1,468 research outputs found
Solving 1ODEs with functions
Here we present a new approach to deal with first order ordinary differential
equations (1ODEs), presenting functions. This method is an alternative to the
one we have presented in [1]. In [2], we have establish the theoretical
background to deal, in the extended Prelle-Singer approach context, with
systems of 1ODEs. In this present paper, we will apply these results in order
to produce a method that is more efficient in a great number of cases.
Directly, the solving of 1ODEs is applicable to any problem presenting
parameters to which the rate of change is related to the parameter itself.
Apart from that, the solving of 1ODEs can be a part of larger mathematical
processes vital to dealing with many problems.Comment: 31 page
Solving the Helmholtz Equation for the Neumann Boundary Condition for the Pseudosphere by the Galerkin Method
In this paper, the Helmholtz equation for the exterior Neumann boundary condition for the pseudosphere in three dimensions using the global Galerkin method is studied. The Galerkin method will be used to solve Jones’ modified integral equation approach (modified as a series of radiating waves will be added to the fundamental solution) for the Neumann problem for the Helmholtz equation, which uses a series of double sums to approximate the integral. A Fortran 77 program is used and some required subroutines from the Naval Warfare Center are called to help increase ouraccuracy since these boundary integrals are difficult to solve. The solutions obtained arecompared to the true solution for the Neumann problem to understand how well the method converges. The lower errors obtained show that the method for complete reflection of the sound waves off of the pseudosphere is accurate and successful. Also presented in this paper are both computational and theoretical details of the method ofdifferent values of k for the pseudosphere
The quantum integrable system
The quantum integrable system is a 3D system with rational potential
related to the non-crystallographic root system . It is shown that the
gauge-rotated Hamiltonian as well as one of the integrals, when written
in terms of the invariants of the Coxeter group , is in algebraic form: it
has polynomial coefficients in front of derivatives. The Hamiltonian has
infinitely-many finite-dimensional invariant subspaces in polynomials, they
form the infinite flag with the characteristic vector \vec \al\ =\ (1,2,3).
One among possible integrals is found (of the second order) as well as its
algebraic form. A hidden algebra of the Hamiltonian is determined. It is
an infinite-dimensional, finitely-generated algebra of differential operators
possessing finite-dimensional representations characterized by a generalized
Gauss decomposition property. A quasi-exactly-solvable integrable
generalization of the model is obtained. A discrete integrable model on the
uniform lattice in a space of -invariants "polynomially"-isospectral to
the quantum model is defined.Comment: 32 pages, 3 figure
Integrability of Some Charged Rotating Supergravity Black Hole Solutions in Four and Five Dimensions
We study the integrability of geodesic flow in the background of some
recently discovered charged rotating solutions of supergravity in four and five
dimensions. Specifically, we work with the gauged multicharge
Taub-NUT-Kerr-(Anti) de Sitter metric in four dimensions, and the
gauged charged-Kerr-(Anti) de Sitter black hole solution of N = 2 supergravity
in five dimensions. We explicitly construct the Killing tensors that permit
separation of the Hamilton-Jacobi equation in these spacetimes. These results
prove integrability for a large class of previously known supergravity
solutions, including several BPS solitonic states. We also derive first-order
equations of motion for particles in these backgrounds and examine some of
their properties. Finally, we also examine the Klein-Gordon equation for a
scalar field in these spacetimes and demonstrate separability.Comment: 17 Pages, updated bibliography, accepted for publication by Physics
Letters
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
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