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

| Peer-Reviewed

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

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

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), 2021. 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

[1]	Nečasová G. et. al. Numerical solution of wave equation using higher-order methods. In AIP Conference Proceedings, AIP Publishing PLC. 2018; 1978: 360005.
[2]	Awawdeh F., Alomari AK., A numerical scheme for the one-dimensional wave Equation subject to an integral conservation condition. In AIP Conference Proceedings, AIP Publishing LLC, 2016; 1739: 020019.
[3]	Bartel LC., et. al. Graded boundary simulation of air/Earth interfaces in finite-difference elastic wave modeling. In SEG Technical Program Expanded, Society of Exploration Geophysicists Abstracts. 2000; 2000: 2444-2447.
[4]	Shakeri F., Dehghan M. The method of lines for the solution of the - Dimensional wave equation Subject to an integral conservation condition. Computers & Mathematics with Applications. 2008; 56: 2175-2188.
[5]	Zin SM. et. al. Cubic Trigonometric B-spline Approach to Numerical Solution of Wave Equation. International Journal of Mathematical, Computational, Physical and Quantum Engineering. 2014; 8 1212-1216.
[6]	S. N. Kihara. Heat Equations and their Applications, Kenya, School of Mathematics University of Nairobi, Master’s Thesis. 2009.
[7]	Caglar, HIKMET et. al. Caglar, and Muge Iseri.. A non- Polynomial spline solution of the one-dimensional wave equation subject to an integral Conservation condition. I Proceedings of the 9^th Wseas international conference on applied computer and applied computational science, Hangzhou. 2010; 27-30.
[8]	Dean GD. Solutions of Partial Differential Equations, India, New Delhi: CBS Publishers and Distributors, 1988.
[9]	Mulder, WA. Huiskes MJ. A simple finite-difference scheme for handling topography with the first-order wave equation. Geophysical Journal International. 2017; 210: 482-499.
[10]	Dehghan, Mehdi, Ali Shokri. A mesh less method for the numerical solution of the one-dimensional wave equation with an integral condition using radial basis functions. Numerical Algorithms, 2009, 52: 461.
[11]	Shortley GH., Weller R. The numerical solution of Laplace's equation. Journal of Applied Physics. 1938; 9: 334-348.
[12]	Haney MM. Generalization of von Neumann analysis for a model of two discrete Half-spaces: The Acoustic case. Geophysics. 2007; 72: SM35-SM46.
[13]	Everink J.. Numerically solving the wave equation using the finite element the method, Faculty of Science, Utrecht University. 2018.
[14]	Rop E C., Oduor O. M.. Solution of Third Order Viscous Wave Equation Using Finite Difference Method, Nonlinear Analysis and Differential Equations. 2019; 7: 53 – 68.
[15]	Kedir Aliyi Koroche. Numerical Solution for One Dimensional Linear Types of Parabolic Partial Differential Equation and Application to Heat Equation. Mathematics and Computer Science. 2020; 5: 76-85.
[16]	Goh J, et. al. Numerical method using cubic B-spline for the heat and wave equation. Computers & Mathematics with Applications. 2011; 62: 4492-4498.
[17]	Wang S. et. al. Lagrangian meshfree finite difference particle method with variable smoothing length for solving wave equations. Advances in Mechanical Engineering. 2018; 10.
[18]	Jain MK,. et. al. computational methods for partial differential equations, New age international publishers, 2007.
[19]	M. W. Oday. Stability and Convergence for nonlinear partial differential equations, Master of Science in Mathematics thesis, Boise State University. 2012.
[20]	Dostart N., Liu Y., Popović MA. Acoustic Waveguide Eigenmode Solver based on a Staggered-Grid Finite-Difference Method. Scientific reports. 2017; 7: 1-11.
[21]	Dehghan M. On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation. Numerical Methods for Partial Differential Equations. 2005; 21: 24-40
[22]	Ang WT. A numerical method for the wave equation subject to a non-local conservation Condition, Appl. Numer. Math. 2006; 56: 1054–1060.
[23]	Seriani G., Oliveira SP. Numerical modeling of mechanical wave propagation. La Rivista del Nuovo Cimento. 2020; 43: 459–514.
[24]	Rashidinia J., Mohsenyzade M.. Numerical Solution of One-Dimensional Heat and Wave Equation by Non-Polynomial Quintic Spline, International Journal of Mathematical Modeling & Computations. 2015; 5: 291- 305.
[25]	M. Dugassa, G. File, T. Aga. Numerical Solution of Second-Order One Dimensional Linear Hyperbolic Telegraph Equation. Ethiopian Journal of Education and Sciences. 2018; 14: 39-52.
[26]	Shokofeh S., Rashidinia J.. Numerical solution of the hyperbolic telegraph the Equation by cubic B-spline collocation method. Applied Mathematics and Computation. 2016; 281: 28–38.
[27]	Saadatmandi A., Dehghan M. Numerical solution of the one-dimensional wave equation with an integral condition, Numerical. Method Partial Differential Equations. 2007; 23: 282–292.
[28]	Ramezani M. et al. Combined finite difference and spectral methods for the numerical solution of the hyperbolic equation with an integral condition. Numerical Methods for Partial Differential Equations: An International Journal. 2008; 24: 1–8.
[29]	Beilin SA. Existence of solutions for one-dimensional wave equations with nonlocal conditions. Electronic Journal of Differential Equations (EJDE) [electronic only]. 2001; 76: 1–8.
[30]	Noye, JN. Nonlinear Partial Differential Equations in Engineering. Conference in Queens College, University of Melbourne, North-Holland Publishing Company, Australia, 1981.
[31]	Mittal RC., Kumar SK.. Numerical Study of Fisher's Equation by Wavelet Galerkin Method. International Journal of Computer Mathematics. 2006; 83: 287-298.
[32]	Bastani M., Salkuyeh DK.. A highly accurate method to solve Fisher’s equation. Pramana, 2012; 78: 335-346.
[33]	Liu E HL. Fundamental methods of numerical extrapolation with Applications. MIT Open Courseware, Massachusetts Institute of Technology 209 (2006).
[34]	Morton KW., Mayers DF. Numerical Solution of Partial Differential Equations. An Introduction, Second Edition, Cambridge University Press, New York, 2005.
[35]	Zafar ZU et al. Finite Element Model for Linear Second Order One Dimensional Inhomogeneous Wave Equation. Pakistan Journal of Engineering and Applied Sciences. 2016; 17: 58–63.
[36]	Sheng Wang, Yong Ou Zhang, Jing Ping Wu. "Lagrangian mesh free finite difference particle method with variable smoothing length for solving wave equations", Advances in Mechanical Engineering, 2018.
[37]	Hu, L.. "Positive solutions of nonlinear singular two point boundary value problems for second-order impulsive differential equations", Applied Mathematics and Computation, 20080301.
[38]	Kadalbajoo, Mohan K., Alpesh Kumar, and Lok Pati Tripathi. "A radial basis functions based finite differences method for wave equation with an integral condition", Applied Mathematics and Computation, 2015.

Cite This Article

Plain Text BibTeX RIS

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

Copy | Download

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

Copy | Download

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

Copy | Download

@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

Author Information

Kedir Aliyi Koroche

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

Download PDF

Sections

Plain Text BibTeX RIS

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

Copy | Download

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

Copy | Download

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

Copy | Download

@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