118 research outputs found
Single polymer dynamics in elongational flow and the confluent Heun equation
We investigate the non-equilibrium dynamics of an isolated polymer in a
stationary elongational flow. We compute the relaxation time to the
steady-state configuration as a function of the Weissenberg number. A strong
increase of the relaxation time is found around the coil-stretch transition,
which is attributed to the large number of polymer configurations. The
relaxation dynamics of the polymer is solved analytically in terms of a central
two-point connection problem for the singly confluent Heun equation.Comment: 9 pages, 6 figure
Ince's limits for confluent and double-confluent Heun equations
We find pairs of solutions to a differential equation which is obtained as a
special limit of a generalized spheroidal wave equation (this is also known as
confluent Heun equation). One solution in each pair is given by a series of
hypergeometric functions and converges for any finite value of the independent
variable , while the other is given by a series of modified Bessel functions
and converges for , where denotes a regular singularity.
For short, the preceding limit is called Ince's limit after Ince who have used
the same procedure to get the Mathieu equations from the Whittaker-Hill ones.
We find as well that, when tends to zero, the Ince limit of the
generalized spheroidal wave equation turns out to be the Ince limit of a
double-confluent Heun equation, for which solutions are provided. Finally, we
show that the Schr\"odinger equation for inverse fourth and sixth-power
potentials reduces to peculiar cases of the double-confluent Heun equation and
its Ince's limit, respectively.Comment: Submitted to Journal of Mathmatical Physic
Teukolsky-Starobinsky Identities - a Novel Derivation and Generalizations
We present a novel derivation of the Teukolsky-Starobinsky identities, based
on properties of the confluent Heun functions. These functions define
analytically all exact solutions to the Teukolsky master equation, as well as
to the Regge-Wheeler and Zerilli ones. The class of solutions, subject to
Teukolsky-Starobinsky type of identities is studied. Our generalization of the
Teukolsky-Starobinsky identities is valid for the already studied linear
perturbations to the Kerr and Schwarzschild metrics, as well as for large new
classes of of such perturbations which are explicitly described in the present
article. Symmetry of parameters of confluent Heun's functions is shown to stay
behind the behavior of the known solutions under the change of the sign of
their spin weights. A new efficient recurrent method for calculation of
Starobinsky's constant is described.Comment: 8 pages, LaTeX file, no figures, final versio
Novel relations and new properties of confluent Heun's functions and their derivatives of arbitrary order
The present article reveals important properties of the confluent Heun's
functions. We derive a set of novel relations for confluent Heun's functions
and their derivatives of arbitrary order. Specific new subclasses of confluent
Heun's functions are introduced and studied. A new alternative derivation of
confluent Heun's polynomials is presented.Comment: 8 pages, no figures, LaTeX file, final versio
Exact Solutions of Regge-Wheeler Equation and Quasi-Normal Modes of Compact Objects
The well-known Regge-Wheeler equation describes the axial perturbations of
Schwarzschild metric in the linear approximation. From a mathematical point of
view it presents a particular case of the confluent Heun equation and can be
solved exactly, due to recent mathematical developments. We present the basic
properties of its general solution. A novel analytical approach and numerical
techniques for study the boundary problems which correspond to quasi-normal
modes of black holes and other simple models of compact objects are developed.Comment: latex file, 25 pages, 4 figures, new references, new results and new
Appendix added, some comments and corrections in the text made. Accepted for
publication in Classical and Quantum Gravity, 2006, simplification of
notations, changes in the norm in some formulas, corrections in reference
Transformations of Heun's equation and its integral relations
We find transformations of variables which preserve the form of the equation
for the kernels of integral relations among solutions of the Heun equation.
These transformations lead to new kernels for the Heun equation, given by
single hypergeometric functions (Lambe-Ward-type kernels) and by products of
two hypergeometric functions (Erd\'elyi-type). Such kernels, by a limiting
process, also afford new kernels for the confluent Heun equation.Comment: This version was published in J. Phys. A: Math. Theor. 44 (2011)
07520
Solutions for the General, Confluent and Biconfluent Heun equations and their connection with Abel equations
In a recent paper, the canonical forms of a new multi-parameter class of Abel
differential equations, so-called AIR, all of whose members can be mapped into
Riccati equations, were shown to be related to the differential equations for
the hypergeometric 2F1, 1F1 and 0F1 functions. In this paper, a connection
between the AIR canonical forms and the Heun General (GHE), Confluent (CHE) and
Biconfluent (BHE) equations is presented. This connection fixes the value of
one of the Heun parameters, expresses another one in terms of those remaining,
and provides closed form solutions in terms of pFq functions for the resulting
GHE, CHE and BHE, respectively depending on four, three and two irreducible
parameters. This connection also turns evident what is the relation between the
Heun parameters such that the solutions admit Liouvillian form, and suggests a
mechanism for relating linear equations with N and N-1 singularities through
the canonical forms of a non-linear equation of one order less.Comment: Original version submitted to Journal of Physics A: 16 pages, related
to math.GM/0002059 and math-ph/0402040. Revised version according to
referee's comments: 23 pages. Sign corrected (June/17) in formula (79).
Second revised version (July/25): 25 pages. See also
http://lie.uwaterloo.ca/odetools.ht
Classes of Exact Solutions to the Teukolsky Master Equation
The Teukolsky Master Equation is the basic tool for study of perturbations of
the Kerr metric in linear approximation. It admits separation of variables,
thus yielding the Teukolsky Radial Equation and the Teukolsky Angular Equation.
We present here a unified description of all classes of exact solutions to
these equations in terms of the confluent Heun functions. Large classes of new
exact solutions are found and classified with respect to their characteristic
properties. Special attention is paid to the polynomial solutions which are
singular ones and introduce collimated one-way-running waves. It is shown that
a proper linear combination of such solutions can present bounded
one-way-running waves. This type of waves may be suitable as models of the
observed astrophysical jets.Comment: 27 pages, LaTeX file, no figures. Final versio
- …