Approximation of fractional integrals by means of derivatives

TL;DR

This paper introduces a novel series decomposition of Riemann-Liouville fractional integrals using derivatives, enabling improved numerical approximations and applications in fractional calculus problems.

Contribution

It presents new formulas expressing fractional integrals as derivatives series, valid for $C^n$ functions, with error estimates for numerical methods.

Findings

New series formulas for fractional integrals using derivatives

Numerical approximation methods with error bounds

Applications demonstrated in fractional equations and calculus of variations

Abstract

We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to develop suitable numerical approximations with known estimations for the error. The usefulness of the obtained results, in solving fractional integral equations and fractional problems of the calculus of variations, is illustrated.