Introduction

When the governing equations of a system are nonlinear, one of the analytical methods for obtaining the solution is the perturbation technique. In such a situation the system is reduced to a non-dimensional form by referring different physical variables to characteristic scales. The dependent variables are expanded asymptotically in terms of small parameters which occur in a natural way depending on the physics of the problem. It is assumed that each term of the perturbation series is smaller in magnitude than its preceding term throughout the region of interest. If this condition breaks down in any part of the region, the asymptotic expansion is no longer valid. An important reason why perturbation solutions are often not uniformly valid is concerned with the ‘large integrated effect’.