Higher order solutions to singularly perturbed problems in one dimension

The construction of higher order approximation solutions to singularly perturbed differential equations is our concern in this lecture. Often, in the literature such solutions are given in the form of a formal asymptotic expansion. Usually these expansions are not easy-to compute, at least, beyond the first term. The current lecture presents some results obtained by the author about some families of singularly perturbed problems which provide easy to compute higher order approximation solutions. We provide uniform approximations which are valid throughout the geometric domain of study including the boundary layers.

The background theory is that of Hilbert spaces. The main tool built and used to reach our goal is a corrector. We start by locating precisely the boundary layer. Then we use the well-known WKB expansion to derive the multiscale equations leading to a corrector. The corrector attached to a qth order approximation is an analytical function obtained from q equations. From a usual outer expansion the action of the corrector helps achieving our goal which is to construct an easy-to-compute uniform approximation of any arbitrary and prescribed order. This uniform approximation is valid inside and outside the boundary layer