This article is cited in 30 papers

Differential Equations and Mathematical Physics

Fluctuations of a beam with clamped ends

K. B. Sabitov

Samara State University of Architecture and Construction, Samara, 443001, Russian Federation

Abstract: In this paper we study the initial problem for the equation of a beam with clamped ends. Uniqueness, existence and stability theorems are proved for the problem in the classes of regular and generalized solutions. Solution of the initial-boundary value problem is constructed in the form of a series in the system of eigenfunctions of one-dimensional spectral problem. We found the spectral problem eigenvalues as roots of the transcendental equation and the corresponding system of eigenfunctions. It is shown that the system of eigenfunctions is orthogonal and complete in $L_2$. On the basis of the completeness of the eigenfunctions the uniqueness theorem for the initial-boundary value problem for the equation of the beam is obtained. The generalized solution is defined as the limit of a sequence of regular solutions of the mean-square norm on the space variable.

Keywords: equation beams, initial-boundary value problem, spectral method, uniqueness, existence, series resistance.

UDC: 517.951; 517.958:531.12

MSC: 35G16

Original article submitted 06/II/2015
revision submitted – 26/III/2015

DOI: 10.14498/vsgtu1406

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