Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition

Kedir Nebi Habib

doi:doi:10.11648/j.ml.20210702.11

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Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition

Kedir Nebi Habib

Published in Mathematics Letters (Volume 7, Issue 2)

Received: 24 October 2020 Accepted: 3 May 2021 Published: 26 May 2021

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Abstract

There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. The wave equation is a second order linear hyperbolic partial differential equation that describes the propagation of variety of waves, such as sound or water waves. In this paper we consider the convergence analysis of the explicit schemes for solving one dimensional, time-dependent wave equation with Drichlet and Neumann boundary condition. Taylor's series expansion is used to expand the finite difference approximations in the explicit scheme. We present the derivation of the schemes and develop a computer program to implement it We use spectral radius of Matrix obtained from discretization and Von Neumann stability condition to determine stability, and consistence of the method from truncated error from discretized method. Using Lax Equivalence Theorem, convergence of the methods was described by testing consistency and stability of the methods. And it is found out that the scheme is stable with the Drichlet boundary and conditionally stable with Derivative boundary condition.

Published in	Mathematics Letters (Volume 7, Issue 2)
DOI	10.11648/j.ml.20210702.11
Page(s)	19-24
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

Wave Equation, Explicit Method, Convergence, Stability

References

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[2]	D. M. Causon; Professor C. G. Mingham (2010). Introductory Finite Differences for PDEs.
[3]	Harwinder Kaur (2012). MSc. Thesis on Numerical Solutions of Some Differantial Equations Using B-Spline Collection Method. School of Mathematics and Computer Applications, Thapar University.
[4]	J. W. Thomas (1998) numerical partia equation: finite difference methods.
[5]	Michael T. Heath (2002). Scientific Computing: An Introductotry Survey Chapter 11- Partial DifferantialEquations.
[6]	Michael P. Lamoureux (2006). The mathematics of PDEs and the wave equation.
[7]	2012 Peter J. Olver Vibration and Diffusion in One–Dimensional Media.
[8]	2012. Rishu SinglaThesis paper of finite difference equation for parabolic equationHans De and Paul Ullrich (2007-2009). Introductionto Computational PDEs Department of Applied Mathematics University of Waterloo.
[9]	Randall J. Leveque (2006). A Finite difference Methods for Differrential Equations. AMath 585, Winter Quarter 2006 University of Washington.
[10]	Rishu Singla (2012). Numerical solution of Some Parabolic Partial Differantial Equations Using Finite Difference Methods.
[11]	S. Larsson and V. Theme. Partial Differential Equations with Numerical Methods, volume 45 of Texts in Applied Mathematics. Springer, 2003.
[12]	Abel Kurura T. and Charles Otieno N. (2018) Stability and Consistency Analysis for Central Difference Scheme for Advection Diffusion Partial Differential Equation Kisii university, Kenya.
[13]	Doyo K. and Gofe G. (2016). Convergence Rates of Finite Difference Schemes for the Diffusion Equation with Neumann Boundary Conditions. American Journal of Computational and Applied Mathematics, 6 (2): 92102.
[14]	Azad, T. M. A. K., M. Begum and L. S. Andallah. (2015). An explicit finite difference scheme for advection diffusion equation (2015). Jahangirnagar J. Mathematics and Mathematical Sciences 24: 2219-5823.
[15]	Chan, T. F. (1984). Stability analysis of finite difference schemes for the advection diffusion equation. SIAM J. Numer. Anal. 21: 272-284.

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APA Style

Kedir Nebi Habib. (2021). Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition. Mathematics Letters, 7(2), 19-24. https://doi.org/10.11648/j.ml.20210702.11

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ACS Style

Kedir Nebi Habib. Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition. Math. Lett. 2021, 7(2), 19-24. doi: 10.11648/j.ml.20210702.11

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AMA Style

Kedir Nebi Habib. Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition. Math Lett. 2021;7(2):19-24. doi: 10.11648/j.ml.20210702.11

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@article{10.11648/j.ml.20210702.11,
  author = {Kedir Nebi Habib},
  title = {Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition},
  journal = {Mathematics Letters},
  volume = {7},
  number = {2},
  pages = {19-24},
  doi = {10.11648/j.ml.20210702.11},
  url = {https://doi.org/10.11648/j.ml.20210702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20210702.11},
  abstract = {There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. The wave equation is a second order linear hyperbolic partial differential equation that describes the propagation of variety of waves, such as sound or water waves. In this paper we consider the convergence analysis of the explicit schemes for solving one dimensional, time-dependent wave equation with Drichlet and Neumann boundary condition. Taylor's series expansion is used to expand the finite difference approximations in the explicit scheme. We present the derivation of the schemes and develop a computer program to implement it We use spectral radius of Matrix obtained from discretization and Von Neumann stability condition to determine stability, and consistence of the method from truncated error from discretized method. Using Lax Equivalence Theorem, convergence of the methods was described by testing consistency and stability of the methods. And it is found out that the scheme is stable with the Drichlet boundary and conditionally stable with Derivative boundary condition.},
 year = {2021}
}

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TY  - JOUR
T1  - Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition
AU  - Kedir Nebi Habib
Y1  - 2021/05/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ml.20210702.11
DO  - 10.11648/j.ml.20210702.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 19
EP  - 24
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20210702.11
AB  - There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. The wave equation is a second order linear hyperbolic partial differential equation that describes the propagation of variety of waves, such as sound or water waves. In this paper we consider the convergence analysis of the explicit schemes for solving one dimensional, time-dependent wave equation with Drichlet and Neumann boundary condition. Taylor's series expansion is used to expand the finite difference approximations in the explicit scheme. We present the derivation of the schemes and develop a computer program to implement it We use spectral radius of Matrix obtained from discretization and Von Neumann stability condition to determine stability, and consistence of the method from truncated error from discretized method. Using Lax Equivalence Theorem, convergence of the methods was described by testing consistency and stability of the methods. And it is found out that the scheme is stable with the Drichlet boundary and conditionally stable with Derivative boundary condition.
VL  - 7
IS  - 2
ER  -

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Author Information

Kedir Nebi Habib

Department of Mathematics, College of Natural Science, Arba Minch University, Arba Minch, Ethiopia

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APA Style

Kedir Nebi Habib. (2021). Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition. Mathematics Letters, 7(2), 19-24. https://doi.org/10.11648/j.ml.20210702.11

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ACS Style

Kedir Nebi Habib. Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition. Math. Lett. 2021, 7(2), 19-24. doi: 10.11648/j.ml.20210702.11

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AMA Style

Kedir Nebi Habib. Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition. Math Lett. 2021;7(2):19-24. doi: 10.11648/j.ml.20210702.11

Copy | Download

@article{10.11648/j.ml.20210702.11,
  author = {Kedir Nebi Habib},
  title = {Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition},
  journal = {Mathematics Letters},
  volume = {7},
  number = {2},
  pages = {19-24},
  doi = {10.11648/j.ml.20210702.11},
  url = {https://doi.org/10.11648/j.ml.20210702.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20210702.11},
  abstract = {There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. The wave equation is a second order linear hyperbolic partial differential equation that describes the propagation of variety of waves, such as sound or water waves. In this paper we consider the convergence analysis of the explicit schemes for solving one dimensional, time-dependent wave equation with Drichlet and Neumann boundary condition. Taylor's series expansion is used to expand the finite difference approximations in the explicit scheme. We present the derivation of the schemes and develop a computer program to implement it We use spectral radius of Matrix obtained from discretization and Von Neumann stability condition to determine stability, and consistence of the method from truncated error from discretized method. Using Lax Equivalence Theorem, convergence of the methods was described by testing consistency and stability of the methods. And it is found out that the scheme is stable with the Drichlet boundary and conditionally stable with Derivative boundary condition.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Convergence Analysis for Wave Equation by Explicit Finite Difference Equation with Drichlet and Neumann Boundary Condition
AU  - Kedir Nebi Habib
Y1  - 2021/05/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ml.20210702.11
DO  - 10.11648/j.ml.20210702.11
T2  - Mathematics Letters
JF  - Mathematics Letters
JO  - Mathematics Letters
SP  - 19
EP  - 24
PB  - Science Publishing Group
SN  - 2575-5056
UR  - https://doi.org/10.11648/j.ml.20210702.11
AB  - There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. The wave equation is a second order linear hyperbolic partial differential equation that describes the propagation of variety of waves, such as sound or water waves. In this paper we consider the convergence analysis of the explicit schemes for solving one dimensional, time-dependent wave equation with Drichlet and Neumann boundary condition. Taylor's series expansion is used to expand the finite difference approximations in the explicit scheme. We present the derivation of the schemes and develop a computer program to implement it We use spectral radius of Matrix obtained from discretization and Von Neumann stability condition to determine stability, and consistence of the method from truncated error from discretized method. Using Lax Equivalence Theorem, convergence of the methods was described by testing consistency and stability of the methods. And it is found out that the scheme is stable with the Drichlet boundary and conditionally stable with Derivative boundary condition.
VL  - 7
IS  - 2
ER  -

Copy | Download