3,214 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
Well-posedness and regularity for a generalized fractional Cahn-Hilliard system
In this paper, we investigate a rather general system of two operator
equations that has the structure of a viscous or nonviscous Cahn--Hilliard
system in which nonlinearities of double-well type occur. Standard cases like
regular or logarithmic potentials, as well as non-differentiable potentials
involving indicator functions, are admitted. The operators appearing in the
system equations are fractional versions of general linear operators and
, where the latter are densely defined, unbounded, self-adjoint and monotone
in a Hilbert space of functions defined in a smooth domain and have compact
resolvents. We remark that our definition of the fractional power of operators
uses the approach via spectral theory. Typical cases are given by standard
second-order elliptic operators (e.g., the Laplacian) with zero Dirichlet or
Neumann boundary conditions, but also other cases like fourth-order systems or
systems involving the Stokes operator are covered by the theory. We derive
general well-posedness and regularity results that extend corresponding results
which are known for either the non-fractional Laplacian with zero Neumann
boundary condition or the fractional Laplacian with zero Dirichlet condition.
It turns out that the first eigenvalue of plays an important
und not entirely obvious role: if is positive, then the operators
and may be completely unrelated; if, however, ,
then it must be simple and the corresponding one-dimensional eigenspace has to
consist of the constant functions and to be a subset of the domain of
definition of a certain fractional power of . We are able to show general
existence, uniqueness, and regularity results for both these cases, as well as
for both the viscous and the nonviscous system.Comment: 36 pages. Key words: fractional operators, Cahn-Hilliard systems,
well-posedness, regularity of solution
A matrix method for fractional Sturm-Liouville problems on bounded domain
A matrix method for the solution of direct fractional Sturm-Liouville
problems on bounded domain is proposed where the fractional derivative is
defined in the Riesz sense. The scheme is based on the application of the
Galerkin spectral method of orthogonal polynomials. The order of convergence of
the eigenvalue approximations with respect to the matrix size is studied. Some
numerical examples that confirm the theory and prove the competitiveness of the
approach are finally presented
Uniqueness of radial solutions for the fractional Laplacian
We prove general uniqueness results for radial solutions of linear and
nonlinear equations involving the fractional Laplacian with for any space dimensions . By extending a monotonicity
formula found by Cabre and Sire \cite{CaSi-10}, we show that the linear
equation in has at most one radial and
bounded solution vanishing at infinity, provided that the potential is a
radial and non-decreasing. In particular, this result implies that all radial
eigenvalues of the corresponding fractional Schr\"odinger operator
are simple. Furthermore, by combining these findings on
linear equations with topological bounds for a related problem on the upper
half-space , we show uniqueness and nondegeneracy of ground
state solutions for the nonlinear equation in for arbitrary space dimensions and all
admissible exponents . This generalizes the nondegeneracy and
uniqueness result for dimension N=1 recently obtained by the first two authors
in \cite{FrLe-10} and, in particular, the uniqueness result for solitary waves
of the Benjamin--Ono equation found by Amick and Toland \cite{AmTo-91}.Comment: 38 pages; revised version; various typos corrected; proof of Lemma
8.1 corrected; discussion of case \kappa_* =1 in the proof of Theorem 2
corrected with new Lemma A.2; accepted for publication in Comm. Pure. Appl.
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