Hölder exponents of irregular signals and local fractional derivatives

Abstract

It has been recognized recently that fractional calculus is useful for handling scaling structures and processes. We begin this survey by pointing out the relevance of the subject to physical situations. Then the essential definitions and formulae from fractional calculus are summarized and their immediate use in the study of scaling in physical systems is given. This is followed by a brief summary of classical results. The main theme of the review rests on the notion of local fractional derivatives. There is a direct connection between local fractional differentiability properties and the dimensions/ local Hölder exponents of nowhere differentiable functions. It is argued that local fractional derivatives provide a powerful tool to analyse the pointwise behaviour of irregular signals and functions.

On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

“…It is then sufficient to expand a theory of approximation of functions by sequences of Mittag-Leffler functions to get the result. Let us point out that the first two terms of this fractional Taylor's series, that is to say the corresponding Rolle's fractional theorem, has been already obtained by Kolwankar and Gangal [17,18] who work with Cantor's sets.…”

“…We used it recently to outline an elementary theory of differential geometry of fractional order, and our purpose herein is to contribute some new results in this approach. For other points of view on fractional calculus, see for instance [7,8,[17][18][19]. After a short background on the definition of the modified Riemann-Liouville derivative and the related fractional Taylor's series, we shall successively display some formulae involving fractional derivative of compounded functions, and some formulae involving integrals with respect to ( ) α , all prerequisite which we shall need for our purpose.…”

“…which can be considered and has been considered [17,18] as a point of departure for fractional calculus.…”

Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order α

“…(vi) More recently Kolwankar and Gangal [35,36] proved the so-called "local fractional Taylor expansion"…”

Approximate Solution to the Fractional Second-Type Lane-Emden Equation

“…There are many kinds of fractional calculus. Such as Riemann-Liouville, Caputo, Kolwankar-Gangal, Oldham and Spanier, Miller and Ross, Cresson's, Grunwald-Letnikov, and modified Riemann-Liouville (Mandelbrot (1982), Kolwankar (1996) and Kolwankar (1997)). …”