46,735 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
Electronic Mach-Zehnder interferometer as a tool to probe fractional statistics
We study transport through an electronic Mach-Zehnder interferometer recently
devised at the Weizmann Institute. We show that this device can be used to
probe statistics of quasiparticles in the fractional quantum Hall regime. We
calculate the tunneling current through the interferometer as the function of
the Aharonov-Bohm flux, temperature and voltage bias, and demonstrate that its
flux-dependent component is strongly sensitive to the statistics of tunneling
quasiparticles. More specifically, the flux-dependent and flux-independent
contributions to the current are related by a power law, the exponent being a
function of the quasiparticle statistics.Comment: 22 pages; 8 figure
Efficient multistep methods for tempered fractional calculus: Algorithms and Simulations
In this work, we extend the fractional linear multistep methods in [C.
Lubich, SIAM J. Math. Anal., 17 (1986), pp.704--719] to the tempered fractional
integral and derivative operators in the sense that the tempered fractional
derivative operator is interpreted in terms of the Hadamard finite-part
integral. We develop two fast methods, Fast Method I and Fast Method II, with
linear complexity to calculate the discrete convolution for the approximation
of the (tempered) fractional operator. Fast Method I is based on a local
approximation for the contour integral that represents the convolution weight.
Fast Method II is based on a globally uniform approximation of the trapezoidal
rule for the integral on the real line. Both methods are efficient, but
numerical experimentation reveals that Fast Method II outperforms Fast Method I
in terms of accuracy, efficiency, and coding simplicity. The memory requirement
and computational cost of Fast Method II are and ,
respectively, where is the number of the final time steps and is the
number of quadrature points used in the trapezoidal rule. The effectiveness of
the fast methods is verified through a series of numerical examples for
long-time integration, including a numerical study of a fractional
reaction-diffusion model
Optimal control of a fractional order epidemic model with application to human respiratory syncytial virus infection
A human respiratory syncytial virus surveillance system was implemented in
Florida in 1999, to support clinical decision-making for prophylaxis of
premature newborns. Recently, a local periodic SEIRS mathematical model was
proposed in [Stat. Optim. Inf. Comput. 6 (2018), no.1, 139--149] to describe
real data collected by Florida's system. In contrast, here we propose a
non-local fractional (non-integer) order model. A fractional optimal control
problem is then formulated and solved, having treatment as the control.
Finally, a cost-effectiveness analysis is carried out to evaluate the cost and
the effectiveness of proposed control measures during the intervention period,
showing the superiority of obtained results with respect to previous ones.Comment: This is a preprint of a paper whose final and definite form is with
'Chaos, Solitons & Fractals', available from
[http://www.elsevier.com/locate/issn/09600779]. Submitted 23-July-2018;
Revised 14-Oct-2018; Accepted 15-Oct-2018. arXiv admin note: substantial text
overlap with arXiv:1801.0963
- …