17,830 research outputs found
Efficient numerical solution of the time fractional diffusion equation by mapping from its Brownian counterpart
The solution of a Caputo time fractional diffusion equation of order
is expressed in terms of the solution of a corresponding integer
order diffusion equation. We demonstrate a linear time mapping between these
solutions that allows for accelerated computation of the solution of the
fractional order problem. In the context of an -point finite difference time
discretisation, the mapping allows for an improvement in time computational
complexity from to , given a
precomputation of . The mapping is applied
successfully to the least-squares fitting of a fractional advection diffusion
model for the current in a time-of-flight experiment, resulting in a
computational speed up in the range of one to three orders of magnitude for
realistic problem sizes.Comment: 9 pages, 5 figures; added references for section
Qualitative dynamics and inflationary attractors in loop cosmology
Qualitative dynamics of three different loop quantizations of spatially flat
isotropic and homogeneous models is studied using effective spacetime
description of the underlying quantum geometry. These include the standard loop
quantum cosmology (LQC), its recently revived modification (referred to as
mLQC-I), and another related modification of LQC (mLQC-II) whose dynamics is
studied in detail for the first time. Various features of LQC, including
quantum bounce and pre-inflationary dynamics, are found to be shared with the
mLQC-I and mLQC-II models. We study universal properties of dynamics for
chaotic inflation, fractional monodromy inflation, Starobinsky potential,
non-minimal Higgs inflation, and an exponential potential. We find various
critical points and study their stability, which reveal various qualitative
similarities in the post-bounce phase for all these models. The pre-bounce
qualitative dynamics of LQC and mLQC-II turns out to be very similar, but is
strikingly different from that of mLQC-I. In the dynamical analysis, some of
the fixed points turn out to be degenerate for which center manifold theory is
used. For all these potentials, non-perturbative quantum gravitational effects
always result in a non-singular inflationary scenario with a phase of
super-inflation succeeded by the conventional inflation. We show the existence
of inflationary attractors, and obtain scaling solutions in the case of the
exponential potential. Since all of the models agree with general relativity at
late times, our results are also of use in classical theory where qualitative
dynamics of some of the potentials has not been studied earlier.Comment: 29 pages, 18 figures. Minor changes. To appear in Phys. Rev.
A generalized family of anisotropic compact object in general relativity
We present model for anisotropic compact star under the general theory of
relativity of Einstein. In the study a 4-dimensional spacetime has been
considered which is embedded into the 5-dimensional flat metric so that the
spherically symmetric metric has class 1 when the condition
is satisfied (
and being the metric potentials along with a constant ). A set of
solutions for the field equations are found depending on the index involved
in the physical parameters. The interior solutions have been matched smoothly
at the boundary of the spherical distribution to the exterior Schwarzschild
solution which necessarily provides values of the unknown constants. We have
chosen the values of as and =10 to 20000 for which interesting and
physically viable results can be found out. The numerical values of the
parameters and arbitrary constants for different compact stars are assumed in
the graphical plots and tables as follows: (i) LMC X-4 : ,
for and , for , (ii) SMC
X-1: , for , and , for . The investigations on the physical features of the model include
several astrophysical issues, like (i) regularity behavior of stars at the
centre, (ii) well behaved condition for velocity of sound, (iii) energy
conditions, (iv) stabilty of the system via the following three techniques -
adiabatic index, Herrera cracking concept and TOV equation, (v) total mass,
effective mass and compactification factor and (vi) surface redshift. Specific
numerical values of the compact star candidates LMC X-4 and SMC X-1 are
calculated for central and surface densities as well as central pressure to
compare the model value with actual observational data.Comment: 20 pages, 9 figures, 2 Table
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
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