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Volume 14, Issue 5
Solving PDEs with a Hybrid Radial Basis Function: Power-Generalized Multiquadric Kernel

Cem Berk Senel, Jeroen van Beeck & Atakan Altinkaynak

Adv. Appl. Math. Mech., 14 (2022), pp. 1161-1180.

Published online: 2022-06

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  • Abstract

Radial Basis Function (RBF) kernels are key functional forms for advanced solutions of higher-order partial differential equations (PDEs). In the present study, a hybrid kernel was developed for meshless solutions of PDEs widely seen in several engineering problems. This kernel, Power-Generalized Multiquadric — Power-GMQ, was built up by vanishing the dependence of $\epsilon$, which is significant since its selection induces severe problems regarding numerical instabilities and convergence issues. Another drawback of $\epsilon$-dependency is that the optimal $\epsilon$ value does not exist in perpetuity. We present the Power-GMQ kernel which combines the advantages of Radial Power and Generalized Multiquadric RBFs in a generic formulation. Power-GMQ RBF was tested in higher-order PDEs with particular boundary conditions and different domains. RBF-Finite Difference (RBF-FD) discretization was also implemented to investigate the characteristics of the proposed RBF. Numerical results revealed that our proposed kernel makes similar or better estimations as against to the Gaussian and Multiquadric kernels with a mild increase in computational cost. Gauss-QR method may achieve better accuracy in some cases with considerably higher computational cost. By using Power-GMQ RBF, the dependency of solution on $\epsilon$ was also substantially relaxed and consistent error behavior were obtained regardless of the selected $\epsilon$ accompanied.

  • Keywords

Meshfree collocation methods, Radial Basis Function (RBF), partial differential equations (PDEs).

  • AMS Subject Headings

65D12, 65N35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-14-1161, author = {Cem Berk and Senel and and 23525 and and Cem Berk Senel and Jeroen van and Beeck and and 23526 and and Jeroen van Beeck and Atakan and Altinkaynak and and 23527 and and Atakan Altinkaynak}, title = {Solving PDEs with a Hybrid Radial Basis Function: Power-Generalized Multiquadric Kernel}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2022}, volume = {14}, number = {5}, pages = {1161--1180}, abstract = {

Radial Basis Function (RBF) kernels are key functional forms for advanced solutions of higher-order partial differential equations (PDEs). In the present study, a hybrid kernel was developed for meshless solutions of PDEs widely seen in several engineering problems. This kernel, Power-Generalized Multiquadric — Power-GMQ, was built up by vanishing the dependence of $\epsilon$, which is significant since its selection induces severe problems regarding numerical instabilities and convergence issues. Another drawback of $\epsilon$-dependency is that the optimal $\epsilon$ value does not exist in perpetuity. We present the Power-GMQ kernel which combines the advantages of Radial Power and Generalized Multiquadric RBFs in a generic formulation. Power-GMQ RBF was tested in higher-order PDEs with particular boundary conditions and different domains. RBF-Finite Difference (RBF-FD) discretization was also implemented to investigate the characteristics of the proposed RBF. Numerical results revealed that our proposed kernel makes similar or better estimations as against to the Gaussian and Multiquadric kernels with a mild increase in computational cost. Gauss-QR method may achieve better accuracy in some cases with considerably higher computational cost. By using Power-GMQ RBF, the dependency of solution on $\epsilon$ was also substantially relaxed and consistent error behavior were obtained regardless of the selected $\epsilon$ accompanied.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2021-0215}, url = {http://global-sci.org/intro/article_detail/aamm/20556.html} }
TY - JOUR T1 - Solving PDEs with a Hybrid Radial Basis Function: Power-Generalized Multiquadric Kernel AU - Senel , Cem Berk AU - Beeck , Jeroen van AU - Altinkaynak , Atakan JO - Advances in Applied Mathematics and Mechanics VL - 5 SP - 1161 EP - 1180 PY - 2022 DA - 2022/06 SN - 14 DO - http://doi.org/10.4208/aamm.OA-2021-0215 UR - https://global-sci.org/intro/article_detail/aamm/20556.html KW - Meshfree collocation methods, Radial Basis Function (RBF), partial differential equations (PDEs). AB -

Radial Basis Function (RBF) kernels are key functional forms for advanced solutions of higher-order partial differential equations (PDEs). In the present study, a hybrid kernel was developed for meshless solutions of PDEs widely seen in several engineering problems. This kernel, Power-Generalized Multiquadric — Power-GMQ, was built up by vanishing the dependence of $\epsilon$, which is significant since its selection induces severe problems regarding numerical instabilities and convergence issues. Another drawback of $\epsilon$-dependency is that the optimal $\epsilon$ value does not exist in perpetuity. We present the Power-GMQ kernel which combines the advantages of Radial Power and Generalized Multiquadric RBFs in a generic formulation. Power-GMQ RBF was tested in higher-order PDEs with particular boundary conditions and different domains. RBF-Finite Difference (RBF-FD) discretization was also implemented to investigate the characteristics of the proposed RBF. Numerical results revealed that our proposed kernel makes similar or better estimations as against to the Gaussian and Multiquadric kernels with a mild increase in computational cost. Gauss-QR method may achieve better accuracy in some cases with considerably higher computational cost. By using Power-GMQ RBF, the dependency of solution on $\epsilon$ was also substantially relaxed and consistent error behavior were obtained regardless of the selected $\epsilon$ accompanied.

Cem Berk Senel, Jeroen van Beeck & Atakan Altinkaynak. (2022). Solving PDEs with a Hybrid Radial Basis Function: Power-Generalized Multiquadric Kernel. Advances in Applied Mathematics and Mechanics. 14 (5). 1161-1180. doi:10.4208/aamm.OA-2021-0215
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