Volume 11, Issue 3
A Meshless Discrete Galerkin Method Based on the Free Shape Parameter Radial Basis Functions for Solving Hammerstein Integral Equations

Pouria Assari & Mehdi Dehghan

Numer. Math. Theor. Meth. Appl., 11 (2018), pp. 540-568.

Published online: 2018-11

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  • Abstract

In the current investigation, we present a numerical technique to solve Fredholm-Hammerstein integral equations of the second kind. The method utilizes the free shape parameter radial basis functions (RBFs) constructed on scattered points as a basis in the discrete Galerkin method to estimate the solution of integral equations. The accuracy and stability of the classical RBFs heavily depend on the selection of shape parameters. But on the other hand, the choice of suitable value for shape parameters is very difficult. Therefore, to get rid of this problem, the free shape parameter RBFs are used in the new method which establish an effective and stable method to estimate an unknown function. We utilize the composite Gauss-Legendre integration rule and employ it to estimate the integrals appeared in the method. Since the scheme does not need any background meshes, it can be identified as a meshless method. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.

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@Article{NMTMA-11-540, author = {}, title = {A Meshless Discrete Galerkin Method Based on the Free Shape Parameter Radial Basis Functions for Solving Hammerstein Integral Equations}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2018}, volume = {11}, number = {3}, pages = {540--568}, abstract = {

In the current investigation, we present a numerical technique to solve Fredholm-Hammerstein integral equations of the second kind. The method utilizes the free shape parameter radial basis functions (RBFs) constructed on scattered points as a basis in the discrete Galerkin method to estimate the solution of integral equations. The accuracy and stability of the classical RBFs heavily depend on the selection of shape parameters. But on the other hand, the choice of suitable value for shape parameters is very difficult. Therefore, to get rid of this problem, the free shape parameter RBFs are used in the new method which establish an effective and stable method to estimate an unknown function. We utilize the composite Gauss-Legendre integration rule and employ it to estimate the integrals appeared in the method. Since the scheme does not need any background meshes, it can be identified as a meshless method. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.2017-OA-0100}, url = {http://global-sci.org/intro/article_detail/nmtma/12444.html} }
TY - JOUR T1 - A Meshless Discrete Galerkin Method Based on the Free Shape Parameter Radial Basis Functions for Solving Hammerstein Integral Equations JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 540 EP - 568 PY - 2018 DA - 2018/11 SN - 11 DO - http://doi.org/10.4208/nmtma.2017-OA-0100 UR - https://global-sci.org/intro/article_detail/nmtma/12444.html KW - AB -

In the current investigation, we present a numerical technique to solve Fredholm-Hammerstein integral equations of the second kind. The method utilizes the free shape parameter radial basis functions (RBFs) constructed on scattered points as a basis in the discrete Galerkin method to estimate the solution of integral equations. The accuracy and stability of the classical RBFs heavily depend on the selection of shape parameters. But on the other hand, the choice of suitable value for shape parameters is very difficult. Therefore, to get rid of this problem, the free shape parameter RBFs are used in the new method which establish an effective and stable method to estimate an unknown function. We utilize the composite Gauss-Legendre integration rule and employ it to estimate the integrals appeared in the method. Since the scheme does not need any background meshes, it can be identified as a meshless method. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates.

Pouria Assari & Mehdi Dehghan. (2020). A Meshless Discrete Galerkin Method Based on the Free Shape Parameter Radial Basis Functions for Solving Hammerstein Integral Equations. Numerical Mathematics: Theory, Methods and Applications. 11 (3). 540-568. doi:10.4208/nmtma.2017-OA-0100
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