Approximate solutions for general pairing Hamiltonian by variational method

1978 ◽  
Vol 21 (9) ◽  
pp. 305-310 ◽  
Author(s):  
A. Kamimura
Keyword(s):  
Variational Method ◽  
Approximate Solutions

The Sector Problem

1957 ◽  
Vol 24 (4) ◽  
pp. 574-581
Author(s):  
G. Horvay ◽  
K. L. Hanson
Keyword(s):  
Variational Method ◽  
Orthogonal Polynomials ◽  
Biharmonic Equation ◽  
Approximate Solutions ◽  
Circular Sector ◽  
Orthonormal Polynomials ◽  
Circular Boundary ◽  
Stress Functions ◽  
Complete Set ◽  
Derivatives Of

Abstract On the basis of the variational method, approximate solutions f k ( r ) h k ( θ ) , f k ( r ) g k ( θ ) , F k ( θ ) H k ( r ) , F k ( θ ) G k ( r ) of the biharmonic equation are established for the circular sector with the following properties: The stress functions fkhk create shear tractions on the radial boundaries; the stress functions fkgk create normal tractions on the radial boundaries; the stress functions FkHk create both shear and normal tractions on the circular boundary, and the stress functions FkGk create normal tractions on the circular boundary. The enumerated tractions are the only tractions which these function sets create on the various boundaries of the sector. The factors fk(r) constitute a complete set of orthonormal polynomials in r into which (more exactly, into the derivatives of which) self-equilibrating normal or shear tractions applied to the radial boundaries of the sector may be expanded; the factors Fk(θ) constitute a complete set of orthonormal polynomials in θ into which shear tractions applied to the circular boundary of the sector may be expanded; and the functions Fk″ + Fk constitute a complete set of non-orthogonal polynomials into which normal tractions applied to the circular boundary of the sector may be expanded. Function tables, to facilitate the use of the stress functions, are also presented.

Approximate Solutions of Compressible Flows Past Bodies of Revolution by Variational Method

1951 ◽  
Vol 18 (3) ◽  
pp. 260-266
Author(s):  
Chi-Teh Wang ◽  
Socrates De Los Santos
Keyword(s):  
Variational Method ◽  
Arbitrary Shape ◽  
Compressible Flows ◽  
Direct Method ◽  
Approximate Solutions ◽  
Body Of Revolution ◽  
Approximate Methods ◽  
Bodies Of Revolution ◽  
Numerical Examples ◽  
Ellipsoid Of Revolution

Abstract Using the direct method of Rayleigh-Ritz in the calculus of variations, the problem of steady irrotational compressible flow past a body of revolution of arbitrary shape is formulated. In order to compare with the analytical solutions obtained by other investigators, two numerical examples have been carried out, namely, compressible flows past a sphere, and an ellipsoid of revolution. The results are found to be in excellent agreement with those computed by other approximate methods.

scholarly journals Optimal Variational Method for Truly Nonlinear Oscillators

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Variational Method ◽  
Numerical Solution ◽  
Periodic Solutions ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Convergence Of Approximate Solutions ◽  
Good Agreement

The Optimal Variational Method (OVM) is introduced and applied for calculating approximate periodic solutions of “truly nonlinear oscillators”. The main advantage of this procedure consists in that it provides a convenient way to control the convergence of approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary. This approach does not depend upon any small or large parameters. A very good agreement was found between approximate and numerical solution, which proves that OVM is very efficient and accurate.

Effect of Quantum Inhomogeneity Corrections in Semi-Statistical Thomas–Fermi Calculations for Atoms

1974 ◽  
Vol 52 (19) ◽  
pp. 1926-1932 ◽  
Author(s):  
J. A. Stauffer ◽  
J. W. Darewych
Keyword(s):  
Dirac Equation ◽  
Variational Method ◽  
Quantum Correction ◽  
Correction Term ◽  
Approximate Solutions ◽  
The Other ◽  
Gradient Term ◽  
Gradient Expansion ◽  
Thomas Fermi ◽  
Correction Terms

Approximate solutions to the Thomas–Fermi equation with so-called 'quantum correction terms' have been obtained by the use of a variational method. The results for krypton support the conclusions of Tomishima and Yonei that the coefficient of the gradient term should be 9/5 of the value derived by Kirzhnits. On the other hand, when the use of these equations is restricted to a region of the atom where the gradient expansion of Kirzhnits might be expected to be valid, the Kirzhnits value of the constant gives the better results, but the best results are obtained with no correction term at all (i.e. with the Thomas–Fermi–Dirac equation).

A Simple Approximation for Plate Deflection Problems

1955 ◽  
Vol 59 (539) ◽  
pp. 778-780
Author(s):  
R. K. Kaul ◽  
V. Cadambe
Keyword(s):  
Variational Method ◽  
Approximate Solutions ◽  
Simultaneous Equations ◽  
Simple Approximation ◽  
Elastic Plates ◽  
Series Form ◽  
Thin Elastic Plates ◽  
Plate Deflection ◽  
Made In

The problem of bending of thin elastic plates has interested mathematicians and engineers for many years. Various methods have been put forward for determining the approximate solutions among which the variational method of Ritz is most common. This procedure leads to a system of linear simultaneous equations, the solution of which is often quite tedious, although generally by taking a few rows and columns quite accurate results are obtained. The purpose of this note is to show that by using bar eigen-functions for plates which are either clamped, or when some edges are freely supported, certain simplifications can be made in the process and the difficulty of solving simultaneous equations can be avoided; and that the deflection surface of the plate can be derived directly in a series form.

Numerical Study of the Structure and Thermodynamics of Colloidal Suspensions by Variational Method

2014 ◽  
pp. 12-26
Author(s):  
Khalid Elhasnaoui ◽  
◽  
A. Maarouf ◽  
M. Badia ◽  
M. Benhamou ◽  
...  
Keyword(s):  
Variational Method ◽  
Numerical Study ◽  
Colloidal Suspensions

Interactive Graphics for the Analysis of Tires

1985 ◽  
Vol 13 (3) ◽  
pp. 127-146 ◽  
Author(s):  
R. Prabhakaran
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Approximate Solutions ◽  
Interactive Graphics ◽  
Element Analysis ◽  
Analysis Process ◽  
Physical Problems ◽  
The Finite Element Analysis ◽  
Analysis Computer ◽  
Discretization Technique

Abstract The finite element method, which is a numerical discretization technique for obtaining approximate solutions to complex physical problems, is accepted in many industries as the primary tool for structural analysis. Computer graphics is an essential ingredient of the finite element analysis process. The use of interactive graphics techniques for analysis of tires is discussed in this presentation. The features and capabilities of the program used for pre- and post-processing for finite element analysis at GenCorp are included.

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