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

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), 2021. Published by Science Publishing Group

Keywords

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

References
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[3] Chen, W., Fu, Z. J., and Chen, C. S. (2014). Recent advances in radial basis function collocation methods, Springer Briefs in Applied Sciences and Technology.
[4] Chenoweth, M. N. (2012). A local radial basis function method for the numerical Solution of partial differential equations. PhD thesis, Marshall University.
[5] Cheong, H. T. (2015). Parallel Localized differential quadrature approach in boundary value problem. Journal of math. Analysis and appl. 31 (2): 127-134.
[6] Ding, H., Shu, C., and Tang, D. B. (2005). Error estimates of local multiquadric-based differential quadrature (LMQDQ) method through numerical experiments. International journal for numerical method in Engin. 63 (11): 1513-1529.
[7] Erfanian, M., and Kosari, S. (2017). Using Chebyshev polynomials zeros as point grid for numerical solution of nonlinear PDEs by differential quadrature-based radial basis functions. Int. Journ. Of Math’s. Modelling and Computations, 7 (1): 67-77.
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[9] Franke, R. (1982). Scattered data interpolation: tests of some methods. Mathematics of computation, 38 (157): 181-200.
[10] Hooshmandasl M. R Haidari, M., and Maalek Ghaini F. M. (2012). Numerical solution of one dimensional heat equation by using Chebyshev Wavelets method. JACM an open access journal, 1 (6), 1-7.
[11] Kalyanil, P., and Rao (2013). Numerical solution of heat equation through double Interpolation. IOSR Journ. Of math. 6 (6): 58-62.
[12] Li, J. R., and Greengard, L. (2007). On the numerical solution of the heat equation I: Fast solvers in free space. Journal of Computational Physics, 226 (2): 1891-1901.
[13] Micchelli, C. A (1984). Interpolation of scattered data: distance matrices and conditionally positive definite functions. In Approximation theory and splinefunctions: 143-145. Springer, Dordrecht.
[14] Mongillo, M. (2011). Choosing basis functions and shape parameters for radial basis function methods. SIAM undergraduate research online, 4 (190-209): 2-6.
[15] Tatari, M. and Dehghan, M., (2010). A method for solving partial differential equations via radial basis functions: Application to the heat equation Engineering Analysis with Boundaryelement 34 (3), 206-212.
[16] Waston, D. (2017). Radial Basis Function Differential Quadrature Method for the numerical solution of partial differential equations. The Aquila Digital Community, Dissertations.
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[18] Masho Jima, Alemayehu Shiferaw, Ali Tsegaye. "Numerical Solution of the Coupled Viscous Burgers’ Equation Using Differential Quadrature Method Based on Fourier. Expansion Basis", Applied Mathematics, 2018.
[19] Zhang, Y., and Zong, Z. (2009). Advanced differential quadrature methods. Chapman and Hall/CRC. USA.
[20] C. Shu, H. Ding, N. Zhao. "Numerical comparison of least square-based finite-difference (LSFD) and radial basis function-based finite-difference (RBFFD) methods, Computers & Mathematics with Applications, 2006.
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  • 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

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

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

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

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