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Initial-Boundary Value Problem for the Beam Vibration Equation in the Multidimensional Case

  • Partial Differential Equations
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Abstract

In the multidimensional case, we study the problem with initial and boundary conditions for the equation of vibrations of a beam with one end clamped and the other hinged. An existence and uniqueness theorem is proved for the posed problem in Sobolev classes. A solution of the problem under consideration is constructed as the sum of a series in the system of eigenfunctions of a multidimensional spectral problem for which the eigenvalues are determined as the roots of a transcendental equation and the system of eigenfunctions is constructed. It is shown that this system of eigenfunctions is complete and forms a Riesz basis in Sobolev spaces. Based on the completeness of the system of eigenfunctions, a theorem about the uniqueness of a solution to the posed initial-boundary value problem is stated.

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Authors and Affiliations

  1. Mirzo Ulugbek National University of Uzbekistan, Tashkent, 100174, Uzbekistan

    Sh. G. Kasimov & U. S. Madrakhimov

Authors
  1. Sh. G. Kasimov
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  2. U. S. Madrakhimov
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Correspondence to Sh. G. Kasimov or U. S. Madrakhimov.

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Russian Text © The Author(s), 2019, published in Differentsial’nye Uravneniya, 2019, Vol. 55, No. 10, pp. 1379–1391.

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Kasimov, S.G., Madrakhimov, U.S. Initial-Boundary Value Problem for the Beam Vibration Equation in the Multidimensional Case. Diff Equat 55, 1336–1348 (2019). https://doi.org/10.1134/S0012266119100094

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  • DOI: https://doi.org/10.1134/S0012266119100094

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