World Applied Sciences Journal 6 (7): 999-1004, 2009 ISSN 1818-4952 © IDOSI Publications, 2009 Corresponding Author: Dr. Syed Tauseef Mohyud-Din, HITEC University Taxila Cantt, Pakistan 999 Modified Variational Iteration Method for Solving Sine-Gordon Equations 1 Syed Tauseef Mohyud-Din, 2 Muhammad Aslam Noor and 2 Khalida Inayat Noor 1 HITEC University Taxila Cantt, Pakistan 2 Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan Abstract: In this paper, we apply the Modified Variational Iteration Method (MVIM) for solving Sine- Gordon equations which arise in differential geometry, propagation of magnetic flux, stability of fluid motions, nonlinear physics and applied sciences. The proposed modification is made by introducing Adomian’s polynomials in the correction functional of VIM. The use of Lagrange multiplier is a clear advantage of this technique over the decomposition method. Numerical results show the efficiency of the suggested algorithm. Key words: Variational iteration method • Lagrange multiplier • Sine-Gordon equations • Adomian’s polynomials INTRODUCTION The Sine-Gordon equations appear in differential geometry, propagation of magnetic flux, stability of fluid motions, nonlinear physics and applied sciences [1-4]. Several techniques including, Backlund transformations, inverse scattering, similarity, variational iteration, homotopy analysis, tanh and decomposition [1-4] have been used for the solution of these equations. It is worth mentioning that Yücel [4] applied Homotopy Analysis Method (HAM) for solving these problems and also proved the compatibility of the obtained results with VIM [2] for h =-1. The standard form of such equations is given by ( ) ( ) 2 tt xx u x,t cu x,t sinu 0 − +α = with initial conditions ( ) ( ) ( ) ( ) t u x,0 fx, u x,0 gx = = He [5-12] developed and formulated the Variational Iteration Method (VIM) for solving various physical problems. The method has been extremely useful for diversified physical problems [1-31]. In a later work, Abbasbandy [1, 13] used Adomian’s’ polynomials in the correction functional for solving Riccati differential and Klein-Gordon equations. Most recently, Noor and Mohyud-Din developed the elegant coupling of Adomian’s polynomials and the correctional functional of VIM calling it as modified variational iteration method(MVIM) and applied this reliable version for solving various singular and nonsingular initial and boundary value problems [14-17]. Inspired and motivated by the ongoing research in this area, we apply the Modified Variational Iteration Method (MVIM) which is formulated by the elegant coupling of Adomian’s polynomials and the correctional functional for solving Sine-Gordon equations. The use of Lagrange multiplier in the MVIM gives it a clear advantage over the decomposition method since it avoids the successive application of the integral operator and hence reduces the computational work to a tangible level. Moreover, the coupling of Adomian’s polynomials makes the technique more compatible with the nonlinearity of the physical problems [14-17]. Numerical results show the complete reliability of the proposed technique. MODIFIED VARIATIONAL ITERATION METHOD (MVIM) To illustrate the basic concept of the MVIM, we consider the following general differential equation Lu Nu g(x) + = (1) where L is a linear operator, N a nonlinear operator and g (x) is the inhomogeneous term. According to variational iteration method [1-31], we can construct a correction functional as follows x n 1 n n n 0 u (x) u (x) (Lu (s) Nu (s) g(s))ds + = + λ + − ∫  (2) where λ is a Lagrange multiplier [5-12], which can be identified optimally via variational iteration method. The subscripts n denote the nth approximation, n u  is considered as a restricted variation. i.e. n u 0; δ =  (2) is