A variety of exact analytical solutions of extended shallow water wave equations via improved (G’/G) -expansion method

Abstract

In this present work, we have established exact solutions for (2+1) and (3+1) dimensional extended shallow-water wave equations involving parameters by applying the improved (G'/G) -expansion method. Abundant traveling wave solutions with arbitrary parameter are successfully obtained by this method, and these wave solutions are expressed in terms of hyperbolic, trigonometric, and rational functions. The improved (G'/G) -expansion method is simple and powerful mathematical technique for constructing traveling wave, solitary wave, and periodic wave solutions of the nonlinear evaluation equations which arise from application in engineering and any other applied sciences. We also present the 3D graphical description of the obtained solutions for different cases with the aid of MAPLE 17.

Families of Travelling Waves Solutions for Fractional-Order Extended Shallow Water Wave Equations, Using an Innovative Analytical Method

“…By using a Maple software and assigning particular values to parameters present in the exact solutions, we have shown the nature of some periodic and solitary solutions in following figures. Remarks 1: The Fig.1 shows soliton profile which is similar to the figure of equation (21) in [28].…”

“…Remarks 3: The Fig.3 shows periodic profile which is similar to the figure of equation 28in [28]. Remarks 4: The Fig.4 shows soliton profile which is similar to the figure of equation 55in [28].…”

“…For example, Faisal and Kumar have solved equation (1)(2) with the help of improved ( G G )-expansion method [28]. The solutions given in [28] are the special cases of the solutions obtained in the present research work, that is for α = β = γ = 1 in the current solutions given in equations (20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30)(31) and (38-49) we get the solutions given in [28]. Similarly, Bekir in [29] have used simple ( G G )-expansion method and obtained only six solutions.…”

“…Bekir and Aksoy in [26] have used exp-function method to solve the equations (1-3) analytically. In [28] Faisal and Kumar have solved equation 1and (2) by using extended form of ( G G )-expansion method. Similarly, Bekir and Aksoy in [29] have used simple ( G G )-expansion method and obtained six solutions of equations (1)(2)(3).…”

New exact solutions of the combined and double combined sinh-cosh-Gordon equations via modified Kudryashov method

“…Nowadays, the concept of reducing nonlinear partial differential equation into ordinary differential has proved a successful way to generate an analytical solution of nonlinear partial differential equation. According to this concept, many powerful method grown up and used for acquiring exact solutions of NPDEs such as, Homogeneous balance method [1], G /G−expansion method [2], Improved G /G−expansion method [3,4], Improved 1/G −expansion and (G /G, 1/G) methods [4], Extended tanh function method [5], Extended F-expansion method [6], Jacobi elliptic function method [7], Transformed rational function method [8], Weierstrass elliptic function expansion method [9], Hirota's bilinear method [10], Generalized Kudryashov method [11], Trial equation method [12], Solitary ansatz method [13], Auxiliary equation method [14], Modified Kudryashov method [15,16,17], Sine-Gordon equation expansion method [17,18], Hyperbolic function method [19] and so on. Recently, researchers easily design and applied these method for reducing the computational difficulty using computational software package like Maple, Mathematica, MATLAB etc.…”