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Abstract
The modeling of wave propagation in microstructure materials should be able to account for the various scales of microstructure. In this paper, the extended trial equation method was modified to construct the traveling wave solutions of the strain wave equation in microstructure solid. Some new different kinds of traveling wave solutions was gotten as, hyperbolic functions, trigonometric functions, Jacobi elliptic functions and rational functional solutions for the nonlinear strain wave equation when the balance number is positive integer. The balance number of this method is not constant and changes by changing the trial equation. These methods allow us to obtain many types of the exact solutions. By using the Maple software package, it was noticed that all the solutions obtained satisfy the original nonlinear strain wave equation. Key words: Strain wave equation, extended trial equation method, exact solutions, balance number, soliton solutions, Jacobi elliptic functions.
Highlights
Nonlinear evolution equations (NLEEs) are very important model equations in mathematical physics and engineering for describing diverse types of physical mechanisms of natural phenomena in the field of applied sciences and engineering
The exact solutions of nonlinear partial differential equation have been investigated by many authors (Ablowitz and Clarkson, 1991; Rogers and Shadwick, 1982; Matveev and Salle, 1991; Li and Chen, 2003; Conte and Musette, 1992; Ebaid and Aly, 2012; Gepreel, 2014; Cariello and Tabor, 1991; Fan, 2000; Fan, 2002; Wang and Li, 2005; Abdou, 2007; Wu and He, 2006; Wu and He, 2008; Li and Wang, 2007; Zheng, 2012; Triki and Wazwaz, 2014; Bibi and Mohyud-Din, 2014; Yu-Bin and Chao, 2009; Zayed and Gepreel, 2009; He, 2006; Gepreel, 2011; Adomian, 1988; Wazwaz, 2007; Liao, 2010; Gepreel and Mohamed, 2013; Wang et al, 2008; Yan, 2003a) who are interested in nonlinear physical phenomena
To integrate Equation 54, we discuss the roots of Equation 55 as the following families: Substituting Equations 60, 58 and 57 into Equation 52, we get the traveling wave solution of nonlinear strain wave Equation 16 taking the following form:
Summary
Nonlinear evolution equations (NLEEs) are very important model equations in mathematical physics and engineering for describing diverse types of physical mechanisms of natural phenomena in the field of applied sciences and engineering. The extended trial equation method was modified to construct the traveling wave solutions of the strain wave equation in microstructure solid.
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