Low-frequency asymmetric vibrations of a tn shell with sign-changing curvature.

Abstract

Most problems in Applied Mathematics involving difficulties such as nonlinear governing equations and boundary conditions, variable coefficients and complex boundary shapes preclude exact solutions. Consequently exact solutions are approximated with ones using numerical techniques, analytical techniques or a combination of both. We need to obtain some

insight into the character ofthe solutions and their dependence on certain parameters. Often one or more of the parameters becomes either very large or very small. Typically these are very difficult situations to treat by straight-forward numerical procedure. The analytical method that can provide an accurate approximation is by asymptotic expansions.

This thesis is largely influenced by Professor M.B.Petrov andP.E. Tovstik[12]paper, which examines the role of turning points on shells of revolution with low frequency vibrations and how the resulting solutions behave.

The thesis focuses on the general idea introduced by Langer, where he realized that any attempt to express the asymptotic expansions of the solutions of turning point problems in

terms of elementary functions must fail in regions containing the turning point. A uniformly valid expansion must be expressed in terms of the solution of non-elementary functions which have the same qualitative features as the equation, for example, Airy equations and the exploration of the shell of revolution with emphasis on the negative Gaussian curvature region where the instability occurs due to low frequency vibrations.