Theoretical and Numerical Studies on Global Stability of Traveling Waves with Oscillations for Time-Delayed Nonlocal Dispersion Equations
Year: 2020
Author: Tianyuan Xu, Shanming Ji, Rui Huang, Ming Mei, Jingxue Yin
International Journal of Numerical Analysis and Modeling, Vol. 17 (2020), Iss. 1 : pp. 68–86
Abstract
This paper is concerned with the global stability of non-critical/critical traveling waves with oscillations for time-delayed nonlocal dispersion equations. We first theoretically prove that all traveling waves, especially the critical oscillatory traveling waves, are globally stable in a certain weighted space, where the convergence rates to the non-critical oscillatory traveling waves are time-exponential, and the convergence to the critical oscillatory traveling waves are time-algebraic. Both of the rates are optimal. The approach adopted is the weighted energy method with the fundamental solution theory for time-delayed equations. Secondly, we carry out numerical computations in different cases, which also confirm our theoretical results. Because of oscillations of the solutions and nonlocality of the equation, the numerical results obtained by the regular finite difference scheme are not stable, even worse to be blow-up. In order to overcome these obstacles, we propose a new finite difference scheme by adding artificial viscosities to both sides of the equation, and obtain the desired numerical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2020-IJNAM-13641
International Journal of Numerical Analysis and Modeling, Vol. 17 (2020), Iss. 1 : pp. 68–86
Published online: 2020-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 19
Keywords: Critical traveling waves time-delay global stability nonlocal dispersion equation oscillations.