A relaxation method for one dimensional traveling waves of singular and nonlocal equations

Weiran Sun; Min Tang

doi:10.3934/dcdsb.2013.18.1459

Article Contents

2013, Volume 18, Issue 5: 1459-1491. Doi: 10.3934/dcdsb.2013.18.1459

This issue Previous Article Dynamics of a limit cycle oscillator with extended delay feedback Next Article $L^\infty$ estimation of the LDG method for 1-d singularly perturbed convection-diffusion problems

A relaxation method for one dimensional traveling waves of singular and nonlocal equations

Weiran Sun^1, and
Min Tang^2,

1.
Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A 4Z2
2.
Department of mathematics and Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240

Received: May 31, 2012

Revised: October 31, 2012

Published: June 2013

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Abstract

Recent models motivated by biological phenomena lead to non-local PDEs or systems with singularities. It has been recently understood that these systems may have traveling wave solutions that are not physically relevant [19]. We present an original method that relies on the physical evolution to capture the ``stable" traveling waves. This method allows us to obtain the traveling wave profiles and their traveling speed simultaneously. It is easy to implement, and it applies to classical differential equations as well as nonlocal equations and systems with singularities. We also show the convergence of the scheme analytically for bistable reaction diffusion equations over the whole space $\mathbb{R}$.

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Mathematics Subject Classification: Primary: 35C07, 35Q92; Secondary: 35K57.

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