30,871 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
Variational problems in fracture mechanics
We present some recent existence results for the variational model of crack
growth in brittle materials proposed by Francfort and Marigo in 1998. These
results, obtained in collaboration with Francfort and Toader, cover the case of
arbitrary space dimension with a general quasiconvex bulk energy and with
prescribed boundary deformations and applied loads.Comment: 9 page
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