Numerical Simulation of the Heat Equation Using RBF Collocation Method

Antoine Filankembo Ouassissou; Den Matouadi; Cordy Jourvel Itoua-Tsele

doi:doi:10.11648/j.ajam.20231103.13

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Numerical Simulation of the Heat Equation Using RBF Collocation Method

Antoine Filankembo Ouassissou, Den Matouadi, Cordy Jourvel Itoua-Tsele

Published in American Journal of Applied Mathematics (Volume 11, Issue 3)

Received: 5 May 2023 Accepted: 5 June 2023 Published: 20 June 2023

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Abstract

For a very long time, finite volume, finite element, or finite difference methods have been used to solve partial differential equations (PDEs) numerically. These techniques have been used by researchers for centuries to solve a wide range of mathematical, physical, or chemical problems. The complexity of these numerical approaches, for the resolution of the PDEs in space dimensions equal to two or higher, can come from the coding, the management, and the good choice of the triangulation or the mesh of the domain in which one wishes to locate the solution. The radial basis function collocation method is a meshless technique used to numerically solve some partial differential equations and is based on the nodes of the domain and a radial basis function is a real-valued function whose value only depends on the separation of its input parameter x from another fixed point, sometimes known as the function's origin or center. This method was introduced by KANSA in the 1990s. In this study, the numerical simulation of the one-dimensional heat equation was carried out using the RBF Collocation Method and particularly the Gaussian function. This model was used to test the accuracy and efficiency of this method by comparing numerical and analytical solutions on rectangular geometry with collocation nodes. The results show that the RBF collocation approximate solution and the exact solution coincided in test case problems 2, 3 and 4.

Published in	American Journal of Applied Mathematics (Volume 11, Issue 3)
DOI	10.11648/j.ajam.20231103.13
Page(s)	52-57
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

RBF Collocation Method, Theta Scheme, RBF, Heat Equation

References

[1]	Antoine Filankembo, Tathy C. (2013). Application of the RBF Collocation Method to solve the Classic Problem of a Meson field. Far East Journal of Applied Mathematics, 84 (1): 37.
[2]	Nawaz Khan M, Ahmad I, Ahmad H. (2020). A Radial Basis Function Collocation Method for Space-dependent‎ Inverse Heat Problems. Journal of Applied and Computational Mechanics. 6 (SI). 1187-1199. doi: 10.22055/JACM.2020.32999.2123.
[3]	Chen, Wen, Zhuo-Jia Fu, and Ching-Shyang Chen. (2014). Recent advances in radial basis function collocation methods. Heidelberg: Springer.
[4]	Fasshauer, Gregory E. (2007). Meshfree approximation methods with MATLAB. Vol. 6. World Scientific.
[5]	Fasshauer, G. E. (1996). Solving partial differential equation by collocation with radial basis function. In Proceeding of Chamonix, volume 1997, pages 1-8. Vanderbilt University Press Nashville, TN.
[6]	Arora, G. and Bhatia, G. S. (2017). Radial basis function methods to solve partial differential equation arising in financial applications-a review. Nonlinear studies, 24 (1).
[7]	Kansa, E. J. (1999). Motivation for using radial basis functions to solve pdes. Lawrence Livermore National Laaboratory and Embry-Riddle Acronatical Univerity.
[8]	Vertink, R. and Sarler, B. (2006). Meshless local radial basis function collocation method for convective-diffusive solid-liquid phase change problems. International Journal of Numerical Methods for Heat & Fluid Flow, 16 (5): 617-640.
[9]	Koepler, G. (2001). Equations aux dérivées partielles, UFR de mathématiques et informatique, Maitrise d’ingénierie Mathématique, Université René Descartes Paris 5.
[10]	Cheng A. HD. (2007). Radial Basis Function Collocation Method. In: Computational Mechanics, Springer, Berlin, Heidelberg, Doi: https://doi.org/10.1007/978-3-540-75999-7_20
[11]	Antoine, O. F. (2006). Application de la méthode de collocation RBF pour la résolution de certains de certaines Equations aux Dérivées Partielles. PhD thesis, Université de Pau et des Pays de l’Adour.
[12]	Li, J. and Chen, Y. T. (2008), Computational Partial Differential Equations Using MATLAB, CRC Press, Taylor & Francis Group.
[13]	Buhmann, M. D. (2000). Radial basis functions. Acta numerica, 9, 1-38.
[14]	Pepper, D. W., Kassab, A. J., & Divo, E. A. (2014). An introduction to finite element, boundary element, and meshless methods: with applications to heat transfer and fluid flow. (No Title).
[15]	Ferreira, A. J., Kansa, E. J., Fasshauer, G. E., & Leitão, V. M. A. (Eds.). (2009). Progress on meshless methods. Dordrecht: Springer.

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Antoine Filankembo Ouassissou, Den Matouadi, Cordy Jourvel Itoua-Tsele. (2023). Numerical Simulation of the Heat Equation Using RBF Collocation Method. American Journal of Applied Mathematics, 11(3), 52-57. https://doi.org/10.11648/j.ajam.20231103.13

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ACS Style

Antoine Filankembo Ouassissou; Den Matouadi; Cordy Jourvel Itoua-Tsele. Numerical Simulation of the Heat Equation Using RBF Collocation Method. Am. J. Appl. Math. 2023, 11(3), 52-57. doi: 10.11648/j.ajam.20231103.13

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AMA Style

Antoine Filankembo Ouassissou, Den Matouadi, Cordy Jourvel Itoua-Tsele. Numerical Simulation of the Heat Equation Using RBF Collocation Method. Am J Appl Math. 2023;11(3):52-57. doi: 10.11648/j.ajam.20231103.13

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@article{10.11648/j.ajam.20231103.13,
  author = {Antoine Filankembo Ouassissou and Den Matouadi and Cordy Jourvel Itoua-Tsele},
  title = {Numerical Simulation of the Heat Equation Using RBF Collocation Method},
  journal = {American Journal of Applied Mathematics},
  volume = {11},
  number = {3},
  pages = {52-57},
  doi = {10.11648/j.ajam.20231103.13},
  url = {https://doi.org/10.11648/j.ajam.20231103.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20231103.13},
  abstract = {For a very long time, finite volume, finite element, or finite difference methods have been used to solve partial differential equations (PDEs) numerically. These techniques have been used by researchers for centuries to solve a wide range of mathematical, physical, or chemical problems. The complexity of these numerical approaches, for the resolution of the PDEs in space dimensions equal to two or higher, can come from the coding, the management, and the good choice of the triangulation or the mesh of the domain in which one wishes to locate the solution. The radial basis function collocation method is a meshless technique used to numerically solve some partial differential equations and is based on the nodes of the domain and a radial basis function is a real-valued function whose value only depends on the separation of its input parameter x from another fixed point, sometimes known as the function's origin or center. This method was introduced by KANSA in the 1990s. In this study, the numerical simulation of the one-dimensional heat equation was carried out using the RBF Collocation Method and particularly the Gaussian function. This model was used to test the accuracy and efficiency of this method by comparing numerical and analytical solutions on rectangular geometry with collocation nodes. The results show that the RBF collocation approximate solution and the exact solution coincided in test case problems 2, 3 and 4.},
 year = {2023}
}

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TY  - JOUR
T1  - Numerical Simulation of the Heat Equation Using RBF Collocation Method
AU  - Antoine Filankembo Ouassissou
AU  - Den Matouadi
AU  - Cordy Jourvel Itoua-Tsele
Y1  - 2023/06/20
PY  - 2023
N1  - https://doi.org/10.11648/j.ajam.20231103.13
DO  - 10.11648/j.ajam.20231103.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 52
EP  - 57
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20231103.13
AB  - For a very long time, finite volume, finite element, or finite difference methods have been used to solve partial differential equations (PDEs) numerically. These techniques have been used by researchers for centuries to solve a wide range of mathematical, physical, or chemical problems. The complexity of these numerical approaches, for the resolution of the PDEs in space dimensions equal to two or higher, can come from the coding, the management, and the good choice of the triangulation or the mesh of the domain in which one wishes to locate the solution. The radial basis function collocation method is a meshless technique used to numerically solve some partial differential equations and is based on the nodes of the domain and a radial basis function is a real-valued function whose value only depends on the separation of its input parameter x from another fixed point, sometimes known as the function's origin or center. This method was introduced by KANSA in the 1990s. In this study, the numerical simulation of the one-dimensional heat equation was carried out using the RBF Collocation Method and particularly the Gaussian function. This model was used to test the accuracy and efficiency of this method by comparing numerical and analytical solutions on rectangular geometry with collocation nodes. The results show that the RBF collocation approximate solution and the exact solution coincided in test case problems 2, 3 and 4.
VL  - 11
IS  - 3
ER  -

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Author Information

Antoine Filankembo Ouassissou

Departement des Sciences Exacts, école Normale Supérieure, Université Marien Ngouabi, Brazzaville, Congo
Den Matouadi

Departement des Sciences Exacts, école Normale Supérieure, Université Marien Ngouabi, Brazzaville, Congo
Cordy Jourvel Itoua-Tsele

Departement des Sciences Exacts, école Normale Supérieure, Université Marien Ngouabi, Brazzaville, Congo

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APA Style

Antoine Filankembo Ouassissou, Den Matouadi, Cordy Jourvel Itoua-Tsele. (2023). Numerical Simulation of the Heat Equation Using RBF Collocation Method. American Journal of Applied Mathematics, 11(3), 52-57. https://doi.org/10.11648/j.ajam.20231103.13

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ACS Style

Antoine Filankembo Ouassissou; Den Matouadi; Cordy Jourvel Itoua-Tsele. Numerical Simulation of the Heat Equation Using RBF Collocation Method. Am. J. Appl. Math. 2023, 11(3), 52-57. doi: 10.11648/j.ajam.20231103.13

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AMA Style

Antoine Filankembo Ouassissou, Den Matouadi, Cordy Jourvel Itoua-Tsele. Numerical Simulation of the Heat Equation Using RBF Collocation Method. Am J Appl Math. 2023;11(3):52-57. doi: 10.11648/j.ajam.20231103.13

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@article{10.11648/j.ajam.20231103.13,
  author = {Antoine Filankembo Ouassissou and Den Matouadi and Cordy Jourvel Itoua-Tsele},
  title = {Numerical Simulation of the Heat Equation Using RBF Collocation Method},
  journal = {American Journal of Applied Mathematics},
  volume = {11},
  number = {3},
  pages = {52-57},
  doi = {10.11648/j.ajam.20231103.13},
  url = {https://doi.org/10.11648/j.ajam.20231103.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20231103.13},
  abstract = {For a very long time, finite volume, finite element, or finite difference methods have been used to solve partial differential equations (PDEs) numerically. These techniques have been used by researchers for centuries to solve a wide range of mathematical, physical, or chemical problems. The complexity of these numerical approaches, for the resolution of the PDEs in space dimensions equal to two or higher, can come from the coding, the management, and the good choice of the triangulation or the mesh of the domain in which one wishes to locate the solution. The radial basis function collocation method is a meshless technique used to numerically solve some partial differential equations and is based on the nodes of the domain and a radial basis function is a real-valued function whose value only depends on the separation of its input parameter x from another fixed point, sometimes known as the function's origin or center. This method was introduced by KANSA in the 1990s. In this study, the numerical simulation of the one-dimensional heat equation was carried out using the RBF Collocation Method and particularly the Gaussian function. This model was used to test the accuracy and efficiency of this method by comparing numerical and analytical solutions on rectangular geometry with collocation nodes. The results show that the RBF collocation approximate solution and the exact solution coincided in test case problems 2, 3 and 4.},
 year = {2023}
}

Copy | Download

TY  - JOUR
T1  - Numerical Simulation of the Heat Equation Using RBF Collocation Method
AU  - Antoine Filankembo Ouassissou
AU  - Den Matouadi
AU  - Cordy Jourvel Itoua-Tsele
Y1  - 2023/06/20
PY  - 2023
N1  - https://doi.org/10.11648/j.ajam.20231103.13
DO  - 10.11648/j.ajam.20231103.13
T2  - American Journal of Applied Mathematics
JF  - American Journal of Applied Mathematics
JO  - American Journal of Applied Mathematics
SP  - 52
EP  - 57
PB  - Science Publishing Group
SN  - 2330-006X
UR  - https://doi.org/10.11648/j.ajam.20231103.13
AB  - For a very long time, finite volume, finite element, or finite difference methods have been used to solve partial differential equations (PDEs) numerically. These techniques have been used by researchers for centuries to solve a wide range of mathematical, physical, or chemical problems. The complexity of these numerical approaches, for the resolution of the PDEs in space dimensions equal to two or higher, can come from the coding, the management, and the good choice of the triangulation or the mesh of the domain in which one wishes to locate the solution. The radial basis function collocation method is a meshless technique used to numerically solve some partial differential equations and is based on the nodes of the domain and a radial basis function is a real-valued function whose value only depends on the separation of its input parameter x from another fixed point, sometimes known as the function's origin or center. This method was introduced by KANSA in the 1990s. In this study, the numerical simulation of the one-dimensional heat equation was carried out using the RBF Collocation Method and particularly the Gaussian function. This model was used to test the accuracy and efficiency of this method by comparing numerical and analytical solutions on rectangular geometry with collocation nodes. The results show that the RBF collocation approximate solution and the exact solution coincided in test case problems 2, 3 and 4.
VL  - 11
IS  - 3
ER  -

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