27,908 research outputs found
Some aspects of fractional diffusion equations of single and distributed order
The time fractional diffusion equation is obtained from the standard
diffusion equation by replacing the first-order time derivative with a
fractional derivative of order . The fundamental solution for
the Cauchy problem is interpreted as a probability density of a self-similar
non-Markovian stochastic process related to a phenomenon of sub-diffusion (the
variance grows in time sub-linearly). A further generalization is obtained by
considering a continuous or discrete distribution of fractional time
derivatives of order less than one.
Then the fundamental solution is still a probability density of a
non-Markovian process that, however, is no longer self-similar but exhibits a
corresponding distribution of time-scales.Comment: 14 pages. International Symposium on "Analytic Function Theory,
Fractional Calculus and Their Applications", University of Victoria (British
Columbia, Canada), 22-27 August 200
Time-fractional diffusion of distributed order
The partial differential equation of Gaussian diffusion is generalized by
using the time-fractional derivative of distributed order between 0 and 1, in
both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general
distribution of time orders we provide the fundamental solution, that is still
a probability density, in terms of an integral of Laplace type. The kernel
depends on the type of the assumed fractional derivative except for the single
order case where the two approaches turn to be equivalent. We consider with
some detail two cases of order distribution: the double-order and the uniformly
distributed order. For these cases we exhibit plots of the corresponding
fundamental solutions and their variance, pointing out the remarkable
difference between the two approaches for small and large times.Comment: 30 pages, 4 figures. International Workshop on Fractional
Differentiation and its Applications (FDA06), 19-21 July 2006, Porto,
Portugal. Journal of Vibration and Control, in press (2007
The M-Wright function in time-fractional diffusion processes: a tutorial survey
In the present review we survey the properties of a transcendental function
of the Wright type, nowadays known as M-Wright function, entering as a
probability density in a relevant class of self-similar stochastic processes
that we generally refer to as time-fractional diffusion processes.
Indeed, the master equations governing these processes generalize the
standard diffusion equation by means of time-integral operators interpreted as
derivatives of fractional order. When these generalized diffusion processes are
properly characterized with stationary increments, the M-Wright function is
shown to play the same key role as the Gaussian density in the standard and
fractional Brownian motions. Furthermore, these processes provide stochastic
models suitable for describing phenomena of anomalous diffusion of both slow
and fast type.Comment: 32 pages, 3 figure
Non-Linear Langevin and Fractional Fokker-Planck Equations for Anomalous Diffusion by Levy Stable Processes
The~numerical solutions to a non-linear Fractional Fokker--Planck (FFP)
equation are studied estimating the generalized diffusion coefficients. The~aim
is to model anomalous diffusion using an FFP description with fractional
velocity derivatives and Langevin dynamics where L\'{e}vy fluctuations are
introduced to model the effect of non-local transport due to fractional
diffusion in velocity space. Distribution functions are found using numerical
means for varying degrees of fractionality of the stable L\'{e}vy distribution
as solutions to the FFP equation. The~statistical properties of the
distribution functions are assessed by a generalized normalized expectation
measure and entropy and modified transport coefficient. The~transport
coefficient significantly increases with decreasing fractality which is
corroborated by analysis of experimental data.Comment: 20 pages 7 figure
Fractional Patlak-Keller-Segel equations for chemotactic superdiffusion
The long range movement of certain organisms in the presence of a
chemoattractant can be governed by long distance runs, according to an
approximate Levy distribution. This article clarifies the form of biologically
relevant model equations: We derive Patlak-Keller-Segel-like equations
involving nonlocal, fractional Laplacians from a microscopic model for cell
movement. Starting from a power-law distribution of run times, we derive a
kinetic equation in which the collision term takes into account the long range
behaviour of the individuals. A fractional chemotactic equation is obtained in
a biologically relevant regime. Apart from chemotaxis, our work has
implications for biological diffusion in numerous processes.Comment: 20 pages, 4 figures, to appear in SIAM Journal on Applied Mathematic
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