Applications to Partial Differential Equations

In this chapter, the methods developed previously are applied to partial differential equations. The plan is the same as for the cases of ordinary differential equations discussed earlier. First, the very simplest case is discussed, in which a singular perturbation problem arises. This is a second-order equation which becomes a first-order one in the limit ε → 0. Following this, various more complicated physical examples of singular perturbations and boundary-layer theory are discussed. Next, the ideas of matching and inner and outer expansions are applied in some problems that are analogous to the singular boundary problems of Section 2.7. The final section deals with multiple variable expansions for partial differential equations, and several applications dealing with different aspects of the procedure are discussed.