Gaussian radial basis functions collocation for fractional PDEs: methodology and error analysis
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914624584024064 |
|---|---|
| author | Tian, Xiaochuan Wu, Yixuan Zhang, Yanzhi |
| author_facet | Tian, Xiaochuan Wu, Yixuan Zhang, Yanzhi |
| contents | The paper introduces a new meshfree pseudospectral method based on Gaussian radial basis functions (RBFs) collocation to solve fractional Poisson equations. Hypergeometric functions are used to represent the fractional Laplacian of Gaussian RBFs, enabling an efficient computation of stiffness matrix entries. Unlike existing RBF-based methods, our approach ensures a Toeplitz structure in the stiffness matrix with equally spaced RBF centers, enabling efficient matrix-vector multiplications using fast Fourier transforms. We conduct a comprehensive study on the shape parameter selection, addressing challenges related to ill-conditioning and numerical stability. The main contribution of our work includes rigorous stability analysis and error estimates of the Gaussian RBF collocation method, representing a first attempt at the rigorous analysis of RBF-based methods for fractional PDEs to the best of our knowledge. We conduct numerical experiments to validate our analysis and provide practical insights for implementation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_17461 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gaussian radial basis functions collocation for fractional PDEs: methodology and error analysis Tian, Xiaochuan Wu, Yixuan Zhang, Yanzhi Numerical Analysis The paper introduces a new meshfree pseudospectral method based on Gaussian radial basis functions (RBFs) collocation to solve fractional Poisson equations. Hypergeometric functions are used to represent the fractional Laplacian of Gaussian RBFs, enabling an efficient computation of stiffness matrix entries. Unlike existing RBF-based methods, our approach ensures a Toeplitz structure in the stiffness matrix with equally spaced RBF centers, enabling efficient matrix-vector multiplications using fast Fourier transforms. We conduct a comprehensive study on the shape parameter selection, addressing challenges related to ill-conditioning and numerical stability. The main contribution of our work includes rigorous stability analysis and error estimates of the Gaussian RBF collocation method, representing a first attempt at the rigorous analysis of RBF-based methods for fractional PDEs to the best of our knowledge. We conduct numerical experiments to validate our analysis and provide practical insights for implementation. |
| title | Gaussian radial basis functions collocation for fractional PDEs: methodology and error analysis |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2312.17461 |