On the Convergence of the Variational Iteration Method for Klein-Gordon Problems with Variable Coefficients II
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| Format: | Preprint |
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2024
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| _version_ | 1866913437961945088 |
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| author | Drabek, Pavel Robinson, Stephen B Siahmazgi, Shohreh Gholizadeh |
| author_facet | Drabek, Pavel Robinson, Stephen B Siahmazgi, Shohreh Gholizadeh |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_14423 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Convergence of the Variational Iteration Method for Klein-Gordon Problems with Variable Coefficients II Drabek, Pavel Robinson, Stephen B Siahmazgi, Shohreh Gholizadeh Numerical Analysis Analysis of PDEs 35 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. |
| title | On the Convergence of the Variational Iteration Method for Klein-Gordon Problems with Variable Coefficients II |
| topic | Numerical Analysis Analysis of PDEs 35 |
| url | https://arxiv.org/abs/2407.14423 |