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.
Highlights
Non-linear and fractal physical phenomena are abundant in nature [1,2]
Examples are given by the De Rham and Neidinger’s singular functions, represented by limits of iterative function systems
Examples of non-linear phenomena can be given by the continuous time random walks resulting in fractional diffusion equations [3], fractional conservation of mass [4] or non-linear viscoelasticity [5,6]
Summary
Examples of non-linear phenomena can be given by the continuous time random walks resulting in fractional diffusion equations [3], fractional conservation of mass [4] or non-linear viscoelasticity [5,6] Such models exhibit global dependence through the action of the nonlinear convolution operator (i.e., differ-integral). Mathematical descriptions of strongly non-linear phenomena necessitate relaxation of the assumption of differentiability [17] While this can be achieved by fractional differ-integrals, or by multi-scale approaches [18], the present work focuses on local descriptions in terms of limits of difference quotients [19] and non-linear scale-space transformations [20]. The relationship between fractional velocities and the localized versions of fractional derivatives in the sense of Kolwankar-Gangal will be demonstrated
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