Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# 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)

How to cite: Luh, L.D. A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation. Preprints 2022, 2022080288 (doi: 10.20944/preprints202208.0288.v1). Luh, L.D. A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation. Preprints 2022, 2022080288 (doi: 10.20944/preprints202208.0288.v1).

## 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.

## Subject

MATHEMATICS & COMPUTER SCIENCE, Numerical Analysis & Optimization

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