Closed form solutions of two nonlinear equation via the enhanced (G′/G)-expansion method

Nonlinear evolution equations have a wide range of applications in science and engineering. In recent years many power-ful methods to construct exact solutions of nonlinear evolution equations. In this paper we present (1G′) expansion method, extended simplest equation method (SEM) and the modification of the truncated expansion (MTEM) method for (2 + 1) dimensional KdV4 equation to establish new exact solutions. So periodic and hyperbolic function solutions are obtained for this equation. The effi-ciency of the these methods for finding travelling wave solutions of the high order nonlinear evolution equations is demonstrated.

This paper is concerned with the exact traveling wave solutions of nonlinear wave equations. Using the tanh function method, we can obtain the accurate expression of the solutions. Further, according to the images of the solutions, we can get the variation depends on the velocity v . Introduction It is well known that nonlinear phenomena are very important in a variety of scientific fields, especially in fluid mechanics, solid state physics, plasma physics, plasma waves, capillary-gravity waves, and chemical physics. Most of these phenomena are described by nonlinear partial differential equations. Analytical solutions of this problems are usually not available, especially when the nonlinear terms are involved. Therefore, finding its travelling solutions is of practical importance. The methods of looking for exact traveling wave solutions of nonlinear evolution equations, has been tremendous development in recent decades, such as inverse scattering method [2], HI Rota’s bilinear technique [5], the Painlve expansion method [13]. In the early nineties of last century, Huibin and Kelin [7] proposed a new method. The main idea of this method is taking hyperbolic tangent function of the power series as possible traveling wave solutions of the nonlinear evolution equations. Then they substituted the power series directly to KdV equation, and obtained the coefficients of the power series. However this method involved very complicated algebra computation. In order to reduce the complex algebra computation, Malfiety [9-11] proposed the tanh-function method. Since all the derivatives of hyperbolic tangent can be expressed by the hyperbolic tangent in itself, this simple translation makes the method can be applied to more nonlinear evolution equations. Fan et al. [3] proposed the extended hyperbolic tangent method, which replace the tanh-function by the solutions of Riccati equation. In [1, 4, 14, 16, and 17], using the tanh function method, they got the exact form of traveling wave solutions of various types of evolution equations. In recent years, the G ′ /G function method [15], the auxiliary function method [6] is based on tanh-function method. This shows that the hyperbolic tangent function method is very effective and direct method when looking for the exact traveling wave solutions of nonlinear evolution equations. The Tanh-function Method Let’s consider the nonlinear partial differential equations ( ) 0 t =   , , , , , xxx xx x u u u u u N (2.1) Where ( ) t x u , is the real function on 2 R ? At first, we assume the traveling wave solutions of (2.1) are the form of (x, t) U( ) U(c(x )), u t ω u = = − (2.2) With the velocity v , and the constant c. Submitted (2.2) into (2.1), we can get the ODEs About ω (U, U , U , U ,...) 0. N ′ ′′ ′′′ = (2.3) Second, we assume the possibly traveling wave solutions can be written International Conference on Information Sciences, Machinery, Materials and Energy (ICISMME 2015) © 2015. The authors Published by Atlantis Press 1325 0 (x, t) U( ) H(Y) , K i i i u a Y ω

Exact travelling wave solutions of nonlinear evolution equations by using the [formula omitted]-expansion 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.

Very recently, Wang et al (2008 Phys. Lett. A 327 417–23) proposed a method, namely the (G′/G)-expansion method, for constructing multiple travelling wave solutions of nonlinear evolution equations arising in mathematical physics. They believe that 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 show that the (G′/G)-expansion method is equivalent to the extended tanh function method.

The Klein equation, Infeld equation, and Sivashinsky equation not only start from realistic physical phenomena but can also be widely used in many physically significant fields such as plasma physics, fluid dynamics, crystal lattice theory, nonlinear circuit theory, and astrophysics. As a consequence, it is a very significant and challenging topic to research the explicit and accurate travelling wave solutions to these three equations. In this paper, in order to solve these three nonlinear partial differential equations (NPDEs), we have made some modifications to the trial function technique proposed by Xie and Tang by bringing in an ansatz solution containing two E-exponential functions. On this basis, we have developed a direct trial function technique to seek the explicit and accurate travelling wave solutions of nonlinear evolution equations (NEEs). We have illustrated its feasibility by applying it to the Klein equation, Infeld equation, and Sivashinsky equation. As a result, a lot of more general explicit and accurate travelling wave solutions of these three equations, including the solitary wave solutions and the singular travelling wave solutions, are successfully constructed in a straightforward and simple manner. The obtained solutions are quite equivalent to those given in the existing references. In addition, compared with the proposed approaches in the existing references, the approach described herein appears to be less calculative. Our technique may provide a novel way of thinking for solving NEEs. It is our firm conviction that the procedure used herein may also be utilized to explore the explicit and accurate travelling wave solutions of other NEEs. We try to generalize this approach to search for the explicit and accurate traveling wave solutions of other NEEs.

In this paper, we find exact solutions of some nonlinear evolution equations by using generalized tanh–coth method. Three nonlinear models of physical significance, i.e. the Cahn–Hilliard equation, the Allen–Cahn equation and the steady-state equation with a cubic nonlinearity are considered and their exact solutions are obtained. From the general solutions, other well-known results are also derived. Also in this paper, we shall compare the generalized tanh–coth method and generalized (G′/G )-expansion method to solve partial differential equations (PDEs) and ordinary differential equations (ODEs). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play important roles in engineering fields. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that the generalized tanh–coth method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear problems.

Nonlinear evolution equations are unavoidable for precisely modelling and understanding nonlinear wave phenomena. The study of nonlinear waves enriches our comprehension of natural phenomena and supports technological advancements across various disciplines. In this work, we have proposed a new expansion method to find the travelling wave solutions of nonlinear evolution equations. This method is named as F μ F + G − expansion method. We applied the proposed technique to construct the exact travelling wave solutions to two well-known nonlinear equations arising in ocean engineering. These equations are extended (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional generalized shallow water wave equation. Propagation of obtained travelling wave solutions are illustrated by surface plots and two-dimensional graphs plotted for suitable parametric values. We observed soliton, kink, breather, lump and periodic wave structures. The results show efficiency and reliability of the proposed method.

Solving two fifth order strong nonlinear evolution equations by using the [formula omitted]-expansion method

Exact solutions of nonlinear evolution equations (NLEEs) play very important role to make known the inner mechanism of compound physical phenomena. In this paper, the new generalized (G'/G)-expansion method is used for constructing the new exact traveling wave solutions for some nonlinear evolution equations arising in mathematical physics namely, the (3+1)-dimensional Zakharov-Kuznetsov equation and the Burgers equation. As a result, the traveling wave solutions are expressed in terms of hyperbolic, trigonometric and rational functions. This method is very easy, direct, concise and simple to implement as compared with other existing methods. This method presents a wider applicability for handling nonlinear wave equations. Moreover, this procedure reduces the large volume of calculations.

The (F/G)-expansion method is firstly proposed, where F=F(ξ) and G = G(ξ) satisfies a first order ordinary differential equation systems (ODEs). We give the exact travelling wave solutions of the variant Boussinesq equations and the KdV equation and by using (F/G)-expansion method. When some parameters of present method are taken as special values, results of the -expansion method are also derived. Hence, -expansion method is sub method of the proposed method. The travelling wave solutions are expressed by three types of functions, which are called the trigonometric functions, the rational functions, and the hyperbolic functions. The present method is direct, short, elementary and effective, and is used for many other nonlinear evolution equations.

The extended (G'/G)-expansion method is significant for finding the exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. In this paper, we enhanced new traveling wave solutions of right-handed non-commutative burgers equations via extended -expansion. Implementation of the method for searching exact solutions of the equation provided many new solutions which can be used to employ some practically physical and mechanical phenomena. Moreover, when the parameters are replaced by special values, the well-known solitary wave solutions of the equation rediscovered from the traveling wave solutions and included free parameters may imply some physical meaningful results in fluid mechanics, gas dynamics, and traffic flow.

Some new exact traveling wave solutions to the simplified MCH equation and the (1 + 1)-dimensional combined KdV–mKdV equations

To seek infinite sequence exact solutions of nonlinear evolution equations, the Böcklund transformation of the solutions to some auxiliary equations and the formula of nonlinear superimposition of solutions are presented for constructing infinite sequence exact solutions to nonlinear evolution equations, which include infinite sequence Jacobi elliptic function solutions, infinite sequence hyperbolic function solutions and infinite sequence trigonomical function solutions. The method is of significance to the search into infinite sequence exact solutions of other nonlinear evolution equations.

General traveling wave solutions of the strain wave equation in microstructured solids via the new approach of generalized (G′/G)-expansion method