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A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation

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  • Lin-Tian Luh

    (Department of Data Science and Big Data Analytics, Providence University, Shalu, Taichung 43310, Taiwan)

Abstract

In this paper, we totally discard the traditional trial-and-error algorithms of choosing the acceptable shape parameter c in the multiquadrics − c 2 + ∥ x ∥ 2 when dealing with differential equations, for example, the Poisson equation, with the RBF collocation method. 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 demonstrate 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 .

Suggested Citation

  • Lin-Tian Luh, 2022. "A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation," Mathematics, MDPI, vol. 10(19), pages 1-18, October.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:19:p:3583-:d:931105
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    Cited by:

    1. Jian Sun & Ling Wang & Dianxuan Gong, 2023. "Model for Choosing the Shape Parameter in the Multiquadratic Radial Basis Function Interpolation of an Arbitrary Sine Wave and Its Application," Mathematics, MDPI, vol. 11(8), pages 1-20, April.

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