Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On the positive square root of the fourth derivative operator


Author: D. L. Russell
Journal: Quart. Appl. Math. 46 (1988), 751-773
MSC: Primary 47E05; Secondary 47A60, 70J99, 73D30, 73K12
DOI: https://doi.org/10.1090/qam/973388
MathSciNet review: 973388
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Abstract: It is only for a special subset of the natural boundary conditions for the operator

$\displaystyle Aw = \frac{{{d^4}w}}{{d{x^4}}}$

that its positive square root is the negative second derivative operator. In this paper we develop a procedure for parametric description of all natural boundary conditions, we show which ones admit $ {A^{1/2}}$ in the form just noted, and we show that in the other cases

$\displaystyle - {D^2}w \equiv - \frac{{{d^2}w}}{{d{x^2}}} = \left[ {I + P} \right]{A^{1/2}}w$

where $ P$ is a bounded, but in general not compact, operator on the Hilbert space $ {L^2}\left[ {0, \pi } \right]$. Possible applications to the theory of the partial differential equation

$\displaystyle \rho \frac{{{\partial ^2}w}}{{\partial {t^2}}} - 2\gamma \frac{{{... ...}{{\partial t\partial {x^2}}} + EI\frac{{{\partial ^4}w}}{{\partial {x^4}}} = 0$

are indicated.

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DOI: https://doi.org/10.1090/qam/973388
Article copyright: © Copyright 1988 American Mathematical Society


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