@Article{IJNAM-17-1,
author = {Xu, Tianyuan and Shanming, Ji and Rui, Huang and Ming, Mei and Yin, Jingxue},
title = {Theoretical and Numerical Studies on Global Stability of Traveling Waves with Oscillations for Time-Delayed Nonlocal Dispersion Equations},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2020},
volume = {17},
number = {1},
pages = {68--86},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2617-8710},
doi = {https://doi.org/2020-IJNAM-13641},
url = {https://global-sci.com/article/83009/theoretical-and-numerical-studies-on-global-stability-of-traveling-waves-with-oscillations-for-time-delayed-nonlocal-dispersion-equations}
}