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

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.

Generalized Differentiability of Continuous Functions

“…Everywhere, will be considered as a small positive variable. The asymptotic small O notation will be used as introduced in [5]. Definition 1.…”

“…What was not recognized at that time was the inherent asymmetry of such definition, which lead to the misplaced expectation that the left and right velocities should be equal. Fractional velocities were used to characterize the growth of some singular functions [5]. Therefore, it is of interests to establish its relationship with the indicial derivative.…”

“…Such an opposition leads to the question whether the ordinary (i.e., Cauchy) derivatives can be generalized locally in a way that can describe such irregular objects. This is certainly achievable for a class of singular functions, such as the De Rham's function, which can be characterized locally in terms of fractional velocity [4,5]. Purely mathematically, the derivatives can be generalized in several ways.…”

Characterization of the Local Growth of Two Cantor-Type Functions

“…It was established that for fractional orders fractional velocity is continuous only if it is zero. The properties of fractional velocity are surveyed in [11].…”

Parametric Identification of Nonlinear Fractional Hammerstein Models

“…FO systems have found wide applications in physics and control [12][13][14]. It has been demonstrated that a singular function can represent the conceptual models of nonlinear physical phenomena accurately [15] and can be extended to processes related with natural phenomena, such as epidemiology [16]. Nevertheless, it is not simple to handle the fractional operator computationally.…”