Exact solutions of nonlinear fractional differential equations by (G'/G)-expansion method

In this paper, we use the fractional complex transform and the （G＇/G）-expansion method to study the nonlinear fractional differential equations and find the exact solutions. The fractional complex transform is proposed to convert a partial fractional differential equation with Jumarie＇s modified Riemann-Liouville derivative into its ordinary differential equation. It is shown that the considered transform and method are very efficient and powerful in solving wide classes of nonlinear fractional order equations.

The (G'/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations

The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.

Exact Solution to Nonlinear Differential Equations of Fractional Order via (<i>G’</i>/<i>G</i>)-Expansion Method

In this article, a new application to find the exact solutions of nonlinear partial time-space fractional differential Equation has been discussed. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential Equations into nonlinear ordinary differential Equations. Afterwards, the (G'/G)-expansion method has been implemented, to celebrate the exact solutions of these Equations, in the sense of modified Riemann-Liouville derivative. As application, the exact solutions of time-space fractional Burgers’ Equation have been discussed.

Comment on “Application of the (G'/G)-Expansion Method for Nonlinear Evolution Equations”[Phys.Lett.A 372 (2008) 3400]

In this paper,some travelling wave solutions involving parameters of the Modified Zakharov-Kuznetsovequation [Phys.Lett.A 372 (2008) 3400] are investigated.We will show that these solutions are not new travellingwave solutions.

(G'/G)-Expansion Method Equivalent to Extended Tanh Function Method

In a recent article [Physics Letters A 372 （2008） 417], Wang et al. proposed a method, which is called the （G′/G）-expansion method, to look for travelling wave solutions of nonlinear evolution equations. The travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations, and the Hirota-Satsuma equations are obtained by using this method. They think the （G′/G）-expansion method is a new method and more travelling wave solutions of many nonlinear evolution equations can be obtained. In this paper, we will show that the （G′/G）-expansion method is equivalent to the extended tanh function method.

An Innovative Solutions for the Generalized FitzHugh-Nagumo Equation by Using the Generalized (G'/G)-Expansion Method

In this paper, the generalized (G'/G)-expansion method is used for construct an innovative explicit traveling wave solutions involving parameter of the generalized FitzHugh-Nagumo equation , for some special parameter where satisfies a second order linear differential equation , , where and are functions of .

General Solution of Two Generalized Form of Burgers Equation by Using the (<i>G</i>^'/<i>G</i>)-Expansion Method

In this work, the (G'/G)-expansion method is proposed for constructing more general exact solutions of two general form of Burgers type equation arising in fluid mechanics namely, Burgers-Korteweg-de Vries (Burgers-KdV) and Burger-Fisher equations. Our work is motivated by the fact that the (G'/G)-expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.

Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method

The novel (G'/G)-expansion method is a powerful and simple technique for finding exact traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, we study explicit exact traveling wave solutions for the (1 + 1)-dimensional combined KdV-mKdV equation by using the novel (G'/G)-expansion method. Consequently, various traveling wave solutions patterns including solitary wave solutions, periodic solutions, and kinks are detected and exhibited.

Exact solutions for the coupled Klein-Gordon-Schrǒdinger equations using the extended F-expansion method

The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations （K-G-S equations） by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.

Exact solutions of the nonlinear differential-difference equations associated with the nonlinear electrical transmission line through a variable-coefficient discrete(G'/G)-expansion method

We investigated exact traveling soliton solutions for the nonlinear electrical transmission line. By applying a concise and straightforward method, the variable-coefficient discrete（G /G）-expansion method, we solve the nonlinear differential–difference equations associated with the network. We obtain some exact traveling wave solutions which include hyperbolic function solution, trigonometric function solution, rational solutions with arbitrary function, bright as well as dark solutions.

Exact Solutions of (2+1)-Dimensional Boiti-Leon-Pempinelle Equation with (G'/G)-Expansion Method

In this paper,we construct exact solutions for the (2+1)-dimensional Boiti-Leon-Pempinelle equation byusing the (G'/G)-expansion method,and with the help of Maple.As a result,non-travelling wave solutions with threearbitrary functions are obtained including hyperbolic function solutions,trigonometric function solutions,and rationalsolutions.This method can be applied to other higher-dimensional nonlinear partial differential equations.

A Generalized Tanh-Function Type Method and the(G'/G) -Expansion Method for Solving

In this paper a generalized tanh-function type method is proposed by using the idea of the transformed rational function method. We show that the (G'/G)?-expansion method is a special case of the generalized tanh-function type method, so the (G'/G)?-expansion method is considered as a special deformation application of the transformed rational function method. We demonstrate that all solutions obtained by the (G'/G)?-expansion method were found by the generalized tanh-function type method. As applications, we consider mKdV equation. Compared with the (G'/G) -expansion method, the generalized tanh-function type method gives new and more abundant solutions.

The Basic (<i>G'/G</i>)-Expansion Method for the Fourth Order Boussinesq Equation

The (G'/G)-expansion method is simple and powerful mathematical tool for constructing traveling wave solutions of nonlinear evolution equations which arise in engineering sciences, mathematical physics and real time application fields. In this article, we have obtained exact traveling wave solutions of the nonlinear partial differential equation, namely, the fourth order Boussinesq equation involving parameters via the (G'/G)-expansion method. In this method, the general solution of the second order linear ordinary differential equation with constant coefficients is implemented. Further, the solitons and periodic solutions are described through three different families. In addition, some of obtained solutions are described in the figures with the aid of commercial software Maple.

Exact Traveling Wave Solutions for the (1 + 1)-Dimensional Compound KdVB Equation via the Novel (G'/G)-Expansion Method

In this work, while applying a new and novel (G'/G)-expansion version technique, we identify four families of the traveling wave solutions to the (1 + 1)-dimensional compound KdVB equation. The exact solutions are derived, in terms of hyperbolic, trigonometric and rational functions, involving various parameters. When the parameters are tuned to special values, both solitary, and periodic wave models are distinguished. State of the art symbolic algebra graphical representations and dynamical interpretations of the obtained solutions physics are provided and discussed. This in turn ends up revealing salient solutions features and demonstrating the used method efficiency.

A novel (G'/G)-expansion method and its application to the Boussinesq equation

In this article, a novel （G＇/G）-expansion method is proposed to search for the traveling wave solutions of nonlinear evolution equations. We construct abundant traveling wave solutions involving parameters to the Boussinesq equation by means of the suggested method. The performance of the method is reliable and useful, and gives more general exact solutions than the existing methods. The new （G＇/G）-expansion method provides not only more general forms of solutions but also cuspon, peakon, soliton, and periodic waves.

The (ω/g)-expansion method and its application to Vakhnenko equation

This paper presents a new function expansion method for finding travelling wave solutions of a nonlinear evolution equation and calls it the （w/g）-expansion method, which can be thought of as the generalization of （G＇/G）-expansion given by Wang et al recently. As an application of this new method, we study the well-known Vakhnenko equation which describes the propagation of high-frequency waves in a relaxing medium. With two new expansions, general types of soliton solutions and periodic solutions for Vakhnenko equation are obtained.

A connection between the(G'/G)-expansion method and the truncated Painlevé expansion method and its application to the mKdV equation

Recently the （G′/G）-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the （G′/G）-expansion method is a special form of the truncated Painlev＇e expansion method by introducing an intermediate expansion method. Then the generalized （G′/G）-（G/G′） expansion method is naturally derived from the standpoint of the nonstandard truncated Painlev＇e expansion. The application of the generalized method to the mKdV equation shows that it extends the range of exact solutions obtained by using the （ G′/ G）-expansion method.

New Generalized (G'/G)-Expansion Method Applications to Coupled Konno-Oono Equation

The new generalized (G'/G)-expansion method is one of the powerful and competent methods that appear in recent time for establishing exact solutions to nonlinear evolution equations (NLEEs). We apply the new generalized (G'/G)-expansion method to solve exact solutions of the new coupled Konno-Oono equation and construct exact solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. The significance of obtained solutions gives credence to the explanation and understanding of related physical phenomena. As a newly developed mathematical tool, this method efficiency for finding exact solutions has been demonstrated through showing its straightforward nature and establishing its ability to handle nonlinearities prototyped by the NLEEs whether in applied mathematics, physics, or engineering contexts.

Exact Travelling Wave Solutions of Two Nonlinear Schr&#246;dinger Equations by Using Two Methods

The special kind of (G’/G)-expansion method and the new mapping method are easy and significant mathematical methods. In this paper, exact travelling wave solutions of the higher order dispersive Cubic-quintic nonlinear Schr&#246;dinger equation and the generalized nonlinear Schr&#246;dinger equation are studied by using the two methods. Finally, the solitary wave solutions, singular soliton solutions, bright and dark soliton solutions and periodic solutions of the two nonlinear Schr&#246;dinger equations are obtained. The results show that this method is effective for solving exact solutions of nonlinear partial differential equations.

General Solution of Generalized (2+1)–Dimensional Kadomtsev-Petviashvili (KP) Equation by Using the –Expansion Method

In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the (G,/G)---expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.