Fractional velocity as a tool for the study of non-linear problems

TL;DR

This paper explores fractional velocity as a novel mathematical tool to analyze nonlinear phenomena, especially singular and Holder functions, highlighting its unique properties and potential applications in modeling instantaneous interactions.

Contribution

It demonstrates the applicability of fractional velocity to characterize singular functions and investigates its relation to local fractional derivatives, offering new insights into nonlinear analysis.

Findings

Fractional velocities generalize local derivatives with unique disconnected value sets.

They can model instantaneous interactions like Langevin dynamics.

Conditions for equivalence with local fractional derivatives are established.

Abstract

Singular functions and, in general, H\"older functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocity as a tool to characterize Holder and in particular singular functions. Fractional velocities are defined as limit of the difference quotient of a fractional power and they generalize the local notion of a derivative. On the other hand, their properties contrast some of the usual properties of derivatives. One of the most peculiar properties of these operators is that the set of their non trivial values is disconnected. This can be used for example to model instantaneous interactions, for example Langevin dynamics. Examples are given by the De Rham and Neidinger's functions, represented by iterative function systems. Finally the conditions for equivalence with the Kolwankar-Gangal local fractional derivative are investigated.