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Applications Of Fractional Calculus In Physics
02 Mar 2000-
TL;DR: An introduction to fractional calculus can be found in this paper, where Butzer et al. present a discussion of fractional fractional derivatives, derivatives and fractal time series.
Abstract: An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.
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TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.
7,412 citations
TL;DR: Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes as mentioned in this paper, and a large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker-Planck equation.
Abstract: Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.
2,119 citations
01 Jan 2015
TL;DR: In this article, the authors present a new definition of fractional derivative with a smooth kernel, which takes on two different representations for the temporal and spatial variable, for which it is more convenient to work with the Fourier transform.
Abstract: In the paper, we present a new definition of fractional deriva tive with a smooth kernel which takes on two different representations for the temporal and spatial variable. The first works on the time variables; thus it is suitable to use th e Laplace transform. The second definition is related to the spatial va riables, by a non-local fractional derivative, for which it is more convenient to work with the Fourier transform. The interest for this new approach with a regular kernel was born from the prospect that there is a class of non-local systems, which have the ability to descri be the material heterogeneities and the fluctuations of diff erent scales, which cannot be well described by classical local theories or by fractional models with singular kernel.
1,972 citations
Cites background from "Applications Of Fractional Calculus..."
...[1], Caponetto [2], Caputo [3], Diethelm [4], Hilfer [5], Jiao et al [6], Kilbas et al....
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TL;DR: In this article, the concept of fractional kinetics is reviewed for systems with Hamiltonian chaos, where the notions of dynamical quasi-traps, Poincare recurrences, Levy flights, exit time distributions, phase space topology, etc.
1,478 citations
TL;DR: A survey of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date can be found in this article, where the authors report some of the most important documents and major events.
1,267 citations
References
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