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

[摘要] 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.

[学科分类] 数值分析

[关键词] fractional calculus;non-differentiable functions;Hölder classes;pseudo-differential operators