The method of matched asymptotic expansions

The method of matched asymptotic expansions can be used to solve singular perturbation problems. One speaks of a singular perturbation problem when a straightforward perturbation solution is not uniformly valid through the interval or field. In these cases the solution is composed of two or more individual asymptotic expansions which must be properly linked together by a matching procedure. The method can be considered a generalization of Prandtl's boundary layer theory where the two expansions refer to the outer inviscid flow and to the boundary layer flow. The method of matched asymptotic expansions is described using illustrative examples of problems that arise in fluid dynamics. For simplicity, problems were chosen where the mathematical analysis leads in most cases to ordinary differential equations containing one or two small perturbation parameters.