Anil D. Gangal

LOCAL FRACTIONAL FOKKER-PLANCK EQUATION

We propose a new class of differential equations, which we call local fractional differential equations. They involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time. A local fractional analog of the Fokker-Planck equation has been derived starting from the Chapman-Kolmogorov condition. We solve the equation with a specific choice of the transition probability and show how subdiffusive behavior can arise.

Hölder exponents of irregular signals and local fractional derivatives

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 analyze the pointwize behaviour of irregular signals and functions.

Local Fractional Calculus: a Calculus for Fractal Space-Time

Recently, new notions such as local fractional derivatives and local fractional differential equations were introduced. Here we argue that these developments provide a possible calculus to deal with phenomena in fractal space-time. We show how the usual calculus is generalized to deal with non Lipschitz functions. We also indicate how a definition of a fractal measure arises from these developments much the same way as the Lebesgue measure from ordinary calculus.

Fractal differential equations and fractal-time dynamical systems

AbstractDifferential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate framework. A new calculus calledFα-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 < α ≤ 1. It involves integral and derivative of order α, calledFα-integral andFα-derivative respectively. TheFα-integral is suitable for integrating functions with fractal support of dimension α, while theFα-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions ofFα-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems.We discuss construction and solutions of some fractal differential equations of the form

CALCULUS ON FRACTAL SUBSETS OF REAL LINE — II: CONJUGACY WITH ORDINARY CALCULUS

D_{F,t}^\alpha x = h(x,t),

Effect of shape anisotropy on transport in a two-dimensional computational model: numerical simulations showing experimental features observed in biomembranes

whereh is a vector field andDF,tα is a fractal differential operator of order α in timet. We also consider some equations of the form

Fractional differentiability of nowhere differentiable functions and dimensions

D_{F,t}^\alpha W(x,t) = L[W(x,t)],