Layer-type Problems. Partial Differential Equations

Abstract

We have seen in Chapter II that even restricting ourselves to a few relatively simple-looking ordinary differential equations we get a great variety of types of expansions when applying singular perturbation techniques. Also, as seen in Chapter II, Section 4, replacing the middle term in eu″ + u _x — u = 0 by the quasi-linear term uu _x greatly increases the variety of solutions for two-point boundary value problems. Obviously, we expect the variety of solutions and the techniques necessary to be very large when we consider partial differential equations.

In this chapter we shall use some terminology and theorems which are standard in the theory of partial differential equations. Since there are many textbooks on this subject no particular reference is given.