Strategies for treating equations with multiple perturbation parameters

Abstract

Differential equations having more than one small parameter are considered. One of the widespread methods is to express all the small parameters in terms of one small parameter and construct a perturbation expansion in terms of this single parameter. The other approach is to employ expansions containing several small parameters. Both approaches are discussed on example problems and some specific guidelines to follow are given depending on the nature of the problem. A third option which is rarely employed is also discussed in which one parameter is enough to simplify the equation, the other small parameter(s) are assumed to be not small although they are small and hence a single perturbation expansion is sufficient to construct the solution. Example equations from nonlinear dynamics as well as boundary layer type equations are treated to exploit the ideas.