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Volume 10, Issue 4
On the Selection of a Good Shape Parameter of the Localized Method of Approximated Particular Solutions

Hui Zheng, Guangming Yao, Lei-Hsin Kuo & Xinxiang Li

Adv. Appl. Math. Mech., 10 (2018), pp. 896-911.

Published online: 2018-07

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  • 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.

  • AMS Subject Headings

65M70, 35J05, 65N99

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COPYRIGHT: © Global Science Press

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@Article{AAMM-10-896, author = {Zheng , HuiYao , GuangmingKuo , Lei-Hsin and Li , Xinxiang}, title = {On the Selection of a Good Shape Parameter of the Localized Method of Approximated Particular Solutions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {4}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0167}, url = {http://global-sci.org/intro/article_detail/aamm/12501.html} }
TY - JOUR T1 - On the Selection of a Good Shape Parameter of the Localized Method of Approximated Particular Solutions AU - Zheng , Hui AU - Yao , Guangming AU - Kuo , Lei-Hsin AU - Li , Xinxiang JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 896 EP - 911 PY - 2018 DA - 2018/07 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2017-0167 UR - https://global-sci.org/intro/article_detail/aamm/12501.html KW - Method of approximated particular solutions, shape parameter, radial basis functions, collocation methods, Kansa's method. AB -

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.

Hui Zheng, Guangming Yao, Lei-Hsin Kuo & Xinxiang Li. (2020). On the Selection of a Good Shape Parameter of the Localized Method of Approximated Particular Solutions. Advances in Applied Mathematics and Mechanics. 10 (4). 896-911. doi:10.4208/aamm.OA-2017-0167
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