Fractional differentiability of nowhere differentiable functions and dimensions

Abstract

Weierstrass's everywhere continuous but nowhere differentiable function is shown to be locally continuously fractionally differentiable everywhere for all orders below the "critical order" 2-s and not so for orders between 2-s and 1, where s, 1<s<2 is the box dimension of the graph of the function. This observation is consolidated in the general result showing a direct connection between local fractional differentiability and the box dimension/local Holder exponent. Levy index for one dimensional Levy flights is shown to be the critical order of its characteristic function. Local fractional derivatives of multifractal signals (non-random functions) are shown to provide the local Holder exponent. It is argued that Local fractional derivatives provide a powerful tool to analyze pointwise behavior of irregular signals. (c) 1996 American Institute of Physics.

Local Fractional Derivatives of Differentiable Functions are Integer-order Derivatives or Zero

“…Recently, so-called local fractional derivative has been proposed in [12,13,14,15] by a motivation of describing the properties of fractal objects and processes. The definition of the local fractional derivative is obtained from the Riemann-Liouville fractional derivative by introducing two changes such as a subtraction of finite number of terms of the Taylor series and the taking the limit.…”

Fractional Newtonian mechanics

“…The fractional operators are non-local, therefore they are suitable for constructing models possessing memory effect. Using models involving local scaling properties the fractional derivatives were renormalized to construct local fractional differential operators [11,12]. The physical interpretation…”

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)). …”