On the Convergence of the Variational Iteration Method for Klein-Gordon Problems with Variable Coefficients II
<|reference_start|>On the Convergence of the Variational Iteration Method for Klein-Gordon Problems with Variable Coefficients II: 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.<|reference_end|>
@article{drabek2024on,
    title={On the Convergence of the Variational Iteration Method for Klein-Gordon
  Problems with Variable Coefficients II},
    author={Pavel Drabek, Stephen B Robinson, Shohreh Gholizadeh Siahmazgi},
    journal={arXiv preprint arXiv:2407.14423},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.14423},
    primaryClass={math.NA cs.NA math.AP}
}