15,968 research outputs found
Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains
Many PDEs involving fractional Laplacian are naturally set in unbounded
domains with underlying solutions decay very slowly, subject to certain power
laws. Their numerical solutions are under-explored. This paper aims at
developing accurate spectral methods using rational basis (or modified mapped
Gegenbauer functions) for such models in unbounded domains. The main building
block of the spectral algorithms is the explicit representations for the
Fourier transform and fractional Laplacian of the rational basis, derived from
some useful integral identites related to modified Bessel functions. With these
at our disposal, we can construct rational spectral-Galerkin and direct
collocation schemes by pre-computing the associated fractional differentiation
matrices. We obtain optimal error estimates of rational spectral approximation
in the fractional Sobolev spaces, and analyze the optimal convergence of the
proposed Galerkin scheme. We also provide ample numerical results to show that
the rational method outperforms the Hermite function approach
Veech surfaces and simple closed curves
We study the SL(2,R)-infimal lengths of simple closed curves on
half-translation surfaces. Our main result is a characterization of Veech
surfaces in terms of these lengths. We also revisit the "no small virtual
triangles" theorem of Smillie and Weiss and establish the following dichotomy:
the virtual triangle area spectrum of a half-translation surface either has a
gap above zero or is dense in a neighborhood of zero. These results make use of
the auxiliary polygon associated to a curve on a half-translation surface, as
introduced by Tang and Webb.Comment: 12 pages. v2: added proof of continuity of infimal length functions
on quadratic differential space; 16 pages, one figure; to appear in Israel J.
Mat
Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations
Recent results in the literature provide computational evidence that
stabilized semi-implicit time-stepping method can efficiently simulate phase
field problems involving fourth-order nonlinear dif- fusion, with typical
examples like the Cahn-Hilliard equation and the thin film type equation. The
up-to-date theoretical explanation of the numerical stability relies on the
assumption that the deriva- tive of the nonlinear potential function satisfies
a Lipschitz type condition, which in a rigorous sense, implies the boundedness
of the numerical solution. In this work we remove the Lipschitz assumption on
the nonlinearity and prove unconditional energy stability for the stabilized
semi-implicit time-stepping methods. It is shown that the size of stabilization
term depends on the initial energy and the perturba- tion parameter but is
independent of the time step. The corresponding error analysis is also
established under minimal nonlinearity and regularity assumptions
Efficient routing strategies in scale-free networks with limited bandwidth
We study the traffic dynamics in complex networks where each link is assigned
a limited and identical bandwidth. Although the first-in-first-out (FIFO)
queuing rule is widely applied in the routing protocol of information packets,
here we argue that if we drop this rule, the overall throughput of the network
can be remarkably enhanced. We proposed some efficient routing strategies that
do not strictly obey the FIFO rule. Comparing with the routine shortest path
strategy, the throughput for both Barab\'asi-Albert (BA) networks and the real
Internet, the throughput can be improved more than five times. We calculate the
theoretical limitation of the throughput. In BA networks, our proposed strategy
can achieve 88% of the theoretical optimum, yet for the real Internet, it is
about 12%, implying that we have a huge space to further improve the routing
strategy for the real Internet. Finally we discuss possibly promising ways to
design more efficient routing strategies for the Internet.Comment: 5 pages, 4 figure
Gradient bounds for a thin film epitaxy equation
We consider a gradient flow modeling the epitaxial growth of thin films with
slope selection. The surface height profile satisfies a nonlinear diffusion
equation with biharmonic dissipation. We establish optimal local and global
wellposedness for initial data with critical regularity. To understand the
mechanism of slope selection and the dependence on the dissipation coefficient,
we exhibit several lower and upper bounds for the gradient of the solution in
physical dimensions
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