Proposal for Use of the Fractional Derivative of Radial Functions in Interpolation Problems

This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions for use in interpolation problems. Furthermore, a technique is employed to precondition the matrices generated in the presented problems through QR decomposition. Similarly, a method is introduced to define two different types of abelian groups for any fractional operator defined in the interval [0,1), among which the Riemann-Liouville fractional integral, Riemann-Liouville fractional derivative, and Caputo fractional derivative are worth mentioning. Finally, a form of radial interpolant is suggested for application in solving fractional differential equations using the asymmetric collocation method, and examples of its implementation in differential operators utilizing the aforementioned fractional operators are shown.