Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 

 

A discrete KPP-theory for Fisher's equation


Author: Bengt Hakberg
Journal: Math. Comp. 82 (2013), 781-802
MSC (2010): Primary 65M06
DOI: https://doi.org/10.1090/S0025-5718-2012-02642-1
Published electronically: August 21, 2012
MathSciNet review: 3008838
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is to extend the theory by Kolmogorov, Petrowsky and Piscunov (KPP) for Fisher's equation, to a discrete solution. We approximate the time derivative in Fisher's equation by an explicit Euler scheme and the diffusion operator by a symmetric difference scheme of second order. We prove that the discrete solution converges towards a traveling wave, under restrictions in the time- and space-widths, as the number of time steps increases to infinity. We also prove that the flame velocity can be determined as a solution to an optimization problem.


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

  • 1. Fisher, R. A. (1937), ``The wave of advance of advantageous genes'', Ann. Eugenics, 7 , 355-369.
  • 2. Hakberg, B. (1997), ``A critical study of the Bray-Moss-Libby model'', Combustion Science and Technology, 125, 25-45.
  • 3. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
  • 4. Kolmogorov, A. N., Petrowsky, I. G., and Piskunow, N. S. (1937), ``Study of the diffusion equation with growth of the quantity of matter and its application to a biology problem'', Bull. Moskov. Gos. Univ. Math. Mekh., 1, no. 6, 1-25 (Russian); English translation in Dynamics of Curved Fronts, Pelcé, P. (editor), Academic Press, 1988, 105-130.
  • 5. H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal. 13 (1982), no. 3, 353–396. MR 653463, https://doi.org/10.1137/0513028
  • 6. B. Zinner, G. Harris, and W. Hudson, Traveling wavefronts for the discrete Fisher’s equation, J. Differential Equations 105 (1993), no. 1, 46–62. MR 1237977, https://doi.org/10.1006/jdeq.1993.1082

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65M06

Retrieve articles in all journals with MSC (2010): 65M06


Additional Information

Bengt Hakberg
Affiliation: Department of Mathematical Sciences, Chalmers and Gothenburg University, S-41296 Goteborg, Sweden
Email: hakberg@chalmers.se

DOI: https://doi.org/10.1090/S0025-5718-2012-02642-1
Received by editor(s): December 6, 2010
Received by editor(s) in revised form: October 11, 2011
Published electronically: August 21, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.