Symbolic computation of hyperbolic tangent solutions for nonlinear differential–difference equations

Abstract

A new algorithm is presented to find exact traveling wave solutions of differentialdifference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the equations might admit polynomial solutions in tanh.Examples illustrate the key steps of the algorithm. Parallels are drawn through discussion and example to the tanh-method for partial differential equations.The new algorithm is implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute traveling wave solutions of nonlinear polynomial differential-difference equations. Use of the package, implementation issues, scope, and limitations of the software are addressed. Program summary

The discrete (G′/G)‐expansion method applied to the differential‐difference Burgers equation and the relativistic Toda lattice system

“…Then, clearing the denominator and setting the coefficients of Remark 5. We note that our solution (37) is not derived in Baldwin et al [5] and Yaxuan et al [32].…”

“…Hu and Ma [4] applied Hirota's bilinear method to the Toeplitz lattice equation. Baldwin et al [5], with the aid of MATHEMATICA, presented an algorithm to find exact traveling wave solutions of NDDEs in terms of the hyperbolic tangent function. Their work can be considered as a breakthrough for solving NDDEs via symbolic computation.…”

Construction of exact solutions for fractional-type difference-differential equations via symbolic computation

“…Substituting (9) and (13a)-(13c) together with (10) into (8), equating the coefficients of w 0 ðnnÞ wðn n Þ l ðl ¼ 0; 1; 2; . .…”

“…Their important role has motivated investigators to develop a number of integrable DDEs since the original work of Fermi, Pasta and Ulam [1]. To name a few; Volterra lattice equation [2], discrete KdV equation [3], Toda lattice equation [4], Ablowitz-Ladik lattice equation [5], discrete sine-Gordon equation [6], discrete modified KdV equation [7], see [8] for a list of DDEs. These DDEs are of (or can be converted to) the form _ u n ¼ Pð.…”

Discrete exact solutions to some nonlinear differential-difference equations via the (G′/G)-expansion method

“…To mention some of the research made on this direction, Hu and Ma [24] used Hirota's bilinear method to construct special soliton-like solutions of the Toeplitz lattice. With the development of computer algebra systems, Baldwin and his co-workers [25] devised an algorithm for discrete nonlinear models in terms of a tanh function. Their work can be thought as a breakthrough for solving NDDEs symbolically.…”