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

## TL;DR

This paper introduces a new class of radial functions based on fractional derivatives of radial basis functions, aiming to improve interpolation and differential equation solving methods.

## Contribution

It proposes a novel approach to apply fractional derivatives to radial basis functions and introduces a preconditioning technique for interpolation matrices.

## Key findings

- New radial functions emulating thin plate splines
- Fractional derivatives applied to improve interpolation
- Preconditioning matrices with QR decomposition enhances stability

## 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.