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

TL;DR

This work establishes rigorous convergence results for the Variational Iteration Method (VIM) applied to a linear Klein-Gordon equation with a variable coefficient. By formulating the VIM with a Lagrange multiplier $λ(r,s)$ expressed as a power series, the authors prove uniform convergence of the iterates to the known exact solution on any compact interval, and show convergence when $λ$ is replaced by partial sums $λ_N$. Two distinct proofs are developed: a standard contraction-style argument for the full multiplier and a novel coefficient-recursion approach that reveals the iterates' structure in terms of Airy coefficients. The results provide theoretical guarantees for the reliability of VIM on variable-coefficient wave-type problems and clarify how multiplier approximations impact convergence, informing efficient numerical implementations.

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}, $λ(r,s)$, 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 $λ(r,s)$ 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.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof