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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A Prüfer transformation for the equation of the vibrating beam


Authors: D. O. Banks and G. J. Kurowski
Journal: Trans. Amer. Math. Soc. 199 (1974), 203-222
MSC: Primary 34C10
MathSciNet review: 0350111
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Abstract: In this paper, the oscillatory properties of the eigenfunctions of an elastically constrained beam are studied. The method is as follows.

The eigenfunction and its first three derivatives are considered as a four-dimensional vector, $ (u,u',pu'',(pu'')')$. This vector is projected onto two independent planes and polar coordinates are introduced in each of these two planes. The resulting transformation is then used to study the oscillatory properties of the eigenfunctions and their derivatives in a manner analogous to the use of the Prüfer transformation in the study of second order Sturm-Liouville systems. This analysis yields, for a given set of boundary conditions, the number of zeros of each of the derivatives, $ u',pu'',(pu'')'$ and the relation of these zeros to the $ n - 1$ zeros of the $ n$th eigenfunction. The method also can be used to establish comparison theorems of a given type.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0350111-3
Keywords: Eigenvalue, eigenfunctions, oscillation theory, fourth order linear differential equations, boundary value problems, vibrations, beams, Prüfer transformation, comparison theorems
Article copyright: © Copyright 1974 American Mathematical Society