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A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations

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  • Cheng-Yu Ku

    (Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
    Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan)

  • Jing-En Xiao

    (Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan)

  • Chih-Yu Liu

    (Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan)

Abstract

In this article, a novel radial–based meshfree approach for solving nonhomogeneous partial differential equations is proposed. Stemming from the radial basis function collocation method, the novel meshfree approach is formulated by incorporating the radial polynomial as the basis function. The solution of the nonhomogeneous partial differential equation is therefore approximated by the discretization of the governing equation using the radial polynomial basis function. To avoid the singularity, the minimum order of the radial polynomial basis function must be greater than two for the second order partial differential equations. Since the radial polynomial basis function is a non–singular series function, accurate numerical solutions may be obtained by increasing the terms of the radial polynomial. In addition, the shape parameter in the radial basis function collocation method is no longer required in the proposed method. Several numerical implementations, including homogeneous and nonhomogeneous Laplace and modified Helmholtz equations, are conducted. The results illustrate that the proposed approach may obtain highly accurate solutions with the use of higher order radial polynomial terms. Finally, compared with the radial basis function collocation method, the proposed approach may produce more accurate solutions than the other.

Suggested Citation

  • Cheng-Yu Ku & Jing-En Xiao & Chih-Yu Liu, 2020. "A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations," Mathematics, MDPI, vol. 8(2), pages 1-22, February.
  • Handle: RePEc:gam:jmathe:v:8:y:2020:i:2:p:270-:d:322036
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    References listed on IDEAS

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    1. Huaiqing Zhang & Yu Chen & Chunxian Guo & Zhihong Fu, 2014. "Application of Radial Basis Function Method for Solving Nonlinear Integral Equations," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-8, October.
    2. F. X. Sun & C. Liu & Y. M. Cheng, 2014. "An Improved Interpolating Element-Free Galerkin Method Based on Nonsingular Weight Functions," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-13, March.
    3. 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.
    4. Yang Cao & Jun-Liang Dong & Lin-Quan Yao, 2014. "A Modification of the Moving Least-Squares Approximation in the Element-Free Galerkin Method," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-13, March.
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