Asymptotic approximation boundary layers, singularly perturbed problems

In this chapter, we will consider the boundary layer function method (or, in short, boundary function method), which gives the possibility of constructing a uniform asymptotic approximation (or, briefly, uniform asymptotics) for the solution u e) of the singularly perturbed problem (i.e., an asymptotic approximation in the whole domain D). 

The three chapters in this volume deal with various aspects of singular perturbations and their numerical solution. The first chapter is concerned with the analysis of some singular perturbation problems that arise in chemical kinetics. In it the matching method is applied to find asymptotic solutions of some dynamical systems of ordinary differential equations whose solutions have multiscale time dependence. The second chapter contains a comprehensive overview of the theory and application of asymptotic approximations for many different kinds of problems in chemical physics, with boundary and interior layers governed by either ordinary or partial differential equations. In the final chapter the numerical difficulties arising in the solution of the problems described in the previous chapters are discussed. In addition, rigorous criteria are proposed for... 

In [157] the authors present an initial-value methodology for the numerical approximation of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations. These problems have a boundary layer at one end (left or right) point. The techniaque which used by the authors is to reduce the original problem to an asymptotically equivalent first order initial-value problem. This is done with the... 

This system and other problems for singularly perturbed ordinary differential equations will be investigated in Sections II-V. Solutions with boundary and/or interior layers will be considered. Our main goal will be the construction of an approximation to the solution valid outside the boundary (interior) layer as well as within the boundary (interior) layer, that is, so-called uniform approximation in the entire t domain. This approximation will have an asymptotic character. The definition of an asymptotic approximation with respect to a small parameter will be introduced in Section LB. 

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