Variational iterative method and initial-value problems. (English) Zbl 1175.65118

Summary: The variational iterative method is revisited for initial-value problems in ordinary or partial differential equation. A distributional characterization of the Lagrange multiplier – the keystone of the method – is proposed, that may be interpreted as a retarded Green function. Such a formulation makes possible the simplification of the iteration formula into a Picard iterative scheme, and facilitates the convergence analysis. The approximate analytical solution of a nonlinear Klein-Gordon equation with inhomogeneous initial data is proposed.


65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
Full Text: DOI


[1] Inokuti, M.; Sekine, H.; Mura, T., General use of the Lagrange multiplier in nonlinear mathematical physics, (), 156-162
[2] He, J.-H., A new approach to nonlinear partial differential equations, Commun. nonlinear sci. numer. simul., 2, 230-235, (1997)
[3] He, J.-H., Variational iteration method – a kind of nonlinear analytical technique: some examples, Int. J. non-linear mech., 34, 699-708, (1999) · Zbl 1342.34005
[4] He, J.-H., Some asymptotics methods for strongly nonlinear equations, Int. J. modern phys., 20, 1141-1199, (2006) · Zbl 1102.34039
[5] Ramos, J.I., On the variational iteration method and other iterative techniques for nonlinear differential equations, Appl. math. comput., 199, 39-69, (2008) · Zbl 1142.65082
[6] Schwartz, L., Theorie des distributions, (1966), Hermann Paris
[7] Morse, P.M.; Feshbach, H., Methods of theoretical physics, part I, (1953), McGraw-Hill Inc. · Zbl 0051.40603
[8] Biazar, J.; Ghazvini, H., An analytical approximation to the solution of a wave equation by a variational iteration method, Appl. math. lett., 21, 780-785, (2008) · Zbl 1156.35305
[9] El-sayed, S.M., The decomposition method for studying the klein – gordon equation, Chaos solitons fract., 18, 1025-1030, (2003) · Zbl 1068.35069
[10] Yusufoğlu, E., The variational iteration method for studying the klein – gordon equation, Appl. math. lett., 21, 669-674, (2008) · Zbl 1152.65475
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.