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The exp(-φ(ξ))-Expansion Method and Its Application for Solving Nonlinear Evolution Equations  [PDF]
Mahmoud A. E. Abdelrahman, Emad H. M. Zahran, Mostafa M. A. Khater
International Journal of Modern Nonlinear Theory and Application (IJMNTA) , 2015, DOI: 10.4236/ijmnta.2015.41004
Abstract: The exp(-φ(ξ))-expansion method is used as the first time to investigate the wave solution of a nonlinear dynamical system in a new double-Chain model of DNA and a diffusive predator-prey system. The proposed method also can be used for many other nonlinear evolution equations.
The exp(-j(x)) Method and Its Applications for Solving Some Nonlinear Evolution Equations in Mathematical Physics  [PDF]
Maha S. M. Shehata
American Journal of Computational Mathematics (AJCM) , 2015, DOI: 10.4236/ajcm.2015.54041
Abstract:

The ?exp(-j(x))?method is employed to find the exact traveling wave solutions involving parameters for nonlinear evolution equations. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the ?exp(-j(x))??method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.

Analytical and Traveling Wave Solutions to the Fifth Order Standard Sawada-Kotera Equation via the Generalized exp(-Φ(ξ))-Expansion Method  [PDF]
M. Y. Ali, M. G. Hafez, M. K. H. Chowdury, M. T. Akter
Journal of Applied Mathematics and Physics (JAMP) , 2016, DOI: 10.4236/jamp.2016.42033
Abstract: In this article, we propose a generalized exp(-Φ(ξ))-expansion method and successfully implement it to find exact traveling wave solutions to the fifth order standard Sawada-Kotera (SK) equation. The exact traveling wave solutions are established in the form of trigonometric, hyperbolic, exponential and rational functions with some free parameters. It is shown that this method is standard, effective and easily applicable mathematical tool for solving nonlinear partial differential equations arises in the field of mathematical physics and engineering.
Exact Solutions of Equation Using Lie Symmetry Approach along with the Simplest Equation and Exp-Function Methods  [PDF]
Hossein Jafari,Nematollah Kadkhoda,Chaudry Masood Khalique
Abstract and Applied Analysis , 2012, DOI: 10.1155/2012/350287
Abstract: This paper obtains the exact solutions of the equation. The Lie symmetry approach along with the simplest equation method and the Exp-function method are used to obtain these solutions. As a simplest equation we have used the equation of Riccati in the simplest equation method. Exact solutions obtained are travelling wave solutions. 1. Introduction The research area of nonlinear equations has been very active for the past few decades. There are several kinds of nonlinear equations that appear in various areas of physics and mathematical sciences. Much effort has been made on the construction of exact solutions of nonlinear equations as they play an important role in many scientific areas, such as, in the study of nonlinear physical phenomena [1, 2]. Nonlinear wave phenomena appear in various scientific and engineering fields, such as fluid mechanics, plasma physics, optical fiber, biology, oceanology [3], solid state physics, chemical physics, and geometry. In recent years, many powerful and efficient methods to find analytic solutions of nonlinear equation have drawn a lot of interest by a diverse group of scientists. These methods include, the tanh-function method, the extended tanh-function method [2, 4, 5], the sine-cosine method [6], and the -expansion method [7, 8]. In this paper, we study the equation, namely, The purpose of this paper is to use the Lie symmetry method along with the simplest equation method (SEM) and the Exp-function method to obtain exact solutions of the equation. The simplest equation method was developed by Kudryashov [9–12] on the basis of a procedure analogous to the first step of the test for the Painlevé property. The Exp-function method is a very powerful method for solving nonlinear equations. This method was introduced by He and Wu [13] and since its appearance in the literature it has been applied by many researchers for solving nonlinear partial differential equations. See for example, [14, 15]. The outline of this paper is as follows. In Section 2 we discuss the methodology of Lie symmetry analysis and obtain the Lie point symmetries of the equation. We then use the translation symmetries to reduce this equation to an ordinary differential equation (ODE). In Section 3 we describe the SEM and then we obtain the exact solutions of the reduced ODE using SEM. In Section 4 we explain the basic idea of the Exp-function method and obtain exact solutions of the reduced ODE using the Exp-function method. Concluding remarks are summarized in Section 5. 2. Lie Symmetry Analysis We recall that a Lie point symmetry of a partial
A Generalized and Improved -Expansion Method for Nonlinear Evolution Equations
M. Ali Akbar,Norhashidah Hj. Mohd. Ali,E. M. E. Zayed
Mathematical Problems in Engineering , 2012, DOI: 10.1155/2012/459879
Abstract: A generalized and improved (/)-expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation and the strain wave equation in microstructured solids. Abundant exact travelling wave solutions of these equations are obtained, which include the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also it is shown that the proposed method is efficient for solving nonlinear evolution equations in mathematical physics and in engineering.
Function expansion methods for solving autonomous nonlinear partial differential equations  [PDF]
Mahouton Norbert Hounkonnou,Pascal Alain Dkengne Sielenou
Physics , 2011,
Abstract: In this paper, we propose some algorithms for analytical solution construction to nonlinear polynomial partial differential equations with constant function coefficients. These schemes are based on one-(single), two- (double) or three- (triple) function expansion methods. Most of the existing expansion function methods are well recovered from the mentioned schemes. The effectiveness of these methods has been tested on some nonlinear partial differential equations (NLPDEs) describing important phenomena in physics.
The -Expansion Method for Abundant Traveling Wave Solutions of Caudrey-Dodd-Gibbon Equation  [PDF]
Hasibun Naher,Farah Aini Abdullah,M. Ali Akbar
Mathematical Problems in Engineering , 2011, DOI: 10.1155/2011/218216
Abstract: We construct the traveling wave solutions of the fifth-order Caudrey-Dodd-Gibbon (CDG) equation by the -expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, the trigonometric, and the rational functions. It is shown that the -expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations. 1. Introduction The investigation of exact traveling wave solutions of nonlinear partial differential equations (NLPDEs) plays an important role in the analysis of complex physical phenomena. The NLPDEs appear in physical sciences, various scientific and engineering problems, such as, fluid mechanics, plasma physics, optical fibers, biology, solid state physics, chemical kinematics, chemical physics, chemistry and many others. In recent years, to obtain exact traveling wave solutions of NLPDEs, many effective and powerful methods have been presented in the literature, such as the Backlund transformation [1], the tanh function method [2], the extended tanh function method [3], the variational iteration method [4], the Adomian decomposition method [5, 6], the homotopy perturbation method [7], the F-expansion method [8], the Hirota’s bilinear method [9], the exp-function method [10], the Cole-Hopf transformation [11], the general projective Riccati equation method [12] and others [13–20], Nowaday, searching analytical solutions of the NLPDEs has become more lucrative partly due to the accessibility computer symbolic systems, like Maple, Mathematica, and Matlab, which help us to calculate the complicated and wearisome algebraic calculations on computer. Recently, Wang et al. [21] introduced a new direct method called the -expansion method to seek traveling wave solutions of the nonlinear evolution equations. Abazari [22] implemented the -expansion method to NLEEs related to fluid mechanics. Zheng [23] used the method for getting exact traveling wave solutions of two different types of equations. Feng et al. [24] used the method to seek solutions of the Kolmogorov-Petrovskii-Piskunov equation. Liu et al. [25] concerned the method to simplified MCH equation and Gardner equation. Feng [26] applied the method to the seventh-order Sawada-Kotera equation. At present time, Guo and Zhou [27] first proposed the extended -expansion method based on new ansatz. They applied the method to the Whitham-Broer-Kaup-Like equations and couple Hirota-Satsuma KdV equations. Zayed et al. [28] used extended
Modified $(G'/G)$-expansion method for solving nonlinear partial differential equations  [PDF]
Mahouton Norbert Hounkonnou,Rolland Finangnon Kanfon
Physics , 2013,
Abstract: This paper addresses a modified ($G'/G$)-expansion method to obtain new classes of solutions to nonlinear partial differential equations (NPDEs). The cases of Burgers, KdV and Kadomtsev-Petviashvili NPDEs are exhaustively studied. Relevant graphical representations are shown in each case.
New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the -Dimensional Evolution Equation  [PDF]
Hasibun Naher,Farah Aini Abdullah
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/486458
Abstract: The generalized Riccati equation mapping is extended with the basic -expansion method which is powerful and straightforward mathematical tool for solving nonlinear partial differential equations. In this paper, we construct twenty-seven traveling wave solutions for the (2+1)-dimensional modified Zakharov-Kuznetsov equation by applying this method. Further, the auxiliary equation is executed with arbitrary constant coefficients and called the generalized Riccati equation. The obtained solutions including solitons and periodic solutions are illustrated through the hyperbolic functions, the trigonometric functions, and the rational functions. In addition, it is worth declaring that one of our solutions is identical for special case with already established result which verifies our other solutions. Moreover, some of obtained solutions are depicted in the figures with the aid of Maple. 1. Introduction The study of analytical solutions for nonlinear partial differential equations (PDEs) has become more imperative and stimulating research fields in mathematical physics, engineering sciences, and other technical arena [1–47]. In the recent past, a wide range of methods have been developed to construct traveling wave solutions of nonlinear PDEs such as, the inverse scattering method [1], the Backlund transformation method [2], the Hirota bilinear transformation method [3], the bifurcation method [4, 5], the Jacobi elliptic function expansion method [6–8], the Weierstrass elliptic function method [9], the direct algebraic method [10], the homotopy perturbation method [11, 12], the Exp-function method [13–17], and others [18–28]. Recently, Wang et al. [29] presented a widely used method, called the ( )-expansion method to obtain traveling wave solutions for some nonlinear evolution equations (NLEEs). Further, in this method, the second-order linear ordinary differential equation is implemented, as an auxiliary equation, where and are constant coefficients. Afterwards, many researchers investigated many nonlinear PDEs to construct traveling wave solutions via this powerful ( )-expansion method. For example, Feng et al. [30] applied the same method for obtaining exact solutions of the Kolmogorov-Petrovskii-Piskunov equation. In [31], Naher et al. concerned about this method to construct traveling wave solutions for the higher-order Caudrey-Dodd-Gibbon equation. Zayed and Al-Joudi [32] studied some nonlinear partial differential equations to obtain analytical solutions by using the same method whereas Gepreel [33] executed this method and found exact solutions of
Remarks on the first integral method for solving nonlinear evolution equations  [PDF]
Aparna Saha,B. Talukdar Umapada Das,Supriya Chatterjee
Physics , 2015,
Abstract: We point out that use of the first integral method ( J.Phys. A :Math. Gen. 35 (2002) 343 ) for solving nonlinear evolution equations gives only particular solutions of equations that model conservative systems. On the other hand, for dissipative dynamical systems, the method leads to incorrect solutions of the equations.
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