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On the Selection of a Good Shape Parameter of the Localized Method of Approximated Particular Solutions

On the Selection of a Good Shape Parameter of the Localized Method of Approximated Particular Solutions
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Year:    2018

Author:    Hui Zheng, Guangming Yao, Lei-Hsin Kuo, Xinxiang Li

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 4 : pp. 896–911

Abstract

In this paper, we propose a new approach for selecting suitable shape parameters of radial basis functions (RBFs) in the context of the localized method of approximated particular solutions. Traditionally, there are no direct connections on choosing good shape parameters and choosing interior and boundary nodes using the local collocation methods. As a result, the approximations of derivative functions are less accurate and the stability is also an issue. One of the focuses of this study is to select the interior and boundary nodes in a special way so that they are correlated. Furthermore, a test differential equation with known exact solution is selected and a good shape parameter for the given differential equation can be selected through a good shape parameter for the test differential equation. Three numerical examples, including a Poison's equation and an eigenvalue problem, are tested. Uniformly distributed node arrangement is compared with the proposed cross knot distribution with Dirichlet boundary conditions and mixed boundary conditions. The numerical results show some potentials for the proposed node arrangements and shape parameter selections.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2017-0167

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 4 : pp. 896–911

Published online:    2018-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    16

Keywords:    Method of approximated particular solutions shape parameter radial basis functions collocation methods Kansa's method.

Author Details

Hui Zheng

Guangming Yao

Lei-Hsin Kuo

Xinxiang Li

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