# American Institute of Mathematical Sciences

July  2013, 18(5): 1459-1491. doi: 10.3934/dcdsb.2013.18.1459

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

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

Received  June 2012 Revised  November 2012 Published  March 2013

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 $\mathbb{R}$.
Citation: Weiran Sun, Min Tang. A relaxation method for one dimensional traveling waves of singular and nonlocal equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1459-1491. doi: 10.3934/dcdsb.2013.18.1459
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