ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.8(2009) No.1,pp.11-18 First Integral Method for the Improved Modiﬁed KdV Equation A. A. Soliman *1,3 K. R. Raslan **2,4 1 Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish 45111, Egypt 2 Department of Mathematics, Faculty of Science, Al-Azhar University,Nasr-City, Cairo, Egypt 3 Department of Mathematics, Bisha Teachers’ College, King Khalid University,Bisha, P. O. Box. 551, Knigdom of Saudi Arabia 4 Current address: Community College in Riyadh, King Saud University, Saudi Arabia (Received 17 November 2008, accepted 27 February 2009) Abstract: New exact solutions of one important partial differential equations are obtained by using the ﬁrst integral method. The efﬁciency of the method is demonstrated by applying it for the improved modiﬁed KdV (IMKdV) equation. Key words: First integral method; IMKdV equation; Nonlinear partial differential equations 1 Introduction The investigation of the exact solution of nonlinear partial differential equations play an important role in the study of the nonlinear science. The exact solutions of the nonlinear equations facilitates the veri- ﬁcation of the numerical solvers and aids in the stability analysis of solutions. In the past decade, there has been signiﬁcant progress in the development of these methods such as Hirota bilinear method [1-2], homogenous balance method [3], variational iteration method [4-9], the Riccati expansion method with constant coefﬁcients[10], the modiﬁed extended tanh-function method [11-12], the generalized hyperbolic function [13-14], cosine-function method [15], and the variable separation method [16-17]. The study of numerical methods for the solution of partial differential equations (PDEs) has enjoyed an intense period of activity over the last 40 years from both theoretical and practical points of view. Improvements in nu- merical techniques, together with the rapid advance in computer technology, have meant that many of the PDEs arising from engineering and scientiﬁc applications, which were previously intractable, can now be routinely solved [18]. In ﬁnite difference methods differential operators are approximated and difference equations are solved. In the ﬁnite element method the continuous domain is represented as a collection of a ﬁnite number N of sub-domains known as elements. The collection of elements is called the ﬁnite ele- ment mesh. The differential equations for time dependent problems are approximated by the ﬁnite element method to obtain a set of ordinary differential equations (ODEs) in time. These differential equations are solved approximately by ﬁnite difference methods. In all ﬁnite difference and ﬁnite elements it is necessary to have boundary and initial conditions. However, the Adomian decomposition method, which has been developed by George Adomian [19], depends only on the initial conditions and obtains a solution in series which converges to the exact solution of the problem. The aim of this paper is extended the ﬁrst integral method proposed by Feng [20-23] to ﬁnd the exact solutions of the the improved modiﬁed KdV equation [24-28]. * Corresponding author. E-mail address: asoliman 99@yahoo.com ** E-mail address: kamal raslan@yahoo.com Copyright c World Academic Press, World Academic Union IJNS.2008.8.15/238