A closed form solution of the one‐dimensional Born–Green–Yvon equation for a hard‐rod fluid

1992 ◽  
Vol 33 (11) ◽  
pp. 3907-3910 ◽  
Author(s):  
D. J. Kaup ◽  
Gregory H. Paine
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
One Dimensional ◽  
The One

New exact results for the defective square lattice

1976 ◽  
Vol 80 (2) ◽  
pp. 365-381 ◽  
Author(s):  
G. Ronca
Keyword(s):  
Closed Form Solution ◽  
Exact Result ◽  
Form Solution ◽  
Square Lattice ◽  
Zero Temperature ◽  
Real Crystal ◽  
One Dimensional ◽  
Resonance Modes ◽  
The One ◽  
Dimensional Crystal

Since the publication of the fundamental papers by Lifshitz (1, 2) and Montroll and Potts (3, 4) many authors have investigated the effect of an isotopic impurity on the lattice vibrations of a harmonic crystal at zero temperature. A fairly broad knowledge is now available on scattering amplitudes, localized modes and resonance modes (6, 7). Nevertheless, as pointed out by Maradudin and Montroll (see (7), p. 430), a closed form solution to the problem has been found only for the one-dimensional crystal, the work done on two and three-dimensional crystals being predominantly numerical. Unfortunately the one-dimensional crystal, as an approximation for a real crystal is an oversimplified model, incapable as it is of exhibiting resonance modes. To the author's knowledge the most significant exact result concerning the classical behaviour at zero temperature of crystals having a dimensionality higher than one is the connexion, calculated by Mahanty et al. (5) between localized mode frequency and impurity mass for the case of a square lattice undergoing planar vibrations.

A Closed Form Solution for a One-Dimensional Multi-Layered Neutron Transport Problem by Analytical Discrete Ordinates Method

2017 ◽  
Vol 372 ◽  
pp. 50-59
Author(s):  
João Francisco Prolo Filho ◽  
Marco Paulsen Rodrigues
Keyword(s):  
Closed Form ◽  
Heterogeneous Media ◽  
Neutron Transport ◽  
Closed Form Solution ◽  
Homogeneous Solution ◽  
Form Solution ◽  
Discrete Ordinates ◽  
Test Problems ◽  
Discrete Ordinates Method ◽  
One Dimensional

In this work, the Analytical Discrete Ordinates Method (ADO method) is used to provide a closed form solution for a class of one-dimensional neutron transport problems in Cartesian geometry, considering heterogeneous media with linearly anisotropic scattering effects. In this context, the mathematical model will describe a steady-state phenomenon, with neutron sources located inside and on the boundaries of the domain of interest. In the process, the integro-differential transport equation is transformed into an ODE system by the SN angular discretization, which homogeneous solution is obtained with a quadratic eigenvalues problem with reduced order. A particular solution in terms of constants is used. To validate the code, the method and provide benchmark results, test problems will be treated and results will be discussed.

An exact closed-form solution for a multilayered one-dimensional orthorhombic quasicrystal plate

2015 ◽  
Vol 226 (11) ◽  
pp. 3611-3621 ◽  
Author(s):  
Lian-Zhi Yang ◽  
Yang Gao ◽  
Ernian Pan ◽  
Natalie Waksmanski
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
One Dimensional

Stability of Vlasov equilibria. Part 2. One non-ignorable co-ordinate

1982 ◽  
Vol 27 (1) ◽  
pp. 25-35 ◽  
Author(s):  
H. Ralph Lewis ◽  
Charles E. Seyler
Keyword(s):  
Closed Form ◽  
Vlasov Equation ◽  
Fourier Expansion ◽  
Closed Form Solution ◽  
Form Solution ◽  
One Dimensional ◽  
Equilibrium Orbit ◽  
Explicit Formulae

The solution of the linearized Vlasov equation is given for an arbitrary equilibrium Hamiltonian in which there is only one non-ignorable co-ordinate. The solution written in terms of integrals with respect to time which only extend over the bounce period of an equilibrium orbit in its equivalent one-dimensional potential. A closed-form solution and a solution based on a Fourier expansion are given. Explicit formulae are presented for Cartesian and cylindrical co-ordinates.

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