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Mat. Sb. (N.S.), 1987, Volume 134(176), Number 1(9), Pages 3–27 (Mi msb2637)  

This article is cited in 1 scientific paper (total in 1 paper)

Asymptotics of the solution of the Dirichlet problem for the system of elasticity theory in the exterior of a thin body of revolution

G. V. Zhdanova


Abstract: The asymptotics is found for a solution of the system of equations
$$ A(\partial_x)\mathbf u(x)+\omega^2\rho\mathbf u(x)=0,\quad x\in D_\varepsilon,\qquad \mathbf u(x)=\mathbf f(x),\quad x\in S_\varepsilon, $$
of steady-state elastic vibrations of an isotropic medium. Here $x\in\mathbf R^3$, $\varepsilon>0$ is a small parameter, $S_\varepsilon$ is a bounded closed surface given in spheroidal coordinates by the equation $\xi=1+\varepsilon g(\eta,\varepsilon)$, and $D_\varepsilon$ is the exterior of $S_\varepsilon$. The vector-valued function $\mathbf u(x)$ satisfies a radiation condition. The asymptotics of the solution of the problem is found up to $O(\varepsilon^m)$, $m>0$ arbitrary, in the case where the boundary condition does not depend on the polar angle $\varphi$, and up to $O(\varepsilon^2\ln\varepsilon)$ in the case of boundary conditions which are not axially symmetric. The formulas obtained are valid everywhere near the body (including neighborhoods of the end points) and far from it.
Bibliography: 12 titles.

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English version:
Mathematics of the USSR-Sbornik, 1989, 62:1, 1–27

Bibliographic databases:

UDC: 531.262
MSC: Primary 73D30; Secondary 35B40
Received: 30.05.1986

Citation: G. V. Zhdanova, “Asymptotics of the solution of the Dirichlet problem for the system of elasticity theory in the exterior of a thin body of revolution”, Mat. Sb. (N.S.), 134(176):1(9) (1987), 3–27; Math. USSR-Sb., 62:1 (1989), 1–27

Citation in format AMSBIB
\Bibitem{Zhd87}
\by G.~V.~Zhdanova
\paper Asymptotics of the solution of the Dirichlet problem for the system
of elasticity theory in the exterior of a~thin body of revolution
\jour Mat. Sb. (N.S.)
\yr 1987
\vol 134(176)
\issue 1(9)
\pages 3--27
\mathnet{http://mi.mathnet.ru/msb2637}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=912408}
\zmath{https://zbmath.org/?q=an:0679.73013|0658.73003}
\transl
\jour Math. USSR-Sb.
\yr 1989
\vol 62
\issue 1
\pages 1--27
\crossref{https://doi.org/10.1070/SM1989v062n01ABEH003223}


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    This publication is cited in the following articles:
    1. Ivan I. Argatov, Federico J. Sabina, “Acoustic diffraction by a thin soft torus”, Wave Motion, 45:6 (2008), 846  crossref  mathscinet  zmath
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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