Complementary variational principles for free molecular channel flow

1979 ◽  
Vol 82 (3-4) ◽  
pp. 211-223 ◽  
Author(s):  
R. J. Cole
Keyword(s):  
Integral Equations ◽  
Channel Flow ◽  
Lower Bounds ◽  
Variational Principles ◽  
Transmission Probability ◽  
Numerical Calculations ◽  
Upper And Lower Bounds ◽  
Gas Flows ◽  
Rectangular Duct ◽  
Molecular Channel

SynopsisA collisionless gas flows through a finite rectangular duct which reflects molecules diffusely. The transmission probability Q of the duct involves the solution of a pair of coupled integral equations. Complementary variational principles have been employed which supply upper and lower bounds to Q. Numerical calculations have been made for a variety of duct shapes and compared, where appropriate, to those of other authors.

Complementary variational principles in neutron diffusion theory

1967 ◽  
Vol 298 (1452) ◽  
pp. 97-101 ◽  
Keyword(s):  
Lower Bounds ◽  
Variational Principles ◽  
Diffusion Theory ◽  
Upper And Lower Bounds ◽  
Neutron Diffusion ◽  
Diffusion In Solids ◽  
Absorption Probability

Complementary variational principles associated with neutron diffusion in solids are presented. The resulting formulae are used to derive new expressions which provide upper and lower bounds for the absorption probability.

Growth rates of bending KdV solitons

1982 ◽  
Vol 28 (3) ◽  
pp. 469-484 ◽  
Author(s):  
E. W. Laedke ◽  
K. H. Spatschek
Keyword(s):  
Growth Rate ◽  
Lower Bounds ◽  
Acoustic Waves ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Type Equation ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Upper And Lower Bounds ◽  
Ion Acoustic Waves

Nonlinear ion-acoustic waves in magnetized plasmas are investigated. In strong magnetic fields they can be described by a Korteweg-de Vries (KdV) type equation. It is shown here that these plane soliton solutions become unstable with respect to bending distortions. Variational principles are derived for the maximum growth rate γ as a function of the transverse wavenumber k of the perturbations. Since the variational principles are formulated in complementary form, the numerical evaluation yields upper and lower bounds for γ. Choosing appropriate test functions and increasing the accuracy of the computations we find very close upper and lower bounds for the γ(k) curve. The results show that the growth rate peaks at a certain value of k and a cut-off kc exists. In the region where the γ(k) curve was not predicted numerically with high accuracy, i.e. near the cut-off, we find very precise analytical estimates. These findings are compared with previous results. For k≥kc, stability with respect to transverse perturbations is proved.

scholarly journals Semidefinite Optimization Providing Guaranteed Bounds on Linear Functionals of Solutions of Linear Integral Equations with Smooth Kernels

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Guangming Zhou ◽  
Chao Deng ◽  
Kun Wu
Keyword(s):  
Integral Equations ◽  
Lower Bounds ◽  
Semidefinite Optimization ◽  
Upper And Lower Bounds ◽  
Optimization Approach ◽  
Linear Functionals ◽  
Smooth Kernels ◽  
Smooth Kernel ◽  
Linear Integral ◽  
Linear Integral Equations

Based on recent progress on moment problems, semidefinite optimization approach is proposed for estimating upper and lower bounds on linear functionals defined on solutions of linear integral equations with smooth kernels. The approach is also suitable for linear integrodifferential equations with smooth kernels. Firstly, the primal problem with smooth kernel is converted to a series of approximative problems with Taylor polynomials obtained by expanding the smooth kernel. Secondly, two semidefinite programs (SDPs) are constructed for every approximative problem. Thirdly, upper and lower bounds on related functionals are gotten by applying SeDuMi 1.1R3 to solve the two SDPs. Finally, upper and lower bounds series obtained by solving two SDPs, respectively infinitely approach the exact value of discussed functional as approximative order of the smooth kernel increases. Numerical results show that the proposed approach is effective for the discussed problems.

Complementary variational principles in perturbation theory

1968 ◽  
Vol 303 (1475) ◽  
pp. 503-509 ◽  
Keyword(s):  
Perturbation Theory ◽  
Lower Bound ◽  
Harmonic Oscillator ◽  
Lower Bounds ◽  
Upper Bound ◽  
Variational Principles ◽  
Upper And Lower Bounds ◽  
Quantum Mechanical ◽  
Simple Application ◽  
Mechanical Perturbation

Complementary upper and lower bounds are derived for second-order quantum-mechanical perturbation energies. The upper bound is equivalent to that of Hylleraas. The lower bound appears to be new, but reduces to that of Prager & Hirschfelder if a certain constraint is applied. A simple application to a perturbed harmonic oscillator is presented.

scholarly journals Potential of several arbitrarily located disks

1988 ◽  
Vol 29 (3) ◽  
pp. 342-351 ◽  
Author(s):  
V. I. Fabrikant
Keyword(s):  
Integral Equations ◽  
Lower Bounds ◽  
Electrostatic Field ◽  
Upper And Lower Bounds ◽  
Total Charge ◽  
Algebraic Equations ◽  
Geometric Characteristics ◽  
Linear Algebraic Equations ◽  
Integral Characteristics ◽  
Circular Disks

AbstractThe electrostatic field of a set of arbitrarily located circular disks is considered. A set of governing integral equations is derived by a new method. It is shown that some integral characteristics can be found without solving the integral equations. The upper and lower bounds for the total charge are found from a set of linear algebraic equations whose coefficients are defined by simple geometric characteristics of the system. Examples considered show sufficient sharpness of the estimations.

Extremum principles for electrostatic and magnetostatic problems

1969 ◽  
Vol 66 (2) ◽  
pp. 433-436 ◽  
Author(s):  
A. M. Arthurs ◽  
P. D. Robinson
Keyword(s):  
Lower Bounds ◽  
Variational Principles ◽  
Upper And Lower Bounds ◽  
Adjoint Operators

AbstractComplementary variational principles are derived for a class of electrostatic and magnetostatic problems using the pairs of adjoint operators (grad, – div) and (curl, curl). This theory unifies the principles of Dirichlet and Thomson in electrostatics and of Schrader in magnetostatics. The results are illustrated by deriving upper and lower bounds for the capacity of a surface, and it is shown how such bounds can be systematically improved by Ritz procedures.

Upper and lower bounds for the torsional stiffness of a prismatic bar

1972 ◽  
Vol 328 (1573) ◽  
pp. 295-299 ◽  
Keyword(s):  
Lower Bound ◽  
Cross Section ◽  
Lower Bounds ◽  
Upper Bound ◽  
Variational Principles ◽  
Circular Cross Section ◽  
Torsional Stiffness ◽  
Upper And Lower Bounds ◽  
Steady Creep

Upper and lower bounds for the torsional stiffness of a prismatic bar in steady creep are derived in a unified manner from the theory of complementary variational principles. The lower bound is known in the literature, but the upper bound appears to be new. The results are illustrated with calculations for a bar with circular cross-section.

Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds

1989 ◽  
Vol 206 ◽  
pp. 25-46 ◽  
Author(s):  
Jacob Rubinstein ◽  
S. Torquato
Keyword(s):  
Lower Bounds ◽  
Variational Principles ◽  
Mathematical Formulation ◽  
Upper And Lower Bounds ◽  
Random Boundary ◽  
Slow Viscous Flow ◽  
Fluid Permeability ◽  
Different Types ◽  
Rigorous Bounds ◽  
Random Porous Medium

The problem of the slow viscous flow of a fluid through a random porous medium is considered. The macroscopic Darcy's law, which defines the fluid permeability k, is first derived in an ensemble-average formulation using the method of homogenization. The fluid permeability is given explicitly in terms of a random boundary-value problem. General variational principles, different to ones suggested earlier, are then formulated in order to obtain rigorous upper and lower bounds on k. These variational principles are applied by evaluating them for four different types of admissible fields. Each bound is generally given in terms of various kinds of correlation functions which statistically characterize the microstructure of the medium. The upper and lower bounds are computed for flow interior and exterior to distributions of spheres.

Sign in / Sign up

Export Citation Format

Share Document