Lie theory and separation of variables. 9. Orthogonal R-separable coordinate systems for the wave equation ψtt−Δ2ψ=0

1976 ◽  
Vol 17 (3) ◽  
pp. 331 ◽  
Author(s):  
E. G. Kalnins
Keyword(s):  
Wave Equation ◽  
Separation Of Variables ◽  
Lie Theory ◽  
Coordinate Systems

The wave equation, O(2, 2), and separation of variables on hyperboloids

1978 ◽  
Vol 79 (3-4) ◽  
pp. 227-256 ◽  
Author(s):  
E. G. Kalnins ◽  
W. Miller
Keyword(s):  
Wave Equation ◽  
Harmonic Analysis ◽  
Separation Of Variables ◽  
Beltrami Operator ◽  
Eigenvalue Equation ◽  
Coordinate Systems ◽  
Group Manifold ◽  
Orthogonal Systems

SynopsisWe classify group-theoretically all separable coordinate systems for the eigenvalue equation of the Laplace-Beltrami operator on the hyperboloid = 1, finding 71 orthogonal and 3 non-orthogonal systems. For a number of cases the explicit spectral resolutions are worked out. We show that our results have application to the problem of separation of variables for the wave equation and to harmonic analysis on the hyperboloid and the group manifold SL(2, R). In particular, most past studies of SL(2, R) have employed only 6 of the 74 coordinate systems in which the Casimir eigenvalue equation separates.

scholarly journals Lie theory and separation of variables. 4. The groups SO (2,1) and SO (3)

1974 ◽  
Vol 15 (8) ◽  
pp. 1263-1274 ◽  
Author(s):  
E. G. Kalnins ◽  
Willard Miller
Keyword(s):  
Separation Of Variables ◽  
Lie Theory

scholarly journals Shapovalov Wave-Like Spacetimes

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1372 ◽  
Author(s):  
Konstantin Osetrin ◽  
Evgeny Osetrin
Keyword(s):  
Jacobi Equation ◽  
Eikonal Equation ◽  
Separation Of Variables ◽  
Complete Classification ◽  
Complete Separation ◽  
Einstein Equations ◽  
Coordinate Systems ◽  
Test Particles ◽  
Hamilton Jacobi Equation

A complete classification of space-time models is presented, which admit the privileged coordinate systems, where the Hamilton–Jacobi equation for a test particle is integrated by the method of complete separation of variables with separation of the isotropic (wave) variable, on which the metric depends (wave-like Shapovalov spaces). For all types of Shapovalov spaces, exact solutions of the Einstein equations with a cosmological constant in vacuum are found. Complete integrals are presented for the eikonal equation and the Hamilton–Jacobi equation of motion of test particles.

A Perturbation Method for Low-Frequency Fluid-Structure Interaction Problems

1973 ◽  
Vol 40 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Y. K. Lou
Keyword(s):  
Wave Equation ◽  
Perturbation Method ◽  
Electromagnetic Scattering ◽  
Perturbation Methods ◽  
Separation Of Variables ◽  
Fluid Structure Interaction ◽  
Low Frequency ◽  
Approximate Solutions ◽  
Fluid Structure ◽  
Structure Interaction

Perturbation methods have been used for electromagnetic scattering and diffraction problems in recent years. A similar method suitable for low-frequency fluid-structure interaction problems is presented. The essence of the method lies in the fact that approximate solutions for fluid-structure interaction problems can be obtained from a set of Poisson’s equations, rather than from the reduced wave equation. The method is particularly useful for those problems where the Poisson’s equation may be solved by the method of separation of variables while the reduced wave equation cannot. As an illustrative example, the vibrations of a submerged spherical shell is studied using the perturbation method and the accuracy of the method is demonstrated.

Separation of variables in the wave equation. Sets of the type (1.1) and Schr�dinger equation

1991 ◽  
Vol 34 (2) ◽  
pp. 122-126
Author(s):  
V. G. Bagrov ◽  
B. F. Samsonov ◽  
A. V. Shapovalov
Keyword(s):  
Wave Equation ◽  
Separation Of Variables ◽  
Dinger Equation
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