127 research outputs found
Symmetry-preserving discrete schemes for some heat transfer equations
Lie group analysis of differential equations is a generally recognized
method, which provides invariant solutions, integrability, conservation laws
etc. In this paper we present three characteristic examples of the construction
of invariant difference equations and meshes, where the original continuous
symmetries are preserved in discrete models. Conservation of symmetries in
difference modeling helps to retain qualitative properties of the differential
equations in their difference counterparts.Comment: 21 pages, 4 ps figure
A heat transfer with a source: the complete set of invariant difference schemes
In this letter we present the set of invariant difference equations and
meshes which preserve the Lie group symmetries of the equation
u_{t}=(K(u)u_{x})_{x}+Q(u). All special cases of K(u) and Q(u) that extend the
symmetry group admitted by the differential equation are considered. This paper
completes the paper [J. Phys. A: Math. Gen. 30, no. 23 (1997) 8139-8155], where
a few invariant models for heat transfer equations were presented.Comment: arxiv version is already officia
Lie point symmetries of difference equations and lattices
A method is presented for finding the Lie point symmetry transformations
acting simultaneously on difference equations and lattices, while leaving the
solution set of the corresponding difference scheme invariant. The method is
applied to several examples. The found symmetry groups are used to obtain
particular solutions of differential-difference equations
Gravitationally distorted P-Cygni profiles from outflows near compact objects
We consider resonant absorption in a spectral line in the outflowing plasma
within several tens of Schwarzschild radii from a compact object. We take into
account both Doppler and gravitational shifting effects and re-formulate the
theory of P-Cygni profiles in these new circumstances. It is found that a
spectral line may have multiple absorption and emission components depending on
how far the region of interaction is from the compact object and what is the
distribution of velocity and opacity. Profiles of spectral lines produced near
a neutron star or a black hole can be strongly distorted by Doppler blue-, or
red-shifting, and gravitational red-shifting. These profiles may have both red-
and blue-shifted absorption troughs. The result should be contrasted with
classical P-Cygni profiles which consist of red-shifted emission and
blue-shifted absorption features.
We suggest this property of line profiles to have complicated narrow
absorption and emission components in the presence of strong gravity may help
to study spectroscopically the innermost parts of an outflow.Comment: 34 pages, 12 figures, accepted by MNRAS; minor correction
Difference schemes with point symmetries and their numerical tests
Symmetry preserving difference schemes approximating second and third order
ordinary differential equations are presented. They have the same three or
four-dimensional symmetry groups as the original differential equations. The
new difference schemes are tested as numerical methods. The obtained numerical
solutions are shown to be much more accurate than those obtained by standard
methods without an increase in cost. For an example involving a solution with a
singularity in the integration region the symmetry preserving scheme, contrary
to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure
Star-Jet Interactions and Gamma-Ray Outbursts from 3C454.3
We propose a model to explain the ultra-bright GeV gamma-ray flares observed
from the blazar 3C454.3. The model is based on the concept of a relativistic
jet interacting with compact gas condensations produced when a star (red giant)
crosses the jet close to the central black hole. The study includes an
analytical treatment of the evolution of the envelop lost by the star within
the jet, and calculations of the related high-energy radiation. The model
readily explains the day-long, variable on timescales of hours, GeV gamma-ray
flare from 3C454.3, observed during November 2010 on top of a weeks-long
plateau. In the proposed scenario, the plateau state is caused by a strong wind
generated by the heating of the star atmosphere by nonthermal particles
accelerated at the jet-star interaction region. The flare itself could be
produced by a few clouds of matter lost by the red giant after the initial
impact of the jet. In the framework of the proposed scenario, the observations
constrain the key model parameters of the source, including the mass of the
central black hole: , the total jet power:
, and the Doppler factor of the
gamma-ray emitting clouds, . Whereas we do not specify the
particle acceleration mechanisms, the potential gamma-ray production processes
are discussed and compared in the context of the proposed model. We argue that
synchrotron radiation of protons has certain advantages compared to other
radiation channels of directly accelerated electrons.Comment: 16 pages, 5 figures, submitted to Ap
Lie Symmetries and Exact Solutions of First Order Difference Schemes
We show that any first order ordinary differential equation with a known Lie
point symmetry group can be discretized into a difference scheme with the same
symmetry group. In general, the lattices are not regular ones, but must be
adapted to the symmetries considered. The invariant difference schemes can be
so chosen that their solutions coincide exactly with those of the original
differential equation.Comment: Minor changes and journal-re
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