Exact Solutions of Equation Generated by the Jaulent-Miodek Hierarchy by(G’/G)-Expansion Method

Abstract

The(G’/G)-expansion method is proposed for constructing more general exact solutions of the nonlinear(2+1)-dimensional equation generated by the Jaulent-Miodek Hierarchy. As a result, when the parameters are taken at special values, some new traveling wave solutions are obtained which include solitary wave solutions which are based from the hyperbolic functions, trigonometric functions, and rational functions. We find in this work that the(G’/G)-expansion method give some new results which are easier and faster to compute by the help of a symbolic computation system. The results obtained were compared with tanh method.

Exact Solutions for(4+1)-Dimensional Nonlinear Fokas Equation Using ExtendedF-Expansion Method and Its Variant

“…In the recent years, much efforts have been spent on this task and many significant methods have been established such as inverse scattering transform [1], Backlund and Darboux transform [2], Hirota [3], homogeneous balance method [4], symmetry reductions method [5][6][7][8], Jacobi elliptic function method [9], tanh-function method [10], expfunction method [11][12][13], simple equation method [14], the meshless methods [15][16][17][18][19][20], / -expansion method [21][22][23], F-expansion method [24,25], improved F-expansion method [26,27], and extended F-expansion method [28].…”

Explicit Exact Solutions of the (2+1)-Dimensional Integro-Differential Jaulent–Miodek Evolution Equation Using the Reliable Methods

“…As a result, some new exact solutions were obtained including triangular periodic wave solutions, exponential solutions, and complex traveling solutions. In 2012, the (G ′ /G)-expansion method was used to construct some new traveling wave solutions including hyperbolic function, trigonometric function, and rational function solutions of the (2 + 1)-dimensional Jaulent-Miodek equation [40]. Wazwaz [41] used the simplified form of Hirota's direct method to obtain multiple soliton solutions of the (3 + 1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy.…”

Application of the (-expansion method for the generalized Fisher‘s equation and modified equal width equation

“…In recent years, the exact solution of non-linear partial differential equations has been investigated by many authors (Krisnangkura et al, 2012;Wazwaz, 2004Wazwaz, , 2006Shi et al, 2012;Jabbari and Kheiri, 2010;Lee and Sakthivel, 2010a,b, 2011a,b, 2012Parand and Rad, 2012;Elboree, 2012a,b;Abdel Rady et al, 2012;Honga and Lub, 2012;Abbasbandy and Shirzadi, 2010;Taghizadeh et al, 2012;Babolian and Dastani, 2012;Kafash et al, 2013;Mu and Ye, 2011;Ebadi and Biswas, 2010a,b;Kim andSakthivel, 2010, 2011;Malik et al, 2010Malik et al, , 2012Kabir et al, 2011;Feng et al, 2011;Jabbari et al, 2011;Ayhan and Bekir, 2012;Naher and Abdullah, 2012;Kraenkel et al, 2013;Taha and Noorani, 2013;Taha et al, 2013;Wang et al, 2008), who are interested in non-linear physical phenomena in various fields of physics and engineering. Many powerful methods have been presented such as the tanh-method (Krisnangkura et al, 2012;Wazwaz, 2004), sine-cosine method (Shi et al, 2012;Wazwaz, 2006), tanhcoth method (Jabbari and Kheiri, 2010;Lee andSakthivel, 2012, 2011a), exp-function method (Lee andSakthivel, 2010a, 2011b;Parand and Rad, 2012), homogeneous-balance method (Elboree, 2012a; Abdel Rady et al, 2012...…”

“…The objective of this paper is to use a powerful method which is called the G 0 G À Á -expansion method (Ebadi and Biswas, 2010a,b;Kim andSakthivel, 2010, 2011;Malik et al, 2010;Kabir et al, 2011;Feng et al, 2011;Jabbari et al, 2011;Ayhan and Bekir, 2012;Malik et al, 2012;Naher and Abdullah, 2012;Elboree, 2012b;Kraenkel et al, 2013;Taha and Noorani, 2013;Taha et al, 2013), to obtain traveling wave solutions of such equations. The main ideas are that the traveling wave solutions of a non-linear equation can be expressed by a polynomial in G 0…”