700–721.Google Scholar
[16]Li, Y. and Qi, Y., The global dynamics of isothermal chemical systems with critical nonlinearity, Nonlinearity 16 (2003), pp. 1057–1074.Google Scholar
[17]Li, Y. and Wu, Y.P., Stability of traveling front solutions with algebraic spatial decay for some auto-catalytic chemical reaction systems, SIAM J. Math. Anal., 44 (2012), pp. 1474–1521.Google Scholar
[18]Merkin, J.H. and Needham, D.J., The development of traveling waves in a simple isothermal chemical system II. Cubic autocatalysis with quadratic and linear decay, Proc. Roy. Soc. A, 430 (1990), pp. 315–345.Google Scholar
[19]Qi, Y., Dynamics and universality of an isothermal combustion problem in 2D, Rev. Math. Phys., 18, (2006), pp. 285–310.Google Scholar
[20]Qi, Y., The global self-similarity of a chemical reaction system with critical nonlinearity, Proc. Roy. Soc. Edinburgh, 137A (2007), pp. 867–883.Google Scholar
[21]Qi, Y., Existence and non-existence of traveling waves for an isothermal chemical system with decay, J. Differential Equations, 258 (2015), pp. 669–690.Google Scholar
[22]Shi, J. and Wang, X., Hair-triggered instability of radial steady states, spread and extinction in semilinear heat equations, J. Differential Equations, 231 (2006), pp. 235–251.Google Scholar
[23]Smith, H.L. and Zhao, X.Q., Dynamics of a periodically pulsed bio-reactor model, J. Differential Equations, 155 (1999), pp. 368–404.Google Scholar
[24]Smith, H.L. and Zhao, X.Q., Traveling waves in a bio-reactor model, Nonlinear Analysis, Real World Application, 5 (2004), pp. 895–909.Google Scholar
[25]Tsai, J.C., Existence of traveling waves in a simple isothermal chemical system with the same order for autocatalysis and decay, Quarterly App. Math., 69 (2011), pp. 123–146.Google Scholar
[26]Zaikin, A.N. and Zhabotinskii, A.M., Concentration wave propagation in two-dimensional liquid-phase self-organising systems, Nature, 225 (1970), pp. 535–537.CrossRefGoogle Scholar
[27]Zhao, Y., Wang, Y., and Shi, J., Steady states and dynamics of an autocatalytic chemical reaction model with decay, J. Differential Equations, 253 (2012), pp. 533–552.Google Scholar