A Gauss-Jacobi Kernel Compression Scheme for Fractional Differential Equations

TL;DR

This paper introduces a Gauss-Jacobi quadrature-based kernel compression scheme for fractional differential equations, enabling efficient and accurate approximation of fractional kernels with bounded terms across parameters.

Contribution

The paper presents a novel kernel approximation method using composite Gauss-Jacobi quadrature, with proven rapid convergence and bounded terms for all relevant parameters.

Findings

The scheme achieves rapid convergence in the number of quadrature nodes.

The number of terms needed is bounded for all fractional orders in (0,1).

The approximation error can be effectively estimated and controlled.

Abstract

A scheme for approximating the kernel $w$ of the fractional $\alpha$-integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss-Jacobi quadrature rule to an integral representation of $w$. This results in an approximation of $w$ in an interval $[\delta,T]$, with $0<\delta$, which converges rapidly in the number $J$ of quadrature nodes associated with each interval of the composite rule. Using error analysis for Gauss-Jacobi quadratures for analytic functions, an estimate of the relative pointwise error is obtained. The estimate shows that the number of terms required for the approximation to satisfy a prescribed error tolerance is bounded for all $\alpha\in(0,1)$, and that $J$ is bounded for $\alpha\in(0,1)$, $T>0$, and $\delta\in(0,T)$.