Exact Solutions to the (2+1)-Dimensional Boussinesq Equation via exp(?(?))-Expansion Method

Abstract

The exp(Φ(η))-expansion method is applied to find exact traveling wave solutions to the (2+1)-dimensional Boussinesq equation which is an important equation in mathematical physics. The traveling wave solutions are expressed in terms of the exponential functions, the hyperbolic functions, the trigonometric functions and the rational functions. The procedure is simple, direct and constructive without the help of a computer algebra system. The applied method will be used in further works to establish more new solutions for other kinds of nonlinear evolution equations arising in mathematical physics and engineering.

Analytical solutions of Phi-four equation

“…-expansion function method [13][14][15][16]. In order to implement this method to the nonlinear partial differential equations, we handle it as follows:…”

On the numerical investigations to the Cahn-Allen equation by using finite difference method

“…Most recently, some serious methods have been developed in order to solve nonlinear differential equation. For example, (G'/G)-expansion method [5,6], the improved (G'/G)-expansion method [7-9], the modified simple equation method [10], the Sumudu transform method [11][12][13][14], the Bäcklund transform method [15], the homotopy analysis method [16,17], the exponential function method [18][19][20], the modified exponential function method [21], generalized Bernoulli sub-ODE method [22], improved Bernoulli sub-ODE method [24][25][26], weak solutions [27] and galerkin method [28]. In the current work, we consider the Cahn-Allen equation given as:…”

Exact solutions of the classical Boussinesq system

“…To seek the exact solutions of the nonlinear partial differential equations, many methods have been proposed, for instance the inverse scattering transformation (IST) (Ablowitz & Segur, 2000), B€ acklund transformation (Rogers & Schief, 2002), Painlev e analysis method (Chowdhury, 1999), Darboux transformation (DT) (Gu, Hu, & Zhou, 2005), Hirota direct method (Hirota, 2004), the tanh-function method (Parkes & Duffy, 1996;Zhang, Xu, & Li, 2002), the improved F-expansion method (Islam, Khan, Akbar, & Mastroberardino, 2014;Wang & Zhang, 2005), the modified simple equation method (Akter & Akbar, 2015;Khan, Akbar, & Alam, 2013), the (G 0 =G)-expansion and extend (G 0 =G)-expansion method (Akbar & Ali, 2011;Akbar, Ali, & Mohyud-Din, 2013;Alam, Hafez, Belgacem, & Akbar, 2015b;Zayed & Shorog, 2010), the Exp-function method (He & Abdou, 2007;, the generalized Kudryashov method (Khan & Akbar, 2016), the expðÀUðgÞÞ-expansion and expðUðgÞÞ method (Alam, Hafez, Akbar, & Roshid, 2015a;Roshid & Rahman, 2014), the extended three-wave method Li, Dai, & Liu, 2011;Singh & Gupta, 2016;Wang, Dai, & Liang, 2010). In this paper, based on the bilinear form, we consider exact solutions including solitary wave solution, periodic solitary solution and rational solution of the classical Boussinesq (CB) system (see Wu & Zhang, 1996) …”