Exact and numerical soliton solutions to nonlinear wave equations

" Exact and numerical soliton solutions to nonlinear wave equations " 

Kamruzzaman Khan ,   Henk Koppelaar ,  Ali Akbar ... 2016 , No.2  , P. 5 - 22 

 

ِDOI : http://dx.doi.org/1022309/cjcme.2016.06

Abstract

Because most physical systems are inherently nonlinear by nature, mathematical modeling of physical systems often leads to nonlinear evolution equations. Investigating traveling wave solutions of nonlinear evolution equations (NLEEs) plays a significant role in studying such nonlinear physical phenomena. This paper discusses the application of the functional variable method for finding solitary and periodic wave solutions of the longitudinal wave motion NLEE in a nonlinear magneto-electro-elastic (MEE) circular rod. Each of the obtained solutions contains an explicit function of the variables in the considered equations. The applied method yielded a powerful mathematical tool for solving these nonlinear wave equations without the necessity of a computer algebra system; according the widespread view “real mathematicians don not compute”. However, if the obtained exact hyperbolic solutions are executed numerically, then they show an undesirable unknown numerical conditioning problem. This paper contains a warning against it and information about how to prevent this outcome.

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