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A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation
Version 1
: Received: 15 August 2022 / Approved: 16 August 2022 / Online: 16 August 2022 (11:04:30 CEST)
A peer-reviewed article of this Preprint also exists.
Luh, L.-T. A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation. Mathematics 2022, 10, 3583. Luh, L.-T. A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation. Mathematics 2022, 10, 3583.
Abstract
In this paper we totally discard the traditional trial-and-error algorithms of choosing acceptable shape parameter c in the multiquadrics $-\sqrt{c^{2}+\|x\|^{2}}$ when dealing with differential equations. Instead, we choose c directly by the MN-curve theory and hence avoid the time-consuming steps of solving a linear system required by each trial of the c value in the traditional methods. The quality of the c value thus obtained is supported by the newly born choice theory of the shape parameter. Experiments show that the approximation error of the approximate solution to the differential equation is very close to the best approximation error among all possible choices of c.
Keywords
radial basis function; multiquadric; shape parameter; collocation; Poisson equation
Subject
Computer Science and Mathematics, Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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