The vast majority of differential equations that arise as models for real physical systems cannot be solved directly by analytical methods. Often, the only way to proceed is to use a computer to calculate an approximate, numerical solution. However, if one or more small, dimensionless parameters appear in the differential equation, it may be possible to use an asymptotic method to obtain an approximate solution. Moreover, the presence of a small parameter often leads to a singular perturbation problem, which can be difficult, if not impossible, to solve numerically.
Small, dimensionless parameters usually arise when one physical process occurs much more slowly than another, or when one geometrical length in the problem is much shorter than another. Examples occur in many different areas of applied mathematics, and we will meet several in Chapter 12. As we shall see, dimensionless parameters arise naturally when we use dimensionless variables, which we discussed at the beginning of Chapter 5. Some other examples are:
— Waves on the surface of a body of fluid or an elastic solid, with amplitude a and wavelength λ, are said to be of small amplitude if ∈ = a/λ ≪ 1. A simplification of the governing equations based on the fact that ∈ ≪ 1 leads to a system of linear partial differential equations (see, for example, Billingham and King, 2001). This is an example of a regular perturbation problem, where the problem is simplified throughout the domain of solution.