Approximations with Non-Symmetric Green's Kernels and their Application to Fractional Differential Equations

TL;DR

This paper introduces a non-symmetric Green's kernel approach for fractional differential equations, extending kernel methods beyond radial basis functions and establishing optimal convergence in Banach spaces.

Contribution

It develops a novel non-symmetric Green's kernel framework for fractional equations, with theoretical analysis outside traditional RKHS, and proves optimal convergence rates.

Findings

Kernel interpolants achieve optimal order convergence.

Extension beyond radial basis function methods.

Theoretical analysis in Banach spaces confirms effectiveness.

Abstract

Several kernel-based methods for the numerical solution of fractional differential equations have been developed in the recent past; however, these techniques exclusively relied on the use of radial basis function approximations. In the present work, we consider the non-symmetric Green's kernel perspective on fractional order spline interpolation and its application to a kernel Galerkin method for the numerical solution of certain fractional order differential equation. Unfortunately, the reliance on a non-symmetric kernel requires that our theoretical analysis of the kernel interpolants must take place outside the familiar setting of reproducing kernel Hilbert spaces. Nevertheless, we are able to prove that the proposed kernel interpolants obtain optimal order convergence rates in a reproducing kernel Banach space.