On the fractional Laplacian of some positive definite kernels with applications in numerically solving the surface quasi-geostrophic equation as a prominent fractional calculus model

Abstract

This paper provides the Riesz potential and fractional Laplacian (−Δ)s, s∈R of the famous radial kernels, including the Gaussian, multiquadric, Sobolev spline, and mainly focuses on Wendland kernels. We show that (−Δ)s maps these kernels into the kernels constructed by the generalized hypergeometric functions or the Meijer G–functions. An essential application of the results applies to developing recent meshless methods for solving equations involving fractional Laplacian. Compared to the recently proposed Gaussian meshless method, our numerical results show that working with other kernels to solve fractional equations can bring better results. As a real word application, we consider the numerical study of the surface quasi-geostrophic equations.