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

TL;DR

The work introduces TPS-like radial basis kernels and their extensions via fractional derivatives to enhance interpolation and PDE-solving capabilities. By constructing explicit polynomial proxies and CPD radial functions, the authors enable robust radial interpolation with unisolvent centers, while employing Riemann–Liouville and Caputo fractional derivatives to generate flexible kernels. QR-based preconditioning is proposed to address ill-conditioning, and asymmetrical collocation is developed to apply these kernels to differential equations, including fractional operators, with demonstrated accuracy on planar domains. The approach offers a framework for high-accuracy, kernel-based interpolation and fractional PDE solving that leverages CPD theory, fractional calculus, and structured linear systems. The methodology has potential impact for multivariate interpolation and fractional-differential equation modeling in applied sciences.

Abstract

In this document we present the construction of a radial functions that have the objective of emulating the behavior of the radial basis function thin plate spline (TPS), which we will name as function TPS, we propose a way to partially and totally apply the fractional derivative to these functions to be used in interpolation problems, a proposal is presented to precondition the matrices generated in the interpolation problem using the $QR$ decomposition and finally is proposed the form of a radial interpolant to be used when solving differential equations using the asymmetric collocation method.

Figures (12)

• Figure 1: They are presented with black and red the functions $r^Nlog(r)$ and $r^N-r^{N+1}$ respectively.
• Figure 2: They are presented with black and red the functions $r^Nlog(r)$ and $-\frac{1}{2}r^{N+2}+2r^{N+1}-\frac{3}{2} r^N$ respectively.
• Figure 3: They are presented with black and red the functions $r^Nlog(r)$ and $-2r^{N+2}+4r^{N+1}-2r^N$ respectively.
• Figure 4: They are presented with black and red the functions $r^Nlog(r)$ and $-2r^{N-\alpha+2}+4r^{N+1}-2r^N$, using different values of $\alpha$, respectively.
• Figure 5: They are presented with black and red the functions $r^Nlog(r)$ and $3r^{N+3}-9r^{N-\alpha+2}+9 r^{N+1}-3r^N$ respectively.

Theorems & Definitions (2)

• Definition 1.1
• Example 1.3