Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation

TL;DR

This paper offers a novel geometric and physical interpretation of fractional calculus operators, including Riemann-Liouville, Caputo, Riesz, and Feller potentials, and extends these ideas to Volterra convolution integrals and Stieltjes integrals.

Contribution

It provides the first comprehensive geometric and physical interpretation of fractional integration and differentiation, unifying various operators under a new interpretative framework.

Findings

New geometric and physical interpretations for fractional operators

Extension of interpretations to Volterra convolution integrals

A novel physical perspective on Stieltjes integrals

Abstract

A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral.