On the fractional derivatives of radial basis functions: Theories and applications

The paper provides the fractional integrals and derivatives of the Riemann-Liouville and Caputo type for the five kinds of radial basis functions, including the Powers, Gaussian, Multiquadric, Matern, and Thin-plate splines, in one dimension. It allows to use high-order numerical methods for solving fractional differential equations. The results are tested by solving two test problems. The first test case focuses on the discretization of the fractional differential operator while the second considers the solution of a fractional order differential equation.