Journal of Mathematics Research; Vol. 7, No. 3; 2015 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education New Approach of Generalized exp (−φ (ξ )) Expansion Method and Its Application to Som Nonlinear Partial Differential Equations Khalil Hasan Yahya 1 & Zelal. Amin. Moussa 1 1 Mathematics Department, Faculty of Science, Damascus University, Damascus, Syria Correspondence: Zelal. Amin. Moussa, Mathematics Department, Faculty of Science, Damascus University, Damascus, Syria. E-mail: Zelal.moussa@yahoo.com Received: June 7, 2015 Accepted: June 26, 2015 Online Published: July 13, 2015 doi:10.5539/jmr.v7n3p106 URL: http://dx.doi.org/10.5539/jmr.v7n3p106 Abstract In this article, the new approach of generalized exp (−φ (ξ))expansion method has been successfully implemented to seek traveling wave solutions of the Korteweg-de vries equation and the modified Zakharov-Kuznetsov equation. The result reveals that the method together with the new ordinary differential equation is a very influential and effective tool for solving nonlinear partial differential equations in mathematical physics and engineering. The obtained solutions have been articulated by the hyperbolic functions, trigonometric functions and rational functions with arbitrary constants. Keywords: generalized exp (−φ (ξ)) expansion method, exact solutions, Kdv equation, modified ZK equation 1. Introduction In recent years, the nonlinear partial deferential equations (NPDEs) are widely used to describe many important phenomena and dynamic processes in physics, mechanics, chemistry and biology, etc. With the development of soliton theory, many powerful methods have been presented, such as the Jacobi elliptic function expansion method (M.Inc & M. Ergut, 2005), The ( ´ G G ) expansion method (M. Wang, X. Li & j. Zhang, 2008; M. Bashir & A. Moussa 2014; L.X.Li, E.q.Li & M. Wang 2010),the sine.cosine method (E. Yusufoglu & A. Bekir, 2006), the tanh-coth method (M. Bashir & A. Moussa 2014), the F-expansion method (M. Bashir & L. Alhakim 2013), the Exp-function method (He. J. H & Wu. x. H 2006), the exp(−φ (ξ)) expansion method and others (M. A. Akbar & N. H. Ali, 2014; K. Khan & M. A. Akbar, 2013; N. Rahman, M. N. Alam, H. Roshid, S. Akter & M. Ali Akbar, 2014; M.j. Ablowitz, G. Biondini & S. Lillo, 1997; M. Mirzazadeh, S. Khaleghizadeh, 2013; S.T. Mohyud- Din & M.A. Noor, 2008). the exp (−φ (ξ)) expansion method is powerful to solve nonlinear partial deferential equations (NPDEs) and can help to get many new exact solutions which we have never seen before. The present work is motivated by the desire to generate many new and more general exact traveling wave solutions. For this purpose, we propose new approach of exp (−φ (ξ))-expansion method for investigating NLEEs. To depict the novelty and advantages of the methods, we apply these to the KdV equation and modified ZK Equation. 2. Description of New Approach of Generalized exp (−φ (ξ)) Expansion Method Suppose that we have a nonlinear PDE in the following form : F(u, u t , u x , u tt , u xt , u xx , u xxt , ....) = 0 (2.1) where u = u( x, t) is an unknown function F is a polynomial in u = u( x, t) and its partial derivatives, in which the highest order derivatives and nonlinear terms are involved.The main steps of this method are as follows Step 1: Use the traveling wave transformation : u( x, t) = u(ξ) , ξ = k 1 x + k 2 t (2.2) where k 1 , k 2 are a constants to be determined latter, permits us reducing (2.1) to an ODE for u = u (ξ) in the form P ( u, k 1 ´ u, k 2 ´ u, k 1 k 2 u ′′ , .... ) = 0 (2.3) where P is a polynomial of u = u(ξ) and its total derivatives. 106