Formal Perturbation Methods

Abstract

The discussion in the preceding chapters, in particular spectral perturbation theory, was of a mathematically rigorous character. Unfortunately, many systems of the real world involve perturbations which do no fit within the framework of Chapter V. Typically, this occurs in boundary value problems involving a differential equation with a coefficient e tending to zero in front of the higher order derivatives. As e becomes 0, the order of the equation decreases and some boundary conditions are violated so that the perturbation exhibits singularities at the corresponding boundaries. Physical intuition led theorists and engineers concerned with mechanics to devise formal (heuristic) methods for studying certain types of such perturbations. One of these is the method of matched asymptotic expansions. This technique allows us to study problems where the perturbed solution has different structures in different regions. The typical example is the boundary layer generated by small viscosity; indeed, viscosity effects are only pronounced in the vicinity of a wall and the asymptotic expansions for small viscosity assume different forms in the inner and outer (to the boundary layer) regions. The compatibility of these two regions is expressed by suitable “matching” conditions. This technique provides formal asymptotic expansions in very general perturbation problems, but unfortunately it is mostly based on physical (or geometrical) intuition.