13,866 research outputs found
Pre-asymptotic corrections to fractional diffusion equations
The motion of contaminant particles through complex environments such as
fractured rocks or porous sediments is often characterized by anomalous
diffusion: the spread of the transported quantity is found to grow sublinearly
in time due to the presence of obstacles which hinder particle migration. The
asymptotic behavior of these systems is usually well described by fractional
diffusion, which provides an elegant and unified framework for modeling
anomalous transport. We show that pre-asymptotic corrections to fractional
diffusion might become relevant, depending on the microscopic dynamics of the
particles. To incorporate these effects, we derive a modified transport
equation and validate its effectiveness by a Monte Carlo simulation.Comment: 6 pages, 3 figure
Statistics of non-linear stochastic dynamical systems under L\'evy noises by a convolution quadrature approach
This paper describes a novel numerical approach to find the statistics of the
non-stationary response of scalar non-linear systems excited by L\'evy white
noises. The proposed numerical procedure relies on the introduction of an
integral transform of Wiener-Hopf type into the equation governing the
characteristic function. Once this equation is rewritten as partial
integro-differential equation, it is then solved by applying the method of
convolution quadrature originally proposed by Lubich, here extended to deal
with this particular integral transform. The proposed approach is relevant for
two reasons: 1) Statistics of systems with several different drift terms can be
handled in an efficient way, independently from the kind of white noise; 2) The
particular form of Wiener-Hopf integral transform and its numerical evaluation,
both introduced in this study, are generalizations of fractional
integro-differential operators of potential type and Gr\"unwald-Letnikov
fractional derivatives, respectively.Comment: 20 pages, 5 figure
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