92,853 research outputs found
Developing Clean Technology through Approximate Solutions of Mathematical Models
In this paper, the role of mathematical modeling in the development of clean technology has been considered.
One method each for obtaining approximate solutions of mathematical models by ordinary differential equations
and partial differential equations respectively arising from the modeling of systems and physical phenomena has
been considered. The construction of continuous hybrid methods for the numerical approximation of the solutions
of initial value problems of ordinary differential equations as well as homotopy analysis method, an approximate
analytical method, for the solution of nonlinear partial differential equations are discussed
Advanced Methods in Black-Hole Perturbation Theory
Black-hole perturbation theory is a useful tool to investigate issues in
astrophysics, high-energy physics, and fundamental problems in gravity. It is
often complementary to fully-fledged nonlinear evolutions and instrumental to
interpret some results of numerical simulations. Several modern applications
require advanced tools to investigate the linear dynamics of generic small
perturbations around stationary black holes. Here, we present an overview of
these applications and introduce extensions of the standard semianalytical
methods to construct and solve the linearized field equations in curved
spacetime. Current state-of-the-art techniques are pedagogically explained and
exciting open problems are presented.Comment: Lecture notes from the NRHEP spring school held at IST-Lisbon, March
2013. Extra material and notebooks available online at
http://blackholes.ist.utl.pt/nrhep2/. To be published by IJMPA (V. Cardoso,
L. Gualtieri, C. Herdeiro and U. Sperhake, Eds., 2013); v2: references
updated, published versio
Interpolation Parameter and Expansion for the Three Dimensional Non-Trivial Scalar Infrared Fixed Point
We compute the non--trivial infrared --fixed point by means of an
interpolation expansion in fixed dimension. The expansion is formulated for an
infinitesimal momentum space renormalization group. We choose a coordinate
representation for the fixed point interaction in derivative expansion, and
compute its coordinates to high orders by means of computer algebra. We compute
the series for the critical exponent up to order twenty five of
interpolation expansion in this representation, and evaluate it using \pade,
Borel--\pade, Borel--conformal--\pade, and Dlog--\pade resummation. The
resummation returns as the value of .Comment: 29 pages, Latex2e, 2 Postscript figure
Double power series method for approximating cosmological perturbations
We introduce a double power series method for finding approximate analytical
solutions for systems of differential equations commonly found in cosmological
perturbation theory. The method was set out, in a non-cosmological context, by
Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases
where perturbations are on sub-horizon scales. The FSN method is essentially an
extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding
approximate analytical solutions for ordinary differential equations. The FSN
method we use is applicable well beyond perturbation theory to solve systems of
ordinary differential equations, linear in the derivatives, that also depend on
a small parameter, which here we take to be related to the inverse wave-number.
We use the FSN method to find new approximate oscillating solutions in linear
order cosmological perturbation theory for a flat radiation-matter universe.
Together with this model's well known growing and decaying M\'esz\'aros
solutions, these oscillating modes provide a complete set of sub-horizon
approximations for the metric potential, radiation and matter perturbations.
Comparison with numerical solutions of the perturbation equations shows that
our approximations can be made accurate to within a typical error of 1%, or
better. We also set out a heuristic method for error estimation. A Mathematica
notebook which implements the double power series method is made available
online.Comment: 22 pages, 10 figures, 2 tables. Mathematica notebook available from
Github at https://github.com/AndrewWren/Double-power-series.gi
Approximate symmetry reduction approach: infinite series reductions to the KdV-Burgers equation
For weak dispersion and weak dissipation cases, the (1+1)-dimensional
KdV-Burgers equation is investigated in terms of approximate symmetry reduction
approach. The formal coherence of similarity reduction solutions and similarity
reduction equations of different orders enables series reduction solutions. For
weak dissipation case, zero-order similarity solutions satisfy the Painlev\'e
II, Painlev\'e I and Jacobi elliptic function equations. For weak dispersion
case, zero-order similarity solutions are in the form of Kummer, Airy and
hyperbolic tangent functions. Higher order similarity solutions can be obtained
by solving linear ordinary differential equations.Comment: 14 pages. The original model (1) in previous version is generalized
to a more extensive form and the incorrect equations (35) and (36) in
previous version are correcte
- …