57 research outputs found
Erd\'elyi-Kober Fractional Diffusion
The aim of this Short Note is to highlight that the {\it generalized grey
Brownian motion} (ggBm) is an anomalous diffusion process driven by a
fractional integral equation in the sense of Erd\'elyi-Kober, and for this
reason here it is proposed to call such family of diffusive processes as {\it
Erd\'elyi-Kober fractional diffusion}. The ggBm is a parametric class of
stochastic processes that provides models for both fast and slow anomalous
diffusion. This class is made up of self-similar processes with stationary
increments and it depends on two real parameters: and . It includes the fractional Brownian motion when and , the time-fractional diffusion stochastic processes when , and the standard Brownian motion when . In
the ggBm framework, the Mainardi function emerges as a natural generalization
of the Gaussian distribution recovering the same key role of the Gaussian
density for the standard and the fractional Brownian motion.Comment: Accepted for publication in Fractional Calculus and Applied Analysi
Short note on the emergence of fractional kinetics
In the present Short Note an idea is proposed to explain the emergence and
the observation of processes in complex media that are driven by fractional
non-Markovian master equations. Particle trajectories are assumed to be solely
Markovian and described by the Continuous Time Random Walk model. But, as a
consequence of the complexity of the medium, each trajectory is supposed to
scale in time according to a particular random timescale. The link from this
framework to microscopic dynamics is discussed and the distribution of
timescales is computed. In particular, when a stationary distribution is
considered, the timescale distribution is uniquely determined as a function
related to the fundamental solution of the space-time fractional diffusion
equation. In contrast, when the non-stationary case is considered, the
timescale distribution is no longer unique. Two distributions are here
computed: one related to the M-Wright/Mainardi function, which is Green's
function of the time-fractional diffusion equation, and another related to the
Mittag-Leffler function, which is the solution of the fractional-relaxation
equation
Nonlinear Time-Fractional Differential Equations in Combustion Science
MSC 2010: 34A08 (main), 34G20, 80A25The application of Fractional Calculus in combustion science to model
the evolution in time of the radius of an isolated premixed flame ball is
highlighted. Literature equations for premixed flame ball radius are rederived by a new method that strongly simplifies previous ones. These equations are nonlinear time-fractional differential equations of order 1/2
with a Gaussian underlying diffusion process. Extending the analysis to
self-similar anomalous diffusion processes with similarity parameter ν/2 > 0,
the evolution equations emerge to be nonlinear time-fractional differential
equations of order 1−ν/2 with a non-Gaussian underlying diffusion process
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
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
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