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Open AccessFeature PaperArticle

A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations

by Cheng-Yu Ku 1,2 and Jing-En Xiao 2,*
1
Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
2
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(9), 1419; https://doi.org/10.3390/sym12091419
Received: 31 July 2020 / Revised: 21 August 2020 / Accepted: 23 August 2020 / Published: 26 August 2020
(This article belongs to the Special Issue Polynomials: Special Polynomials and Number-Theoretical Applications)
In this article, a collocation method using radial polynomials (RPs) based on the multiquadric (MQ) radial basis function (RBF) for solving partial differential equations (PDEs) is proposed. The new global RPs include only even order radial terms formulated from the binomial series using the Taylor series expansion of the MQ RBF. Similar to the MQ RBF, the RPs is infinitely smooth and differentiable. The proposed RPs may be regarded as the equivalent expression of the MQ RBF in series form in which no any extra shape parameter is required. Accordingly, the challenging task for finding the optimal shape parameter in the Kansa method is avoided. Several numerical implementations, including problems in two and three dimensions, are conducted to demonstrate the accuracy and robustness of the proposed method. The results depict that the method may find solutions with high accuracy, while the radial polynomial terms is greater than 6. Finally, our method may obtain more accurate results than the Kansa method. View Full-Text
Keywords: multiquadric; radial basis function; radial polynomials; the shape parameter; meshless; Kansa method multiquadric; radial basis function; radial polynomials; the shape parameter; meshless; Kansa method
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Ku, C.-Y.; Xiao, J.-E. A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations. Symmetry 2020, 12, 1419.

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