On the Unambiguous Solvability of a Multidimensional Initial-boundary Value Problem for the Beam Oscillation Equation with Nonlocal Boundary Conditions in Sobolev Classes

被引：0

作者：

Sh. G. Kasimov ^{[1
]}

A. P. Koshanov ^{[1
]}

机构：

[1] National University of Uzbekistan, Faculty of Mathematics, Tashkent

来源：

Lobachevskii Journal of Mathematics | 2024年 / 45卷 / 11期

关键词：

beam equation considering its rotational motion during bending; existence; initial-boundary value problem; Riesz basis; series; spectral method; uniqueness;

D O I：

10.1134/S1995080224606404

中图分类号：

学科分类号：

摘要：

Abstract: In the multidimensional case, the problem with initial and nonlocal boundary conditions for the beam oscillation equation, considering its rotational motion during bending, is studied. A theorem of existence and uniqueness of the posed problem in Sobolev classes is proved. The solution to the considered problem is constructed as a sum of a series based on the system of eigenfunctions of a multidimensional spectral problem, for which its eigenvalues are found as roots of a transcendental equation, and the corresponding 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 of uniqueness of the solution to the posed initial-boundary value problem is obtained. © Pleiades Publishing, Ltd. 2024.

引用

页码：5546 / 5558

页数：12

共 13 条

[1]

Tikhonov L.N., Samarskii L.L., Equations of Mathematical Physics, (1999)

[2]

Korenev B.G., Problems of Calculating Beams and Plates on an Elastic Foundation, (1965)

[3]

Krylov L.N., Vibration of Ships, (1965)

[4]

Klebsch L., Theorie der Elasticital Faster, (1862)

[5]

Donkin W.F., Acoustics, (1870)

[6]

Rayleigh L., The Theory of Sound, (1945)

[7]

Sabitov K.B., Vibrations of a beam with closed ends,” Izv. Samar. Tekh. Univ., Ser.: Fiz.-, Mat. Nauki, 19, pp. 311-324, (2015)

[8]

Sabitov K.B., A remark on the theory of initial-boundary value problems for the equation of rods and beams, Differ. Equat, 53, pp. 86-98, (2017)

[9]

Sabitov K.B., Cauchy problem for the beam vibration equation, Differ. Equat, 53, pp. 658-664, (2017)

[10]

Sabitov K.B., Initial-boundary value problems for the of beam vibration equation with allowance for its rotational motion under bending, Differ. Equat, 57, pp. 342-352, (2021)

← 1 2 →