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, where gin 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')'=lambda ^2y 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.