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

Abstract

Fourth order problems, with the differential equation y (4) − (gy ′ ) ′ = λ 2 y, where g ∈ C 1 [0, a] and 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 ′ ) ′ = 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.