A Prüfer transformation for the equation of the vibrating beam

Banks, D. O.; Kurowski, G. J.

doi:10.1090/S0002-9947-1974-0350111-3

A Prüfer transformation for the equation of the vibrating beam
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by D. O. Banks and G. J. Kurowski PDF

Trans. Amer. Math. Soc. 199 (1974), 203-222 Request permission

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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