Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the evolution of travelling wave solutions of the Burgers-Fisher equation

Authors: J. A. Leach and E. Hanaç
Journal: Quart. Appl. Math. 74 (2016), 337-359
MSC (2010): Primary 35B40, 35K57
DOI: https://doi.org/10.1090/qam/1421
Published electronically: March 16, 2016
MathSciNet review: 3505607
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we consider an initial-value problem for the Burgers-Fisher equation

$\displaystyle u_{t}+ k u u_{x}=u_{xx}+u(1-u), \quad -\infty <x<\infty , \quad t>0,$

where $ x$ and $ t$ represent dimensionless distance and time respectively and $ k\, (\not = 0)$ is a parameter. In particular, we consider the case when the initial data has a discontinuous compressive step, where $ u(x,0)=1$ for $ x \le 0$ and $ u(x,0)=0$ for $ x>0$. The method of matched asymptotic coordinate expansions is used to obtain the large-$ t$ asymptotic structure of the solution to this problem, which exhibits the formation of a permanent form travelling wave propagating in the $ +x$ direction with the minimum possible speed $ c=c^*(k)$, where

$\displaystyle c^{*}(k)=\left \{ \begin {array}{ll} 2, \quad & -\infty < k \le 2, \\ \frac {2}{k}+\frac {k}{2}, & 2<k <\infty . \end{array} \right .$    

The rate of convergence of the solution of the initial-value problem to the permanent form travelling wave is found to be algebraic in $ t$, as $ t \to \infty $, when $ k \in (-\infty ,2]$ and exponential in $ t$, as $ t \to \infty $, when $ k \in (2,\infty )$.

References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. Stegun,
    Handbook of Mathematical Functions.
    Dover (1965).
  • [2] Maury D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math. 31 (1978), no. 5, 531-581. MR 0494541 (58 #13382)
  • [3] R. A. Fisher,
    The wave of advance of advantageous genes.
    Ann. Eugenics 7, 353-369 (1937).
  • [4] A. Kolmogorov, I. Petrovsky and N. Piscounov.
    Etude de l'équation de la diffusion avec croissance de la quantité de la metière et son application à un problèm biologique,
    Moscow University Bulletin of Mathematics (1937) 1, 1-25.
    Ann. Fac. Sci. Tolouse Math. (5), 8:175-203, 1986.
  • [5] D. A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math. 34 (1978), no. 1, 93-103. MR 482838 (80b:35021), https://doi.org/10.1137/0134008
  • [6] J. A. Leach and D. J. Needham, Matched asymptotic expansions in reaction-diffusion theory, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2004. MR 2013330 (2005h:35198)
  • [7] J. A. Leach and D. J. Needham, The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation. I. Initial data has a discontinuous expansive step, Nonlinearity 21 (2008), no. 10, 2391-2408. MR 2439485 (2009k:35267), https://doi.org/10.1088/0951-7715/21/10/010
  • [8] H. P. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Comm. Pure Appl. Math. 28 (1975), no. 3, 323-331. MR 0400428 (53 #4262)
  • [9] J. H. Merkin and D. J. Needham, Propagating reaction-diffusion waves in a simple isothermal quadratic autocatalytic chemical system, J. Engrg. Math. 23 (1989), no. 4, 343-356. MR 1029938 (90k:80024), https://doi.org/10.1007/BF00128907
  • [10] J. D. Murray,
    Nonlinear Differential Equation Models in Biology.
    Clarendon Press, Oxford (1977).
  • [11] J. D. Murray, Mathematical biology, Biomathematics, vol. 19, Springer-Verlag, Berlin, 1989. MR 1007836 (90g:92001)
  • [12] William E. Schiesser and Graham W. Griffiths, A compendium of partial differential equation models, Method of lines analysis with Matlab, Cambridge University Press, Cambridge, 2009. MR 2499504 (2010d:65234)
  • [13] Milton Van Dyke, Perturbation methods in fluid mechanics, Annotated edition, The Parabolic Press, Stanford, Calif., 1975. MR 0416240 (54 #4315)

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Additional Information

J. A. Leach
Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Email: j.a.leach@bham.ac.uk

E. Hanaç
Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Email: hana.esen@gmail.com

DOI: https://doi.org/10.1090/qam/1421
Received by editor(s): September 30, 2014
Published electronically: March 16, 2016
Article copyright: © Copyright 2016 Brown University

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