Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition

Kedir Aliyi Koroche

doi:doi:10.11648/j.ijtam.20210703.11

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Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition

Kedir Aliyi Koroche

Published in International Journal of Theoretical and Applied Mathematics (Volume 7, Issue 3)

Received: 28 March 2021 Accepted: 24 May 2021 Published: 31 May 2021

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Abstract

In this paper, the Lax-Wend off difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion, and the Lax-Wend off difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error, L₂ and L_∞ norm. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 7, Issue 3)
DOI	10.11648/j.ijtam.20210703.11
Page(s)	40-52
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Hyperbolic Equation, One-dimensional Wave Equation, Lax-Wend off Difference Scheme, Taylor Series Methods, Richardson Extrapolation, Stability, and Convergence

References

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APA Style

Kedir Aliyi Koroche. (2021). Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition. International Journal of Theoretical and Applied Mathematics, 7(3), 40-52. https://doi.org/10.11648/j.ijtam.20210703.11

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ACS Style

Kedir Aliyi Koroche. Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition. Int. J. Theor. Appl. Math. 2021, 7(3), 40-52. doi: 10.11648/j.ijtam.20210703.11

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AMA Style

Kedir Aliyi Koroche. Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition. Int J Theor Appl Math. 2021;7(3):40-52. doi: 10.11648/j.ijtam.20210703.11

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@article{10.11648/j.ijtam.20210703.11,
  author = {Kedir Aliyi Koroche},
  title = {Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {7},
  number = {3},
  pages = {40-52},
  doi = {10.11648/j.ijtam.20210703.11},
  url = {https://doi.org/10.11648/j.ijtam.20210703.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20210703.11},
  abstract = {In this paper, the Lax-Wend off difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion, and the Lax-Wend off difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error, L2 and L∞ norm. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.},
 year = {2021}
}

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TY  - JOUR
T1  - Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition
AU  - Kedir Aliyi Koroche
Y1  - 2021/05/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ijtam.20210703.11
DO  - 10.11648/j.ijtam.20210703.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 40
EP  - 52
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20210703.11
AB  - In this paper, the Lax-Wend off difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion, and the Lax-Wend off difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error, L2 and L∞ norm. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.
VL  - 7
IS  - 3
ER  -

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Author Information

Kedir Aliyi Koroche

Department of Mathematics, College of Natural and Computational Sciences, Ambo University, Ambo, Ethiopia

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Plain Text BibTeX RIS

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Kedir Aliyi Koroche. (2021). Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition. International Journal of Theoretical and Applied Mathematics, 7(3), 40-52. https://doi.org/10.11648/j.ijtam.20210703.11

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ACS Style

Kedir Aliyi Koroche. Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition. Int. J. Theor. Appl. Math. 2021, 7(3), 40-52. doi: 10.11648/j.ijtam.20210703.11

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AMA Style

Kedir Aliyi Koroche. Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition. Int J Theor Appl Math. 2021;7(3):40-52. doi: 10.11648/j.ijtam.20210703.11

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@article{10.11648/j.ijtam.20210703.11,
  author = {Kedir Aliyi Koroche},
  title = {Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {7},
  number = {3},
  pages = {40-52},
  doi = {10.11648/j.ijtam.20210703.11},
  url = {https://doi.org/10.11648/j.ijtam.20210703.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20210703.11},
  abstract = {In this paper, the Lax-Wend off difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion, and the Lax-Wend off difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error, L2 and L∞ norm. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Lax-Wend off Difference Scheme with Richardson Extrapolation Method for One Dimensional Wave Equation Subjected to Integral Condition
AU  - Kedir Aliyi Koroche
Y1  - 2021/05/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ijtam.20210703.11
DO  - 10.11648/j.ijtam.20210703.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 40
EP  - 52
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20210703.11
AB  - In this paper, the Lax-Wend off difference scheme has been presented for solving the one-dimensional wave equation with integral boundary conditions. First, the given solution domain is discretized and the derivative involving the spatial variable is replaced by the central finite difference approximation of functional values at each grid point by using Taylor series expansion. Then, for solving the resulting second-order linear ordinary differential equation, the displacement function is discretized in the direction of a temporal variable by using Taylor series expansion, and the Lax-Wend off difference scheme is developed, then it gives a system of algebraic equations. The derivative of the initial condition is also discretized by using the central finite difference method. Then the obtained system of algebraic equations is solved by the matrix inverse method. The stability and convergent analysis of the scheme are investigated. The established convergence of the scheme is further accelerated by applying the Richardson extrapolation which yields fourth-order convergent in spatial variable and sixth-order convergent in a temporal variable. To validate the applicability of the proposed method, three model examples are considered and solved for different values of the mesh sizes in both directions. Numerical results are presented in tables in terms of maximum absolute error, L2 and L∞ norm. The numerical results presented in tables and graphs confirm that the approximate solution is in good agreement with the exact solution.
VL  - 7
IS  - 3
ER  -

Copy | Download