4,487 research outputs found
Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit
The 1D discrete fractional Laplacian operator on a cyclically closed
(periodic) linear chain with finitenumber of identical particles is
introduced. We suggest a "fractional elastic harmonic potential", and obtain
the -periodic fractionalLaplacian operator in the form of a power law matrix
function for the finite chain ( arbitrary not necessarily large) in explicit
form.In the limiting case this fractional Laplacian
matrix recovers the fractional Laplacian matrix ofthe infinite chain.The
lattice model contains two free material constants, the particle mass and
a frequency.The "periodic string continuum limit" of the
fractional lattice model is analyzed where lattice constant and
length of the chain ("string") is kept finite: Assuming finiteness of
the total mass and totalelastic energy of the chain in the continuum limit
leads to asymptotic scaling behavior for of thetwo material
constants,namely and . In
this way we obtain the -periodic fractional Laplacian (Riesz fractional
derivative) kernel in explicit form.This -periodic fractional Laplacian
kernel recovers for the well known 1D infinite space
fractional Laplacian (Riesz fractional derivative) kernel. When the scaling
exponentof the Laplacian takesintegers, the fractional Laplacian kernel
recovers, respectively, -periodic and infinite space (localized)
distributionalrepresentations of integer-order Laplacians.The results of this
paper appear to beuseful for the analysis of fractional finite domain problems
for instance in anomalous diffusion (L\'evy flights), fractional Quantum
Mechanics,and the development of fractional discrete calculus on finite
lattices
Overdetermined problems with fractional Laplacian
Let and . In the present work we characterize bounded
open sets with boundary (\textit{not necessarily connected})
for which the following overdetermined problem \begin{equation*} ( -\Delta)^s u
= f(u) \text{ in ,} \qquad u=0 \text{ in ,} \qquad(\partial_{\eta})_s u=Const. \text{ on }
\end{equation*} has a nonnegative and nontrivial solution, where is the
outer unit normal vectorfield along and for
Under mild assumptions on , we prove that must be a ball. In the
special case , we obtain an extension of Serrin's result in 1971.
The fact that is not assumed to be connected is related to the
nonlocal property of the fractional Laplacian.
The main ingredients in our proof are maximum principles and the method of
moving planes
Fractional generalization of the Ginzburg-Landau equation: An unconventional approach to critical phenomena in complex media
Equations built on fractional derivatives prove to be a powerful tool in the
description of complex systems when the effects of singularity, fractal
supports, and long-range dependence play a role. In this paper, we advocate an
application of the fractional derivative formalism to a fairly general class of
critical phenomena when the organization of the system near the phase
transition point is influenced by a competing nonlocal ordering. Fractional
modifications of the free energy functional at criticality and of the widely
known Ginzburg-Landau equation central to the classical Landau theory of
second-type phase transitions are discussed in some detail. An implication of
the fractional Ginzburg-Landau equation is a renormalization of the transition
temperature owing to the nonlocality present.Comment: 10 pages, improved content, submitted for publication to Phys. Lett.
Fractional isoperimetric Noether's theorem in the Riemann-Liouville sense
We prove Noether-type theorems for fractional isoperimetric variational
problems with Riemann-Liouville derivatives. Both Lagrangian and Hamiltonian
formulations are obtained. Illustrative examples, in the fractional context of
the calculus of variations, are discussed.Comment: Submitted 12-Oct-2012; revised 05-Jan-2013; accepted 23-Jan-2013; for
publication in Reports on Mathematical Physics. arXiv admin note: substantial
text overlap with arXiv:1001.450
Clifford algebras, Fourier transforms and quantum mechanics
In this review, an overview is given of several recent generalizations of the
Fourier transform, related to either the Lie algebra sl_2 or the Lie
superalgebra osp(1|2). In the former case, one obtains scalar generalizations
of the Fourier transform, including the fractional Fourier transform, the Dunkl
transform, the radially deformed Fourier transform and the super Fourier
transform. In the latter case, one has to use the framework of Clifford
analysis and arrives at the Clifford-Fourier transform and the radially
deformed hypercomplex Fourier transform. A detailed exposition of all these
transforms is given, with emphasis on aspects such as eigenfunctions and
spectrum of the transform, characterization of the integral kernel and
connection with various special functions.Comment: Review paper, 39 pages, to appear in Math. Methods. Appl. Sc
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