Explicit upper and lower bounds for the traveling wave solutions ofFisher-Kolmogorov type equations

Armengol Gasull; Hector Giacomini; Joan Torregrosa

doi:10.3934/dcds.2013.33.3567

Article Contents

2013, Volume 33, Issue 8: 3567-3582. Doi: 10.3934/dcds.2013.33.3567

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Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations

1.
Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona
2.
Laboratoire de Mathématique et Physique Théorique, C.N.R.S. UMR 7350, Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont,37200 Tours
3.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona

Received: June 30, 2012

Revised: October 31, 2012

Published: July 2013

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Abstract

It is well-known that the existence of traveling wave solutions for reaction-diffusion partial differential equations can be proved by showing the existence of certain heteroclinic orbits for related autonomous planar differential equations. We introduce a method for finding explicit upper and lower bounds of these heteroclinic orbits. In particular, for the classical Fisher-Kolmogorov equation we give rational upper and lower bounds which allow to locate these solutions analytically and with very high accuracy. These results allow one to construct analytical approximate expressions for the traveling wave solutions with a rigorous control of the errors for arbitrary values of the independent variables. These explicit expressions are very simple and tractable for practical purposes. They are constructed with exponential and rational functions.

Keywords:

Traveling wave,
heteroclinic orbit,
Fisher-Kolmogorov equation,
reaction-diffusion partial differential equation,
invariant manifold.

Mathematics Subject Classification: 34C37, 35C07, 37C29.

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