On the Fractional Derivatives of Radial Basis Functions
Provides mathematical tools for applying RBF-based high-order methods to fractional differential equations, benefiting researchers in numerical analysis and fractional calculus.
The paper derives fractional integrals and derivatives for five types of radial basis functions (RBFs) and demonstrates their use in solving fractional differential equations via RBF collocation and method of lines, achieving accurate numerical solutions.
The paper provides the fractional integrals and derivatives of the Rie\-mann-Liouville and Caputo type for the five kinds of radial basis functions (RBFs), 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 fractional differential equations. The first one is a fractional ODE which is solved by the RBF collocation method and the second one is a fractional PDE which is solved by the method of lines based on the spatial trial spaces spanned by the Lagrange basis associated to the RBFs.