213 research outputs found
Interior penalty discontinuous Galerkin FEM for the -Laplacian
In this paper we construct an "Interior Penalty" Discontinuous Galerkin
method to approximate the minimizer of a variational problem related to the
Laplacian. The function is log H\"{o}lder
continuous and . We prove that the minimizers of the
discrete functional converge to the solution. We also make some numerical
experiments in dimension one to compare this method with the Conforming
Galerkin Method, in the case where is close to one. This example is
motivated by its applications to image processing.Comment: 26 pages, 2 figure
Fractional oscillator process with two indices
We introduce a new fractional oscillator process which can be obtained as
solution of a stochastic differential equation with two fractional orders.
Basic properties such as fractal dimension and short range dependence of the
process are studied by considering the asymptotic properties of its covariance
function. The fluctuation--dissipation relation of the process is investigated.
The fractional oscillator process can be regarded as one-dimensional fractional
Euclidean Klein-Gordon field, which can be obtained by applying the Parisi-Wu
stochastic quantization method to a nonlocal Euclidean action. The Casimir
energy associated with the fractional field at positive temperature is
calculated by using the zeta function regularization technique.Comment: 32 page
Variational Problems with Fractional Derivatives: Euler-Lagrange Equations
We generalize the fractional variational problem by allowing the possibility
that the lower bound in the fractional derivative does not coincide with the
lower bound of the integral that is minimized. Also, for the standard case when
these two bounds coincide, we derive a new form of Euler-Lagrange equations. We
use approximations for fractional derivatives in the Lagrangian and obtain the
Euler-Lagrange equations which approximate the initial Euler-Lagrange equations
in a weak sense
Parameters of the fractional Fokker-Planck equation
We study the connection between the parameters of the fractional
Fokker-Planck equation, which is associated with the overdamped Langevin
equation driven by noise with heavy-tailed increments, and the transition
probability density of the noise generating process. Explicit expressions for
these parameters are derived both for finite and infinite variance of the
rescaled transition probability density.Comment: 5 page
Dynamical Renormalization Group Study for a Class of Non-local Interface Equations
We provide a detailed Dynamic Renormalization Group study for a class of
stochastic equations that describe non-conserved interface growth mediated by
non-local interactions. We consider explicitly both the morphologically stable
case, and the less studied case in which pattern formation occurs, for which
flat surfaces are linearly unstable to periodic perturbations. We show that the
latter leads to non-trivial scaling behavior in an appropriate parameter range
when combined with the Kardar-Parisi-Zhang (KPZ) non-linearity, that
nevertheless does not correspond to the KPZ universality class. This novel
asymptotic behavior is characterized by two scaling laws that fix the critical
exponents to dimension-independent values, that agree with previous reports
from numerical simulations and experimental systems. We show that the precise
form of the linear stabilizing terms does not modify the hydrodynamic behavior
of these equations. One of the scaling laws, usually associated with Galilean
invariance, is shown to derive from a vertex cancellation that occurs (at least
to one loop order) for any choice of linear terms in the equation of motion and
is independent on the morphological stability of the surface, hence
generalizing this well-known property of the KPZ equation. Moreover, the
argument carries over to other systems like the Lai-Das Sarma-Villain equation,
in which vertex cancellation is known {\em not to} imply an associated symmetry
of the equation.Comment: 34 pages, 9 figures. Journal of Statistical Mechanics: Theory and
Experiments (in press
Fractional transport equations for Levy stable processes
The influence functional method of Feynman and Vernon is used to obtain a
quantum master equation for a Brownian system subjected to a Levy stable random
force. The corresponding classical transport equations for the Wigner function
are then derived, both in the limit of weak and strong friction. These are
fractional extensions of the Klein-Kramers and the Smoluchowski equations. It
is shown that the fractional character acquired by the position in the
Smoluchowski equation follows from the fractional character of the momentum in
the Klein-Kramers equation. Connections among fractional transport equations
recently proposed are clarified.Comment: 4 page
Fractional Generalization of Gradient Systems
We consider a fractional generalization of gradient systems. We use
differential forms and exterior derivatives of fractional orders. Examples of
fractional gradient systems are considered. We describe the stationary states
of these systems.Comment: 11 pages, LaTe
Spontaneous emission from a two-level atom in anisotropic one-band photonic crystals: a fractional calculus approach
Spontaneous emission (SE) from a two-level atom in a photonic crystal (PC)
with anisotropic one-band model is investigated using the fractional calculus.
Analytically solving the kinetic equation in terms of the fractional
exponential function, the dynamical discrepancy of SE between the anisotropic
and isotropic systems is discussed on the basis of different photon density of
states (DOS) and the existence of incoherent diffusion field that becomes even
more clearly as the atomic transition frequency lies close to the band edge.
With the same atom-field coupling strength and detuning in the forbidden gap,
the photon-atom bound states in the isotropic system turn into the unbound ones
in the anisotropic system that is consistent with the experimental observation
in \textbf{96}, 243902 (2006). Dynamics along different
wavevectors with various curvatures of dispersion is also addressed with the
changes of the photon DOS and the appearance of the diffusion fields.Comment: 16 pages, 4 figure
Fractional wave equation and damped waves
In this paper, a fractional generalization of the wave equation that
describes propagation of damped waves is considered. In contrast to the
fractional diffusion-wave equation, the fractional wave equation contains
fractional derivatives of the same order both in
space and in time. We show that this feature is a decisive factor for
inheriting some crucial characteristics of the wave equation like a constant
propagation velocity of both the maximum of its fundamental solution and its
gravity and mass centers. Moreover, the first, the second, and the Smith
centrovelocities of the damped waves described by the fractional wave equation
are constant and depend just on the equation order . The fundamental
solution of the fractional wave equation is determined and shown to be a
spatial probability density function evolving in time that possesses finite
moments up to the order . To illustrate analytical findings, results of
numerical calculations and numerous plots are presented.Comment: 21 pages, 10 figure
The Stokes and Poisson problem in variable exponent spaces
We study the Stokes and Poisson problem in the context of variable exponent
spaces. We prove the existence of strong and weak solutions for bounded domains
with C^{1,1} boundary with inhomogenous boundary values. The result is based on
generalizations of the classical theories of Calderon-Zygmund and
Agmon-Douglis-Nirenberg to variable exponent spaces.Comment: 20 pages, 1 figur
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