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On solving elliptic boundary value problems using a meshless method with radial polynomials

Author

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  • Ku, Cheng-Yu
  • Xiao, Jing-En
  • Liu, Chih-Yu
  • Lin, Der-Guey

Abstract

This paper presents the meshless method using radial polynomials with the combination of the multiple source collocation scheme for solving elliptic boundary value problems. In the proposed method, the basis function is based on the radial polynomials, which is different from the conventional radial basis functions that approximate the solution using the specific function such as the multiquadric function with the shape parameter for infinitely differentiable. The radial polynomial basis function is a non-singular series function in nature which is infinitely smooth and differentiable in nature without using the shape parameter. With the combination of the multiple source collocation scheme, the center point is regarded as the source point for the interpolation of the radial polynomials. Numerical solutions in multiple dimensions are approximated by applying the radial polynomials with given terms of the radial polynomials. The comparison of the proposed method with the radial basis function collocation method (RBFCM) using the multiquadric and polyharmonic spline functions is conducted. Results demonstrate that the accuracy obtained from the proposed method is better than that of the conventional RBFCM with the same number of collocation points. In addition, highly accurate solutions with the increase of radial polynomial terms may be obtained.

Suggested Citation

  • Ku, Cheng-Yu & Xiao, Jing-En & Liu, Chih-Yu & Lin, Der-Guey, 2021. "On solving elliptic boundary value problems using a meshless method with radial polynomials," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 153-173.
    • Handle: RePEc:eee:matcom:v:185:y:2021:i:c:p:153-173
    DOI: 10.1016/j.matcom.2020.12.012
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    References listed on IDEAS

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    1. Biazar, Jafar & Hosami, Mohammad, 2017. "An interval for the shape parameter in radial basis function approximation," Applied Mathematics and Computation, Elsevier, vol. 315(C), pages 131-149.
    2. Lin, Ji & Zhao, Yuxiang & Watson, Daniel & Chen, C.S., 2020. "The radial basis function differential quadrature method with ghost points," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 173(C), pages 105-114.
    3. Lei Mu & Zhi-hong He & Shi-kui Dong, 2015. "Reproducing Kernel Particle Method for Radiative Heat Transfer in 1D Participating Media," Mathematical Problems in Engineering, Hindawi, vol. 2015, pages 1-11, March.
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    Cited by:

    1. Chih-Yu Liu & Cheng-Yu Ku, 2023. "A Novel ANN-Based Radial Basis Function Collocation Method for Solving Elliptic Boundary Value Problems," Mathematics, MDPI, vol. 11(18), pages 1-19, September.

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