Exact solutions for a new models of nonlinear partial differential equations Using ( G ′ G 2 )-Expansion Method

In this paper, we present a new model of Kadomtsev–Petviashvili (KP) equation, the KadomtsevPetviashvili–equal width (KP-EW) equation and the Yu–Toda–Sassa– Fukuyama (YTSF) equation. We apply the ( G′ G2 )-expansion method to solve the new models. Exact travelling wave solutions are obtained and expressed in terms of hyperbolic functions, trigonometric functions, rational functionssolutions of this equations from the method, with the aid of the software Maple.


Introduction
Exact solutions of nonlinear partial differential equations (NLPDEs) play a dynamic role in nonlinear sciences. Numerous techniques have been proposed to investigateexact solutions of such equations (NLPDEs). The detailed study of literature reveals some credible contributions in this area. A variety of many authentic methods have been suggested to get the exact solutions of partial differential equations (PDEs) and have beenexpansion-established such as the( ′ ) method [9], the tanh-coth method [8,11], the Jacobi elliptic function expansion method [18], Darbouxtransformation [6],the expfunction method [2], the sine-cosine method [4], Bäcklund transformation method [10], the mapping method [12,14] and the multiple soliton solutions [15].
Recently, Li Wen-An, Chen Hao and Zhang Guo-Cai [17] introduced a new approach, namely the ( ′ 2 )-expansion method, for a reliable treatment of the nonlinear wave equations. The useful ( ′ 2 )-expansion method is then widely used by many authors [3,5,13,17,19].This can be method applied to various nonlinear equations and also gives a few new kinds of solution. Partial differential equations acquired lot of interest and Description of the ( ′ )-Expansion Method attracted attention of many studies due to their frequent occurrence in biochemical, mathematics, viscoelasticity, economics and other areas of science.
(2) In the above equation, ′ denotes to the differentiation with respect to .
Step 2 We suppose that the solution of Eq.(2) has the form where the coefficients 0 , , and are constants to be determinedand( ′ 2 )satisfies a nonlinear ordinary differential equation where and are arbitrary constants, shut that ≠ 1, ≠ 0. The value of positive integer is easy to find by balancing the highest order nonlinear terms with the highest order derivative term appearing in Eq. (2).
Step 3 Substituting Eq. (3) into Eq.(2) and using Eq. (4), collect the coefficients with the same order of( In the above expressions, 1 and 2 are nonzero constants. The multiple exact special solutions of nonlinear partial differential equation (1) are obtained by making use of Eq.

Applications
In this section, we determine the exact traveling wave solutions of the nonlinear (cmKP), (KP-cmEW) and (cmYTSF) equations by using ( ′ 2 )-expansion method.

Conclusions
In this paper, the (