Stability of a flexible missile and asymptotics of the eigenvalues of fourth order boundary value problems

Abstract

Fourth order problems with the differential equation $$y^{(4)}-(gy')'=\lambda ^2y$$ y ( 4 ) - ( g y ′ ) ′ = λ 2 y , where $$g\in C^1[0,a]$$ g ∈ C 1 [ 0 , a ] and $$a>0$$ a > 0 , occur in engineering on stability of elastic rods. They occur as well in aeronautics to describe the stability of a flexible missile. Fourth order Birkhoff regular problems with the differential equation $$y^{(4)}-(gy')'=\lambda ^2y$$ y ( 4 ) - ( g y ′ ) ′ = λ 2 y and eigenvalue dependent boundary conditions are considered. These problems have quadratic operator representations with non-self-adjoint operators. The first four terms of the asymptotics of the eigenvalues of the problems as well as those of the eigenvalues of the problem describing the stability of a flexible missile are evaluated explicitly.