Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs
In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method...
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| Format: | Article |
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MDPI AG
2021-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/13/1535 |
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| author | Chih-Yu Liu Cheng-Yu Ku Li-Dan Hong Shih-Meng Hsu |
| author_facet | Chih-Yu Liu Cheng-Yu Ku Li-Dan Hong Shih-Meng Hsu |
| author_sort | Chih-Yu Liu |
| collection | DOAJ |
| description | In this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required. |
| first_indexed | 2024-03-10T09:55:16Z |
| format | Article |
| id | doaj.art-eafec8ebadc449809bc04e700af69500 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T09:55:16Z |
| publishDate | 2021-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-eafec8ebadc449809bc04e700af695002023-11-22T02:28:21ZengMDPI AGMathematics2227-73902021-06-01913153510.3390/math9131535Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEsChih-Yu Liu0Cheng-Yu Ku1Li-Dan Hong2Shih-Meng Hsu3Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanCenter of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanDepartment of Harbor and River Engineering, School of Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanDepartment of Harbor and River Engineering, School of Engineering, National Taiwan Ocean University, Keelung 20224, TaiwanIn this article, a novel infinitely smooth polyharmonic radial basis function (PRBF) collocation method for solving elliptic partial differential equations (PDEs) is presented. The PRBF with natural logarithm is a piecewise smooth function in the conventional radial basis function collocation method for solving governing equations. We converted the piecewise smooth PRBF into an infinitely smooth PRBF using source points collocated outside the domain to ensure that the radial distance was always greater than zero to avoid the singularity of the conventional PRBF. Accordingly, the PRBF and its derivatives in the governing PDEs were always continuous. The seismic wave propagation problem, groundwater flow problem, unsaturated flow problem, and groundwater contamination problem were investigated to reveal the robustness of the proposed PRBF. Comparisons of the conventional PRBF with the proposed method were carried out as well. The results illustrate that the proposed approach could provide more accurate solutions for solving PDEs than the conventional PRBF, even with the optimal order. Furthermore, we also demonstrated that techniques designed to deal with the singularity in the original piecewise smooth PRBF are no longer required.https://www.mdpi.com/2227-7390/9/13/1535partial differential equationspolyharmonicradial basis functionsource pointcollocation method |
| spellingShingle | Chih-Yu Liu Cheng-Yu Ku Li-Dan Hong Shih-Meng Hsu Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs Mathematics partial differential equations polyharmonic radial basis function source point collocation method |
| title | Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs |
| title_full | Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs |
| title_fullStr | Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs |
| title_full_unstemmed | Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs |
| title_short | Infinitely Smooth Polyharmonic RBF Collocation Method for Numerical Solution of Elliptic PDEs |
| title_sort | infinitely smooth polyharmonic rbf collocation method for numerical solution of elliptic pdes |
| topic | partial differential equations polyharmonic radial basis function source point collocation method |
| url | https://www.mdpi.com/2227-7390/9/13/1535 |
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