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Open AccessArticle

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

1
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
2
Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(2), 270; https://doi.org/10.3390/math8020270
Received: 27 January 2020 / Revised: 12 February 2020 / Accepted: 14 February 2020 / Published: 18 February 2020
(This article belongs to the Special Issue Numerical Modeling and Analysis)
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. View Full-Text
Keywords: radial polynomial; radial basis function; collocation method; meshfree; nonhomogeneous radial polynomial; radial basis function; collocation method; meshfree; nonhomogeneous
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MDPI and ACS Style

Ku, C.-Y.; Xiao, J.-E.; Liu, C.-Y. A Novel Meshfree Approach with a Radial Polynomial for Solving Nonhomogeneous Partial Differential Equations. Mathematics 2020, 8, 270.

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