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Volume 37, Issue 3
Solution of Klein-Gordon Equation by a Method of Lines Using Reproducing Kernel Hilbert Space Method

Kaushal Patel & Gautam Patel

DOI: 10.4208/jpde.v37.n3.2

J. Part. Diff. Eq., 37 (2024), pp. 251-262.

Published online: 2024-08

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  • Abstract

This paper presents a method of lines solution based on the reproducing kernel Hilbert space method to the nonlinear one-dimensional Klein-Gordon equation that arises in many scientific fields areas. Our method uses discretization of the partial derivatives of the space variable to get a system of ODEs in the time variable and then solve the system of ODEs using reproducing kernel Hilbert space method. Consider two examples to validate the proposed method. Compare the results with the exact solution by calculating the error norms $L_2$ and $L_∞$ at various time levels. The results show that the presented scheme is a systematic, effective and powerful technique for the solution of Klein-Gordon equation.

  • Keywords

Method of Lines, reproducing kernel Hilbert space method, Klein-Gordon equation.

  • AMS Subject Headings

65D15, 65Q05, 68Q25, 68U20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-37-251, author = {Patel , Kaushal and Patel , Gautam}, title = {Solution of Klein-Gordon Equation by a Method of Lines Using Reproducing Kernel Hilbert Space Method}, journal = {Journal of Partial Differential Equations}, year = {2024}, volume = {37}, number = {3}, pages = {251--262}, abstract = {

This paper presents a method of lines solution based on the reproducing kernel Hilbert space method to the nonlinear one-dimensional Klein-Gordon equation that arises in many scientific fields areas. Our method uses discretization of the partial derivatives of the space variable to get a system of ODEs in the time variable and then solve the system of ODEs using reproducing kernel Hilbert space method. Consider two examples to validate the proposed method. Compare the results with the exact solution by calculating the error norms $L_2$ and $L_∞$ at various time levels. The results show that the presented scheme is a systematic, effective and powerful technique for the solution of Klein-Gordon equation.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v37.n3.2}, url = {http://global-sci.org/intro/article_detail/jpde/23341.html} }
TY - JOUR T1 - Solution of Klein-Gordon Equation by a Method of Lines Using Reproducing Kernel Hilbert Space Method AU - Patel , Kaushal AU - Patel , Gautam JO - Journal of Partial Differential Equations VL - 3 SP - 251 EP - 262 PY - 2024 DA - 2024/08 SN - 37 DO - http://doi.org/10.4208/jpde.v37.n3.2 UR - https://global-sci.org/intro/article_detail/jpde/23341.html KW - Method of Lines, reproducing kernel Hilbert space method, Klein-Gordon equation. AB -

This paper presents a method of lines solution based on the reproducing kernel Hilbert space method to the nonlinear one-dimensional Klein-Gordon equation that arises in many scientific fields areas. Our method uses discretization of the partial derivatives of the space variable to get a system of ODEs in the time variable and then solve the system of ODEs using reproducing kernel Hilbert space method. Consider two examples to validate the proposed method. Compare the results with the exact solution by calculating the error norms $L_2$ and $L_∞$ at various time levels. The results show that the presented scheme is a systematic, effective and powerful technique for the solution of Klein-Gordon equation.

Patel , Kaushal and Patel , Gautam. (2024). Solution of Klein-Gordon Equation by a Method of Lines Using Reproducing Kernel Hilbert Space Method. Journal of Partial Differential Equations. 37 (3). 251-262. doi:10.4208/jpde.v37.n3.2
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