Variational iterative method and initial-value problems

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.