Multiquadrics without the Shape Parameter for Solving Partial Differential Equations
In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional colloca...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-11-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/11/1813 |
| _version_ | 1797549118890967040 |
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| author | Cheng-Yu Ku Chih-Yu Liu Jing-En Xiao Shih-Meng Hsu |
| author_facet | Cheng-Yu Ku Chih-Yu Liu Jing-En Xiao Shih-Meng Hsu |
| author_sort | Cheng-Yu Ku |
| collection | DOAJ |
| description | In this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy. |
| first_indexed | 2024-03-10T15:09:15Z |
| format | Article |
| id | doaj.art-9e6f6f0c2a144e9c90f0325c7a7bf827 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T15:09:15Z |
| publishDate | 2020-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-9e6f6f0c2a144e9c90f0325c7a7bf8272023-11-20T19:28:41ZengMDPI AGSymmetry2073-89942020-11-011211181310.3390/sym12111813Multiquadrics without the Shape Parameter for Solving Partial Differential EquationsCheng-Yu Ku0Chih-Yu Liu1Jing-En Xiao2Shih-Meng Hsu3School of Engineering, Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanSchool of Engineering, Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanSchool of Engineering, Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanSchool of Engineering, Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanIn this article, we present multiquadric radial basis functions (RBFs), including multiquadric (MQ) and inverse multiquadric (IMQ) functions, without the shape parameter for solving partial differential equations using the fictitious source collocation scheme. Different from the conventional collocation method that assigns the RBF at each center point coinciding with an interior point, we separated the center points from the interior points, in which the center points were regarded as the fictitious sources collocated outside the domain. The interior, boundary, and source points were therefore collocated within, on, and outside the domain, respectively. Since the radial distance between the interior point and the source point was always greater than zero, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Accordingly, the shape parameter was no longer required in the MQ and IMQ RBFs. Numerical examples with the domain in symmetry and asymmetry are presented to verify the accuracy and robustness of the proposed method. The results demonstrated that the proposed method using MQ RBFs without the shape parameter acquires more accurate results than the conventional RBF collocation method with the optimum shape parameter. Additionally, it was found that the locations of the fictitious sources were not sensitive to the accuracy.https://www.mdpi.com/2073-8994/12/11/1813shape parametermultiquadricradial basis functionfictitious source pointmeshless method |
| spellingShingle | Cheng-Yu Ku Chih-Yu Liu Jing-En Xiao Shih-Meng Hsu Multiquadrics without the Shape Parameter for Solving Partial Differential Equations Symmetry shape parameter multiquadric radial basis function fictitious source point meshless method |
| title | Multiquadrics without the Shape Parameter for Solving Partial Differential Equations |
| title_full | Multiquadrics without the Shape Parameter for Solving Partial Differential Equations |
| title_fullStr | Multiquadrics without the Shape Parameter for Solving Partial Differential Equations |
| title_full_unstemmed | Multiquadrics without the Shape Parameter for Solving Partial Differential Equations |
| title_short | Multiquadrics without the Shape Parameter for Solving Partial Differential Equations |
| title_sort | multiquadrics without the shape parameter for solving partial differential equations |
| topic | shape parameter multiquadric radial basis function fictitious source point meshless method |
| url | https://www.mdpi.com/2073-8994/12/11/1813 |
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