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Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domainMethods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eignevalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: the interior of a spherical triangle, and the exterior of a great circle arc extending for less than pi radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.
Document ID
19780027671
Acquisition Source
Legacy CDMS
Document Type
Reprint (Version printed in journal)
External Source(s)
Authors
Walden, H.
(NASA Goddard Space Flight Center Greenbelt, Md., United States)
Date Acquired
August 9, 2013
Publication Date
October 1, 1977
Publication Information
Publication: Journal of Engineering Mathematics
Volume: 11
Subject Category
Numerical Analysis
Accession Number
78A11580
Distribution Limits
Public
Copyright
Other

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