©Freund Publishing House Ltd. International Journal ofNonlinear Sciences and Numerical Simulation, 9(2), 141-156 2008 Variational Iteration Method for Solving Higher-order Nonlinear Boundary Value Problems Using He's Polynomials Muhammad Aslam Noor and Syed Tauseef Mohyud-Din Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan. Emails: noormaslam(a),hotmail.com (M. A. Noor) syedtauseefs&.hotmail. com (S. T. Mohyud-Din) Abstract In this paper, we apply the variational iteration method using He's polynomials (VIMHP) for solving the higher-order boundary value problems. The proposed method is an elegant combination of variational iteration and the homotopy perturbation methods. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The proposed iterative scheme finds the solution without any discritization, linearization or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that the proposed VIMHP solves nonlinear problems without using the Adomian's polynomials can be considered as a clear advantage of this algorithm over the decomposition method. Keywords: Variational iteration method, He's polynomials, nonlinear problems, higher order boundary value problems, Boussinesq equations, error estimates. 1. Introduction This paper is devoted to the study of higher-order boundary value problem which are known to arise in the study of astrophysics, hydrodynamic and hydro magnetic stability, fluid dynamics, astronomy, beam and long wave theory and applied physics, see [3-8, 28, 29, 31, 33-35, 38- 41], A class of characteristic-value problems of higher order (as higher as twenty four) is known to arise in hydrodynamic and hydro magnetic stability. In addition, it is well known that when a uniform magnetic field is applied across the fluid in the same direction as that of gravity than instability may sets in as over stability which may be modeled by a twelfth-order or eighth-order boundary value problem; and if the instability sets in as ordinary convection than it can be modeled by a tenth-order boundary value problem. Moreover, eighth-order boundary value problems occur in the torsinal vibration of uniform beam, see [3-7, 28, 29, 31, 33-35, 38-41] and the references therein. The literature of numerical analysis contains little on the solution of the higher-order boundary value problems. Research in this direction may be considered in its early stages [3-7], Theorems which list the conditions for the existence and uniqueness of solutions of such problems are contained in a comprehensive survey by Agarwal [3]. The boundary value problems of higher order have been investigated due to their mathematical importance and the potential for applications in diversified applied sciences. Several techniques including the finite- difference, polynomial and non polynomial spline and decomposition have been employed for solving such problems, see [4, 7, 8, 38-41] and the references therein. All these techniques have their inbuilt deficiencies, like divergence of the results at the points adjacent to the boundary and calculation of the so-called Adomian's polynomials. Moreover, the performance of most of the methods used so far is well known that they provide the solution at grid points only. Recently, Noor and Mohyud-Din employed homotopy perturbation, variational iteration and the variational decomposition methods for solving higher-order boundary value problems, see [28, 29, 31, 33-35], He [11-21] developed the variational iteration and homotopy perturbation methods for solving linear, nonlinear, initial and boundary value problems. It is worth mentioning Brought to you by | Heinrich Heine Universität Düsseldorf Authenticated | 134.99.128.41 Download Date | 7/29/13 8:47 AM