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

(April 2 1997)(cond-mat/9801138)New kind of differential equations, called local fractional differential equations, has been proposed for the first time. They involve local fractional derivatives introduced recently. Such equations appear to be suitable to deal with phenomena taking place in fractal space and time. A local fractional analog of Fokker-Planck equation has been derived starting from the Chapman-Kolmogorov condition. Such an equation is solved, with a specific choice of the transition probability, and shown to give rise to subdiffusive behavior. PACS number(s): 02.50. Ga, 47.53.+n, 05.40.+j, 05.60.+w Derivatives and integrals of fractional order have found many applications in recent studies of scaling phenomena [1][2][3][4]. The main aim of the most of these papers is to formulate fractional integro-differential equations to describe some scaling process. Modifications of equations governing physical processes such as diffusion equation, wave equation and Fokker-Planck equation have been suggested [5][6][7][8][9][10] which incorporate fractional derivatives with respect to time. Recently Zaslavasky [11] argued that the chaotic Hamiltonian dynamics of particles can be described by using fractional generalization of the Fokker-Planck-Kolmogorov (FPK) equation. However fractional derivatives are nonlocal and hence such equations are not suitable to study local scaling behavior. In the present work we rigorously derive fractional analogs of equations like the FPK equation involving one space variable. Our approach differs from the above mentioned ones since we use local fractional Taylor expansion, which was established only recently [12]. As is argued below, such equations can provide appropriate schemes for describing evolutions (e.g. sub or superdiffusive) normally not obtained from the usual FPK equation.It was realized recently [12] that there is a direct quantitative connection between fractional differentiability properties of continuous but nowhere differentiable functions and the dimensions of their graphs. In order to show this, a new notion of local fractional derivative (LFD) was introduced. The LFD of order q of a function f (y) was defined by

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.

We define new functional spaces designed to measure the fine local regularity of functions. In contrast with classical approaches based on, e. g., Littlewood-Paley or wavelet analysis, these spaces are characterized by conditions expressed in the time domain. This is in some cases simpler and more convenient. In particular, because no pre-processing of the data is necessary, it is possible to obtain robust numerical estimation procedures in the case of sampled signals.

flights over a fractal nest and first-passage statistics on irregular surfaces.