On some geometric interpretations of fractional-order operators

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.

3 pages, 4 figures. This work has been submitted as discussion paper 074 and presented as talk WeB02.2 (Wednesday, July 9, 2024, 16:40-17:00) at \textit {ICFDA'2024 -- 12th IFAC Conference on Fractional Differentiation and its Applications, Bordeaux, France, July 9--12, 2024.}