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Fractal Fract 2018, 2(1), 4; https://doi.org/10.3390/fractalfract2010004

Fractional Velocity as a Tool for the Study of Non-Linear Problems

Environment, Health and Safety, IMEC vzw, Kapeldreef 75, 3001 Leuven, Belgium
Received: 27 December 2017 / Revised: 14 January 2018 / Accepted: 15 January 2018 / Published: 17 January 2018
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)
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Abstract

Singular functions and, in general, Hölder functions represent conceptual models of nonlinear physical phenomena. The purpose of this survey is to demonstrate the applicability of fractional velocities as tools to characterize Hölder and singular functions, in particular. Fractional velocities are defined as limits of the difference quotients 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 singular functions, represented by limits of iterative function systems. Finally, the conditions for equivalence with the Kolwankar-Gangal local fractional derivative are investigated. View Full-Text
Keywords: fractional calculus; non-differentiable functions; Hölder classes; pseudo-differential operators fractional calculus; non-differentiable functions; Hölder classes; pseudo-differential operators
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

Supplementary material

  • Externally hosted supplementary file 1
    Doi: 10.5281/zenodo.570926
    Link: https://www.zenodo.org/record/570926
    Description: Prodanov, Dimiter. (2017). Plots of De Rham's function and its fractional velocity [Data set]. Zenodo. http://doi.org/10.5281/zenodo.570926
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Prodanov, D. Fractional Velocity as a Tool for the Study of Non-Linear Problems. Fractal Fract 2018, 2, 4.

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