On some geometric interpretations of fractional-order operators

TL;DR

This paper explores geometric interpretations of fractional-order operators, highlighting Leibniz's early questions and introducing new geometric insights into Stieltjes and fractal derivatives.

Contribution

It provides the first geometric interpretation of Stieltjes derivatives, including fractal derivatives, expanding understanding of fractional calculus.

Findings

Geometric interpretation of Riemann--Liouville fractional integral

Geometric interpretation of Stieltjes integral

Introduction of geometric interpretation for Stieltjes and fractal derivatives

Abstract

This discussion paper presents some parts of the work in progress. It is shown that G.W. Leibniz was the first who raised the question about geometric interpretation of fractional-order operators. Geometric interpretations of the Riemann--Liouville fractional integral and the Stieltjes integral are explained. Then, for the first time, a geometric interpretation of the Stieltjes derivatives is introduced, which holds also for so-called ``fractal derivatives'', which are a particular case of Stieltjes derivatives.