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

A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order α, 0 < α ≤ 1, called F α -integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called F α -derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, "changing" only on a fractal set. The F α -derivative is local unlike the classical fractional derivative. The F α -calculus retains much of the simplicity of ordinary calculus. Several results including analogues of fundamental theorems of calculus are proved.The integral staircase function, which is a generalisation of the functions like the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, the γ-dimension. F α -differential equations are equations involving F α -derivatives. They can be used to model sublinear dynamical systems and fractal time processes, since sublinear behaviours are associated with staircase-like functions which occur naturally as their solutions. As examples, we discuss a fractal-time diffusion equation, and one dimensional motion of a particle undergoing friction in a fractal medium.

(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

Calculus on fractals, or Fα-calculus, developed in a previous paper, is a calculus based fractals F ⊂ R, and involves Fα-integral and Fα-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The Fα-integral is suitable for integrating functions with fractal support of dimension α, while the Fα-derivative enables us to differentiate functions like the Cantor staircase. Several results in Fα-calculus are analogous to corresponding results in ordinary calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of Fα-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour. In this paper we show that, as operators, the Fα-integral and Fα-derivative are conjugate to the Riemann integral and ordinary derivative respectively. This is accomplished by constructing a map ψ which takes Fα-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes Fα-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in ordinary calculus and Fα-calculus. This conjugacy is useful, among other things, to find solutions to Fα-differential equations: they can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.

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.