Collocation methods for integral fractional Laplacian and fractional PDEs based on radial basis functions
Collocation methods for integral fractional Laplacian and fractional PDEs based on radial basis functions
复制标题
DOI：
10.1016/j.amc.2024.128548
复制
发表时间：
2024-05
期刊：
影响因子：
--
通讯作者：
Zhuang Qiao;A. Heryudono;Fanhai Zeng;Zhongqiang Zhang
中科院分区：
文献类型：
--
作者：
Zhuang Qiao;A. Heryudono;Fanhai Zeng;Zhongqiang Zhang
我们考虑使用径向基函数（RBF）在一般有界域上使用积分分数拉普拉斯算子的分数椭圆方程的配置方法。利用 Hankel 变换，我们开发了 RBF 积分分数拉普拉斯算子的高效数值技术。此外，我们针对实际应用设计了一种搭配公式，与现有公式相比，它有助于使用相对大量的搭配点，同时保持较小的条件数。除了我们关注 Matern RBF 之外，所提出的方法还适用于具有平滑傅立叶变换的一类广泛的正定 RBF。我们证明了我们的方法在解决光滑和非光滑平面域中的几个问题方面的有效性。
We consider collocation methods for fractional elliptic equations with the integral fractional Laplacian on general bounded domains using radial basis functions (RBFs). Leveraging the Hankel transform, we develop highly efficient numerical techniques for the integral fractional Laplacian of RBFs. Furthermore, we devise a collocation formulation toward practical applications that facilitates the use of a relatively large number of collocation points while maintaining smaller condition numbers compared to existing formulations. In addition to our focus on Matern RBFs, the proposed method is applicable to a broad class of positive definite RBFs with smooth Fourier transformations. We demonstrate the effectiveness of our method in solving several problems in both smooth and non-smooth planar domains.