Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation

Kedir Aliy; Alemayehu Shiferaw; Hailu Muleta

doi:doi:10.11648/j.ajmcm.20210602.12

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Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation

Kedir Aliy, Alemayehu Shiferaw, Hailu Muleta

Published in American Journal of Mathematical and Computer Modelling (Volume 6, Issue 2)

Received: 28 March 2021 Accepted: 11 May 2021 Published: 26 May 2021

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Abstract

In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients times functional values at each grid points. Next by using properties of linear independence of vector space and Multiquadratic radial basis function we can find all waiting coefficient at each grid points of solution domain and we obtain first order linear system of ordinary differential equation with N by N square coefficient Matrices. Then, the resulting first order linear ordinary differential equation is solved by fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the shape parameter ‘c’ and mesh sizes in the direction of the temporal variable; t and fixed value of mesh size in the direction of spatial variable, x. Numerical results are presented in tables in terms of root mean square (E²), maximum absolute error (E^∞) and condition number K (A) of the system matrix. The numerical results presented in tables and graphs confirm that the approximate solution is in a good agreement with the exact solution.

Published in	American Journal of Mathematical and Computer Modelling (Volume 6, Issue 2)
DOI	10.11648/j.ajmcm.20210602.12
Page(s)	35-42
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

Heat Equation, Shape Parameter, Multiquadric-Radial Basis Functions, Weighting Coefficients, Mesh-free Method

References

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Kedir Aliy, Alemayehu Shiferaw, Hailu Muleta. (2021). Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation. American Journal of Mathematical and Computer Modelling, 6(2), 35-42. https://doi.org/10.11648/j.ajmcm.20210602.12

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ACS Style

Kedir Aliy; Alemayehu Shiferaw; Hailu Muleta. Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation. Am. J. Math. Comput. Model. 2021, 6(2), 35-42. doi: 10.11648/j.ajmcm.20210602.12

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AMA Style

Kedir Aliy, Alemayehu Shiferaw, Hailu Muleta. Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation. Am J Math Comput Model. 2021;6(2):35-42. doi: 10.11648/j.ajmcm.20210602.12

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@article{10.11648/j.ajmcm.20210602.12,
  author = {Kedir Aliy and Alemayehu Shiferaw and Hailu Muleta},
  title = {Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {6},
  number = {2},
  pages = {35-42},
  doi = {10.11648/j.ajmcm.20210602.12},
  url = {https://doi.org/10.11648/j.ajmcm.20210602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210602.12},
  abstract = {In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients times functional values at each grid points. Next by using properties of linear independence of vector space and Multiquadratic radial basis function we can find all waiting coefficient at each grid points of solution domain and we obtain first order linear system of ordinary differential equation with N by N square coefficient Matrices. Then, the resulting first order linear ordinary differential equation is solved by fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the shape parameter ‘c’ and mesh sizes in the direction of the temporal variable; t and fixed value of mesh size in the direction of spatial variable, x. Numerical results are presented in tables in terms of root mean square (E2), maximum absolute error (E∞) and condition number K (A) of the system matrix. The numerical results presented in tables and graphs confirm that the approximate solution is in a good agreement with the exact solution.},
 year = {2021}
}

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TY  - JOUR
T1  - Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation
AU  - Kedir Aliy
AU  - Alemayehu Shiferaw
AU  - Hailu Muleta
Y1  - 2021/05/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ajmcm.20210602.12
DO  - 10.11648/j.ajmcm.20210602.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 35
EP  - 42
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20210602.12
AB  - In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients times functional values at each grid points. Next by using properties of linear independence of vector space and Multiquadratic radial basis function we can find all waiting coefficient at each grid points of solution domain and we obtain first order linear system of ordinary differential equation with N by N square coefficient Matrices. Then, the resulting first order linear ordinary differential equation is solved by fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the shape parameter ‘c’ and mesh sizes in the direction of the temporal variable; t and fixed value of mesh size in the direction of spatial variable, x. Numerical results are presented in tables in terms of root mean square (E2), maximum absolute error (E∞) and condition number K (A) of the system matrix. The numerical results presented in tables and graphs confirm that the approximate solution is in a good agreement with the exact solution.
VL  - 6
IS  - 2
ER  -

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Author Information

Kedir Aliy

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

Department of Mathematics, Jimma University College of Natural Sciences, Jimma, Ethiopia
Hailu Muleta

Department of Mathematics, Jimma University College of Natural Sciences, Jimma, Ethiopia

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Plain Text BibTeX RIS

APA Style

Kedir Aliy, Alemayehu Shiferaw, Hailu Muleta. (2021). Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation. American Journal of Mathematical and Computer Modelling, 6(2), 35-42. https://doi.org/10.11648/j.ajmcm.20210602.12

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ACS Style

Kedir Aliy; Alemayehu Shiferaw; Hailu Muleta. Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation. Am. J. Math. Comput. Model. 2021, 6(2), 35-42. doi: 10.11648/j.ajmcm.20210602.12

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AMA Style

Kedir Aliy, Alemayehu Shiferaw, Hailu Muleta. Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation. Am J Math Comput Model. 2021;6(2):35-42. doi: 10.11648/j.ajmcm.20210602.12

Copy | Download

@article{10.11648/j.ajmcm.20210602.12,
  author = {Kedir Aliy and Alemayehu Shiferaw and Hailu Muleta},
  title = {Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {6},
  number = {2},
  pages = {35-42},
  doi = {10.11648/j.ajmcm.20210602.12},
  url = {https://doi.org/10.11648/j.ajmcm.20210602.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210602.12},
  abstract = {In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients times functional values at each grid points. Next by using properties of linear independence of vector space and Multiquadratic radial basis function we can find all waiting coefficient at each grid points of solution domain and we obtain first order linear system of ordinary differential equation with N by N square coefficient Matrices. Then, the resulting first order linear ordinary differential equation is solved by fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the shape parameter ‘c’ and mesh sizes in the direction of the temporal variable; t and fixed value of mesh size in the direction of spatial variable, x. Numerical results are presented in tables in terms of root mean square (E2), maximum absolute error (E∞) and condition number K (A) of the system matrix. The numerical results presented in tables and graphs confirm that the approximate solution is in a good agreement with the exact solution.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Radial Basis Functions Based Differential Quadrature Method for One Dimensional Heat Equation
AU  - Kedir Aliy
AU  - Alemayehu Shiferaw
AU  - Hailu Muleta
Y1  - 2021/05/26
PY  - 2021
N1  - https://doi.org/10.11648/j.ajmcm.20210602.12
DO  - 10.11648/j.ajmcm.20210602.12
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 35
EP  - 42
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20210602.12
AB  - In this paper, Radial basis functions based differential quadrature method has been presented for solving one-dimensional heat equation. First, the given solution domain is discretized using uniform discretization grid point in both direction and the derivative involving the spatial variable, x is replaced by the sum of the weighting coefficients times functional values at each grid points. Next by using properties of linear independence of vector space and Multiquadratic radial basis function we can find all waiting coefficient at each grid points of solution domain and we obtain first order linear system of ordinary differential equation with N by N square coefficient Matrices. Then, the resulting first order linear ordinary differential equation is solved by fifth-order Runge-Kutta method. To validate the applicability of the proposed method, one model example is considered and solved for different values of the shape parameter ‘c’ and mesh sizes in the direction of the temporal variable; t and fixed value of mesh size in the direction of spatial variable, x. Numerical results are presented in tables in terms of root mean square (E2), maximum absolute error (E∞) and condition number K (A) of the system matrix. The numerical results presented in tables and graphs confirm that the approximate solution is in a good agreement with the exact solution.
VL  - 6
IS  - 2
ER  -

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