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3. FURTHER APPLICATIONS

























          The ideas and techniques developed in  Chapter 2  have  taken us beyond elementary
          applications,  and  as far as  some  methods  that  enable us to  construct  (asymptotic)
          solutions  of a few types  of ordinary differential equation.  The aim now is  to  extend
          these methods, in particular, to partial differential equations. The first reaction to this
          proposal might be that the move from ordinary to partial differential equations is a very
          big step—and it can certainly be argued thus if we compare the solutions, and methods
          of solution, for these two categories of equation. However, in the context of singular
          perturbation theory, this is a misleading position to adopt. Without doubt we must have
          some skills in the methods of solution of partial differential equations (albeit usually in
          a reduced, simplified form), but the fundamental ideas of singular perturbation theory
          are essentially the same as those developed for ordinary differential equations. The only
          adjustment, because the solution will now sit in a domain of two or more dimensions,
          is that an appropriate scaling may apply, for example, in only one direction and not in
          the others,  or in time and not in space.
            In this chapter we will examine some fairly straightforward problems that are repre-
          sented by partial differential equations, starting with an example of a regular problem.
          The approach that we adopt will emphasise how the methods for ordinary differential
          equations carry over directly to partial differential equations. In addition, we will take
          the opportunity  to write a little  more  about  more advanced  aspects of the  solution
          of ordinary differential equations, in part as a preparation for the very powerful and
          general methods  introduced in  Chapter 4.
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