Determination of Eigenvalues and Eigenfunctions

2005 ◽  
pp. 87-140
Keyword(s):  
Eigenvalues And Eigenfunctions

Eigenvalues and eigenfunctions of finite-difference operators

1961 ◽  
Vol 57 (3) ◽  
pp. 532-546 ◽  
Author(s):  
W. G. Bickley ◽  
John McNamee
Keyword(s):  
Continuous System ◽  
Successive Approximations ◽  
Difference Operators ◽  
Difference System ◽  
Eigenvalues And Eigenfunctions ◽  
First Approximation ◽  
The Difference ◽  
The Given ◽  
Moderate Length

Numerical solution of differential and integral equations is concerned mainly with the determination of the wanted function at a finite number of discrete points which are, in general, uniformly spaced. A first approximation to the solution can be obtained if the given differential or integral system is replaced by a difference system. Any differential or integral operator can be expressed as an infinite series of difference operators and the difference system of the first approximation is obtained by neglecting all but the first few terms of the infinite expansions. We shall distinguish two processes for improving the approximation: the first uses a tabular interval of moderate length but the approximation to the given continuous system is improved by including in the difference system as many terms of the infinite expansions as are necessary or practicable; the second process uses an unvarying difference system of low accuracy, but the tabular interval is reduced in successive approximations, the process being continued until two successive approximations agree within the accuracy required. We regard these processes as essentially distinct. If the solutions obtained by the two processes approach limits, these limits need not coincide.

New approach to the determination of eigenvalues and eigenfunctions for a relativistic two‐fermion equation

1993 ◽  
Vol 34 (6) ◽  
pp. 2089-2106 ◽  
Author(s):  
A. O. Barut ◽  
A. J. Bracken ◽  
S. Komy ◽  
N. Ünal
Keyword(s):  
Eigenvalues And Eigenfunctions ◽  
New Approach

scholarly journals A New Framework for the Determination of the Eigenvalues and Eigenfunctions of the Quantum Harmonic Oscillator

2021 ◽  
Vol 13 (6) ◽  
pp. 20
Author(s):  
Francis T. Oduro ◽  
Amos Odoom
Keyword(s):  
Harmonic Oscillator ◽  
The Other ◽  
Erential Equation ◽  
Quantum Harmonic Oscillator ◽  
Eigenvalues And Eigenfunctions ◽  
Energy Eigenvalues ◽  
Alternative Approach ◽  
Schmidt Orthogonalization ◽  
New Framework

This study was designed to obtain the energy eigenvalues and the corresponding Eigenfunctions of the Quantum Harmonic oscillator through an alternative approach. Starting with an appropriate family of solutions to a relevant linear di erential equation, we recover the Schr¨odinger Equation together with its eigenvalues and eigenfunctions of the Quantum Harmonic Oscillator via the use of Gram Schmidt orthogonalization process in the usual Hilbert space. Significantly, it was found that there exists two separate sequences arising from the Gram Schmidt Orthogonalization process; one in respect of the even eigenfunctions and the other in respect of the odd eigenfunctions.

scholarly journals Chemical Engineering Systems Stability

2020 ◽  
pp. 9-16
Author(s):  
Christo Boyadjiev
Keyword(s):  
Theoretical Analysis ◽  
Bifurcation Theory ◽  
Chemical Engineering ◽  
Engineering Systems ◽  
Unified Approach ◽  
Eigenvalues And Eigenfunctions ◽  
The Stability ◽  
Equilibrium Chemical ◽  
Chemical Engineering Systems

A theoretical analysis of the stability of the non-equilibrium chemical engineering systems is presented. A unified approach is proposed for the creation of the mathematical models of the systems that allows the determination of the velocities at which the systems move to their thermodynamic equilibriums and their use for mathematical analysis of systems velocities stability. For this purpose is used mathematical stability theory, evolution (autonomous) equations, bifurcation theory (stable focuses, stable cycles), parameter eigenvalues and eigenfunctions.

scholarly journals Determination of Reynolds shear stress from the turbulent mean velocity profile and spectral characteristics of Orr-Sommerfeld -Squire equations

2021 ◽  
Vol 2056 (1) ◽  
pp. 012017
Author(s):  
R. S. Selim
Keyword(s):  
Tangential Stress ◽  
Reynolds Shear Stress ◽  
Spectral Characteristics ◽  
Eigenvalues And Eigenfunctions ◽  
Mean Velocity ◽  
Wave Resonance ◽  
System Solution ◽  
Waveguide Model ◽  
Sommerfeld Equation

Abstract Theoretically, based on a waveguide model, the expression of the tangential stress is formulated for steady, two-dimensional incompressible fluid flow over a flat plate in turbulent boundary layer. It is dependent on some factors, one of them, the behaviour of the last damping mode eigenvalues, and eigenfunctions, that are deduced from solution Orr-Sommerfeld equation by spectral Chebyshev collocation Method. Verification of the latter method is investigated by comparison the deduced formula of turbulent tangential stress with experimental data. In addition to, weight factors in this expression are connected to define the condition of dynamical system solution for multiple 3-wave resonance. This system is solved numerically, and the dynamic invariant is normalized to obtain the time average of the square modulus harmonic, and sub harmonics amplitudes by theorem Birkhoff-Khinchin. Comparison is made between the time-averaged and the phase average for the square modulus of harmonic, and sub harmonic amplitudes that defined on the unit sphere, in the state of multiple 3-wave resonance.

Galactic orbits of stars

1966 ◽  
Vol 25 ◽  
pp. 93-97
Author(s):  
Richard Woolley
Keyword(s):  
Globular Cluster ◽  
Potential Field ◽  
High Velocity ◽  
Velocity Vector ◽  
Variable Stars ◽  
The Sun ◽  
Proper Motions ◽  
Rr Lyrae ◽  
The Galaxy

It is now possible to determine proper motions of high-velocity objects in such a way as to obtain with some accuracy the velocity vector relevant to the Sun. If a potential field of the Galaxy is assumed, one can compute an actual orbit. A determination of the velocity of the globular clusterωCentauri has recently been completed at Greenwich, and it is found that the orbit is strongly retrograde in the Galaxy. Similar calculations may be made, though with less certainty, in the case of RR Lyrae variable stars.

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