On a discrete composition of the fractional integral and Caputo derivative

TL;DR

This paper establishes a discrete analogue for the composition of the fractional integral and Caputo derivative, aiding numerical analysis of fractional PDEs with the L1 scheme by using asymptotic sum evaluation techniques.

Contribution

It introduces a discrete analogue for the composition of fractional integral and Caputo derivative, advancing numerical methods for fractional PDEs.

Findings

Provides a discrete analogue relevant for numerical schemes

Uses Euler-Maclaurin formula for asymptotic evaluation

Facilitates improved discretization of fractional derivatives

Abstract

We prove a discrete analogue for the composition of the fractional integral and Caputo derivative. This result is relevant in numerical analysis of fractional PDEs when one discretizes the Caputo derivative with the so-called L1 scheme. The proof is based on asymptotic evaluation of the discrete sums with the use of the Euler-Maclaurin summation formula.