From fractional order equations to integer order equations (PREPRINT)

TL;DR

This paper introduces a novel method to convert certain fractional order integral equations into ordinary integral equations using fractional integral operators, simplifying their solutions and ensuring minimal order in the resulting equations.

Contribution

The paper presents a new, general technique to transform linear fractional integral equations with rational orders into simpler ordinary integral equations, optimizing their order.

Findings

Method successfully converts FOIE to OIE for rational orders.

Construction is applicable to a broad class of fractional integral equations.

Resulting OIE has the minimal possible order.

Abstract

The main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator. After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of "additive" operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is "optimal" in the sense that the OIE that we obtain has the least possible order.