Variational iteration method for solving non-linear partial differential equations A.A. Hemeda Department of Mathematics, Faculty of Science, University of Tanta, Tanta, Egypt Accepted 29 May 2007 Abstract In this paper, we shall use the variational iteration method to solve some problems of non-linear partial differential equations (PDEs) such as the combined KdV–MKdV equation and Camassa–Holm equation. The variational iteration method is superior than the other non-linear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can fined wide application in non-linear problems without linearization or small perturbation. In this method, the problems are initially approximated with possible unknowns, then a correction functional is constructed by a general Lagrange multiplier, which can be identified optimally via the variational theory. Ó 2009 Published by Elsevier Ltd. 1. Introduction Differential equations are widely used to describe physical problems. In most cases, these problems may be too com- plicated to solve exactly. Alternatively, the numerical methods can provide approximate solutions rather than the exact solutions. In fact there are many of methods to solve these problems numerically such as: finite difference methods, multi-grid methods, perturbation methods, but most of these methods are of low accuracy and are of high expensive. In this paper, the variational iteration method has been shown to solve effectively, easily and accurately a large class of linear and non-linear problems with approximations converging rapidly to accurate solutions. In 1978, Inokuti et al. [1] proposed a general Lagrange multiplier method to solve analytically non-linear problems, at first the proposed method used to solve problems in quantum mechanics. The main feature of the proposed method is: the solution of a mathe- matical problem with linearization assumption is used as initial approximation (trial-function), then a more highly pre- cise approximation at some special point can be obtained. More details of this method is illustrated in the following problems. Problem 1 (Duffing equation). This equation is used widely by many perturbation techniques to verify their effectiveness, here we shall use Duffing equation with non-linearity of seventh order to illustrate the general evaluation process of the proposed method. Considering: 0960-0779/$ - see front matter Ó 2009 Published by Elsevier Ltd. doi:10.1016/j.chaos.2007.06.025 E-mail address: aahemeda@yahoo.com Chaos, Solitons and Fractals 39 (2009) 1297–1303 www.elsevier.com/locate/chaos