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Volume 27, Issue 5
Improved RBF Collocation Methods for Fourth Order Boundary Value Problems

C. S. Chen, Andreas Karageorghis & Hui Zheng

DOI: 10.4208/cicp.OA-2019-0163

Commun. Comput. Phys., 27 (2020), pp. 1530-1549.

Published online: 2020-03

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  • Abstract

Radial basis function (RBF) collocation methods (RBFCMs) are applied to fourth order boundary value problems (BVPs). In particular, we consider the classical Kansa method and the method of approximate particular solutions (MAPS). In the proposed approach we include some so-called ghost points which are located inside and outside the domain of the problem. The inclusion of these points is shown to improve the accuracy and the stability of the collocation methods. An appropriate value of the shape parameter in the RBFs used is obtained using either the leave-one-out cross validation (LOOCV) algorithm or Franke's formula. We present and analyze the results of several numerical tests.

  • Keywords

Radial basis functions, Kansa method, method of particular solutions, collocation, fourth order PDEs.

  • AMS Subject Headings

65N35, 65N99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

c.s. chen@usm.edu (C. S. Chen)

andreask@ucy.ac.cy (Andreas Karageorghis)

zhenghui@ncu.edu.cn (Hui Zheng)

  • BibTex
  • RIS
  • TXT
@Article{CiCP-27-1530, author = {S. Chen , C.Karageorghis , Andreas and Zheng , Hui}, title = {Improved RBF Collocation Methods for Fourth Order Boundary Value Problems}, journal = {Communications in Computational Physics}, year = {2020}, volume = {27}, number = {5}, pages = {1530--1549}, abstract = {

Radial basis function (RBF) collocation methods (RBFCMs) are applied to fourth order boundary value problems (BVPs). In particular, we consider the classical Kansa method and the method of approximate particular solutions (MAPS). In the proposed approach we include some so-called ghost points which are located inside and outside the domain of the problem. The inclusion of these points is shown to improve the accuracy and the stability of the collocation methods. An appropriate value of the shape parameter in the RBFs used is obtained using either the leave-one-out cross validation (LOOCV) algorithm or Franke's formula. We present and analyze the results of several numerical tests.

}, issn = {1991-7120}, doi = {https://doi.org/ 10.4208/cicp.OA-2019-0163}, url = {http://global-sci.org/intro/article_detail/cicp/15768.html} }
TY - JOUR T1 - Improved RBF Collocation Methods for Fourth Order Boundary Value Problems AU - S. Chen , C. AU - Karageorghis , Andreas AU - Zheng , Hui JO - Communications in Computational Physics VL - 5 SP - 1530 EP - 1549 PY - 2020 DA - 2020/03 SN - 27 DO - http://doi.org/ 10.4208/cicp.OA-2019-0163 UR - https://global-sci.org/intro/article_detail/cicp/15768.html KW - Radial basis functions, Kansa method, method of particular solutions, collocation, fourth order PDEs. AB -

Radial basis function (RBF) collocation methods (RBFCMs) are applied to fourth order boundary value problems (BVPs). In particular, we consider the classical Kansa method and the method of approximate particular solutions (MAPS). In the proposed approach we include some so-called ghost points which are located inside and outside the domain of the problem. The inclusion of these points is shown to improve the accuracy and the stability of the collocation methods. An appropriate value of the shape parameter in the RBFs used is obtained using either the leave-one-out cross validation (LOOCV) algorithm or Franke's formula. We present and analyze the results of several numerical tests.

S. Chen , C.Karageorghis , Andreas and Zheng , Hui. (2020). Improved RBF Collocation Methods for Fourth Order Boundary Value Problems. Communications in Computational Physics. 27 (5). 1530-1549. doi: 10.4208/cicp.OA-2019-0163
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