On the Convergence of the Variational Iteration Method for Klein-Gordon Problems with Variable Coefficients II

TL;DR

This paper proves the convergence of the Variational Iteration Method (VIM) for solving linear Klein-Gordon equations with variable coefficients, demonstrating uniform convergence and analyzing the structure of iterates.

Contribution

It establishes the convergence of VIM for Klein-Gordon problems with variable coefficients and introduces a new approach for analyzing partial sum approximations of the Lagrange multiplier.

Findings

VIM converges uniformly to the unique solution on compact intervals.

Convergence holds even when using partial sums of the Lagrange multiplier.

A new proof approach provides insights into the structure of the iterates.

Abstract

In this paper we investigate convergence for the Variational Iteration Method (VIM) which was introduced and described in \cite{He0},\cite{He1}, \cite{He2}, and \cite{He3}. We prove the convergence of the iteration scheme for a linear Klein-Gorden equation with a variable coefficient whose unique solution is known. The iteration scheme depends on a {\em Lagrange multiplier}, $\lambda(r,s)$, which is represented as a power series. We show that the VIM iteration scheme converges uniformly on compact intervals to the unique solution. We also prove convergence when $\lambda(r,s)$ is replaced by any of its partial sums. The first proof follows a familiar pattern, but the second requires a new approach. The second approach also provides some detail regarding the structure of the iterates.