Uniform Convergence of Spectral Expansions in the Terms of Root Functions of a Spectral Problem for the Equation of a Vibrating Beam

Ziyatkhan S.Aliyevand Konul F.Abdullayeva

1Department of Mathematical Analysis,Faculty of Mechanics and Mathematics,Baku State University,AZ1148 Baku,Azerbaijan.

2Department of Differential Equations,Institute of Mathematics and Mechanics,Azerbaijan National Academy of Sciences,AZ1141 Baku,Azerbaijan.

3Department of Mathematical Analysis and Theory of Functions,Faculty of Mathematics,Sumgait State University,Sumgait AZ5008,Azerbaijan.

Abstract.In this paper we consider a spectral problem which describes bending vibrations of a homogeneous rod,in cross-sections of which the longitudinal force acts,the left end of which is fixed rigidly and on the right end is concentrated an elastically fixed load.We study the uniform convergence of spectral expansions in terms of root functions of this problem.

Keywords:Ordinary differential equations of fourth order,bending vibrations of a homogeneous rod,root functions,uniform convergence of spectral expansions.

1 Introduction

The small bending vibrations of a homogeneous rod(Euler-Bernoulli beam)of length L and of constant rigidity,in cross sections of which the longitudinal force acts,is described by the equation[7,Ch.8,Section 5,formula(84)]

where U(X,t)is a flexure of the current point of axis of the rod,EJ is the flexural rigidity of the rod,˜Q(X)is longitudinal force.

Moreover,we assume that q(x)is an absolutely continuous function on[0,1].

One of the most common methods for solving partial differential equations is method of separation of variables.The justification of this method is based on the convergence of spectral expansions in the systems of root functions of the corresponding eigenvalue problems in various functional spaces.

The spectral properties of problem(1.1a)-(1.1c)in more general form were investigated in the paper[2].The subject of the present paper is the study of uniform convergence of Fourier series expansions for continuous functions in the subsystems of root functions of problem(1.1a)-(1.1c).

The spectral properties(including the basis properties of the root functions in the space Lp(0,1),1

The structure of this paper is as follows.In Section 2 we give some spectral properties of problem(1.1a)-(1.1c)which are necessary in the future.In Section 3 we refine asymptotic formulas for the eigenvalues and eigenfunctions of problems(1.1a)-(1.1b)(with q(x)≡0),y(1)=0,and(1.1a)-(1.1c),respectively.In Section 4,we establish a sufficient condition,as well as a necessary and sufficient condition,so that the Fourier series expansions of continuous functions to converge uniformly in the subsystems of root functions of the boundary value problem(1.1a)-(1.1c).

2 Preliminary

3 Refined asymptotic formulas for eigenvalues and eigenfunctions of problems(1.1a),(1.1b),(2.1)with q≡0,and(1.1a)-(1.1c)

4 Uniform convergence Fourier series of continuous function in the system of root functions of the boundary value problem(1.1a)-(1.1c)

Acknowledgments

The authors thank the reviewers for their valuable remarks and suggestions,which contributed to a significant improvement in the presentation of the text of the article and the obtained results.