On the variational iteration method and other iterative techniques for nonlinear differential equations

Abstract A variety of iterative methods for the solution of initial- and/or boundary-value problems in ordinary and partial differential equations is presented. These iterative procedures provide the solution or an approximation to it as a sequence of iterates. For initial-value problems, it is shown that these iterative procedures can be written in either an integral or differential form. The integral form is governed by a Volterra integral equation, whereas the differential one can be obtained from the Volterra representation by simply differentiation. It is also shown that integration by parts, variation of parameters, adjoint operators, Green’s functions and the method of weighted residuals provide the same Volterra integral equation and that this equation, in turn, can be written as that of the variational iteration method. It is, therefore, shown that the variational iteration method is nothing else by the Picard–Lindelof theory for initial-value problems in ordinary differential equations and Banach’s fixed-point theory for initial-value problems in partial differential equations, and the convergence of these iterative procedures is ensured provided that the resulting mapping is Lipschitz continuous and contractive. It is also shown that some of the iterative methods for initial-value problems presented here are special cases of the Bellman–Kalaba quasilinearization technique provided that the nonlinearities are differentiable with respect to the dependent variable and its derivatives, but such a condition is not required by the techniques presented in this paper. For boundary-value problems, it is shown that one may use the iterative procedures developed for initial-value problems but the resulting iterates may not satisfy the boundary conditions, and two new iterative methods governed by Fredholm integral equations are proposed. It is shown that the resulting iterates satisfy the boundary conditions if the first one does so. The iterative integral formulation presented here is applied to ten nonlinear oscillators with odd nonlinearities and it is shown that its results coincide with those of (differential) two- and three-level iterative techniques, harmonic balance procedures and standard and modified Linstedt–Poincare techniques. The method is also applied to two boundary-value problems.