Differential Equations

, Volume 55, Issue 10, pp 1336–1348 | Cite as

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

  • Sh. G. KasimovEmail author
  • U. S. MadrakhimovEmail author
Partial Differential Equations


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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© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Mirzo Ulugbek National University of UzbekistanTashkentUzbekistan

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