Collocation methods for integral fractional Laplacian and fractional PDEs based on radial basis functions

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. •Develop efficient methods for fractional Laplacian of radial basis functions with smooth Fourier transformations, e.g. Matern kernel.•Devise a collocation formulation for fractional elliptic problems on complex domains.•The formulation exhibits reduced condition numbers and allows the use of a large number of collocation points.•Various numerical results and comparison among different collocation formulations and Gaussian and Matern radial functions.