Local Fractional Fokker-Planck Equation

Abstract

(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

Fractional Newtonian mechanics

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Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order α

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Exact solutions for fractional DDEs via auxiliary equation method coupled with the fractional complex transform

“…Several local versions of fractional derivatives have been proposed for the investigation of local behavior of fractional models. For example, Cresson's derivative [38], Kolwankar-Gangal local fractional derivative [39], and Jumarie's modified RiemannLiouville derivative [40]. Indeed, Liouville-Caputo derivatives are defined only for differentiable functions, while f can be a continuous (but not necessarily differentiable) function.…”