18,686 research outputs found
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application
We develop further the theory of symmetrization of fractional Laplacian
operators contained in recent works of two of the authors. The theory leads to
optimal estimates in the form of concentration comparison inequalities for both
elliptic and parabolic equations. In this paper we extend the theory for the
so-called \emph{restricted} fractional Laplacian defined on a bounded domain
of with zero Dirichlet conditions outside of .
As an application, we derive an original proof of the corresponding fractional
Faber-Krahn inequality. We also provide a more classical variational proof of
the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297
Non-diffusive transport in plasma turbulence: a fractional diffusion approach
Numerical evidence of non-diffusive transport in three-dimensional, resistive
pressure-gradient-driven plasma turbulence is presented. It is shown that the
probability density function (pdf) of test particles' radial displacements is
strongly non-Gaussian and exhibits algebraic decaying tails. To model these
results we propose a macroscopic transport model for the pdf based on the use
of fractional derivatives in space and time, that incorporate in a unified way
space-time non-locality (non-Fickian transport), non-Gaussianity, and
non-diffusive scaling. The fractional diffusion model reproduces the shape, and
space-time scaling of the non-Gaussian pdf of turbulent transport calculations.
The model also reproduces the observed super-diffusive scaling
Diffusive representations for fractional Laplacian: systems theory framework and numerical issues
Bridging the gap between an abstract definition of pseudo-differential operators, such as (-\Delta)^{\gamma} for - 1/2 < \gamma < 1/2, and a concrete way to represent them has proved difficult; deriving stable numerical schemes for such operators is not an easy task either. Thus, the framework of diffusive representations, as already developed for causal fractional integrals and derivatives,
is being applied to fractional Laplacian: it can be seen as an extension of the Wiener-ÂHopf factorization and splitting techniques to irrational transfer functions
The Variable-Order Fractional Calculus of Variations
This book intends to deepen the study of the fractional calculus, giving
special emphasis to variable-order operators. It is organized in two parts, as
follows. In the first part, we review the basic concepts of fractional calculus
(Chapter 1) and of the fractional calculus of variations (Chapter 2). In
Chapter 1, we start with a brief overview about fractional calculus and an
introduction to the theory of some special functions in fractional calculus.
Then, we recall several fractional operators (integrals and derivatives)
definitions and some properties of the considered fractional derivatives and
integrals are introduced. In the end of this chapter, we review integration by
parts formulas for different operators. Chapter 2 presents a short introduction
to the classical calculus of variations and review different variational
problems, like the isoperimetric problems or problems with variable endpoints.
In the end of this chapter, we introduce the theory of the fractional calculus
of variations and some fractional variational problems with variable-order. In
the second part, we systematize some new recent results on variable-order
fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018).
In Chapter 3, considering three types of fractional Caputo derivatives of
variable-order, we present new approximation formulas for those fractional
derivatives and prove upper bound formulas for the errors. In Chapter 4, we
introduce the combined Caputo fractional derivative of variable-order and
corresponding higher-order operators. Some properties are also given. Then, we
prove fractional Euler-Lagrange equations for several types of fractional
problems of the calculus of variations, with or without constraints.Comment: The final authenticated version of this preprint is available online
as a SpringerBrief in Applied Sciences and Technology at
[https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos,
detected by the authors while reading the galley proofs, were corrected,
SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 201
Stochastic processes via the pathway model
After collecting data from observations or experiments, the next step is to
build an appropriate mathematical or stochastic model to describe the data so
that further studies can be done with the help of the models. In this article,
the input-output type mechanism is considered first, where reaction, diffusion,
reaction-diffusion, and production-destruction type physical situations can fit
in. Then techniques are described to produce thicker or thinner tails (power
law behavior) in stochastic models. Then the pathway idea is described where
one can switch to different functional forms of the probability density
function) through a parameter called the pathway parameter.Comment: 15 pages, 7 figures, LaTe
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