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188 4. The method of multiple scales



          Now that we have seen the method of multiple scales applied to boundary-layer prob-
          lems, it should be evident that this provides the simplest and most direct approach
          to the solution  of this type of problem. Not  only do we avoid the  need to  match—
          although, of course, the  correct selection of fast and slow variables is essential—but
          we also generate a composite expansion directly, which may be used as the basis for
          numerical or graphical representations of the solution. Further examples are given in
          exercises Q4.28–4.35.
            This  concludes our presentation of the method of multiple  scales, and  is  the  last
          technique that we shall describe. In the final chapter, the plan is to work through a
          number of examples  taken from various branches of the mathematical,  physical and
          related sciences, grouped by subject area. These, we hope, will show how our various
          techniques are relevant and important. It is to be hoped that those readers with interests
          in particular fields will find something to excite their curiosity and to point the way
          to the solution of problems that might otherwise appear intractable.


          FURTHER READING
          Most of the texts that we have mentioned earlier discuss the method of multiple scales,
          to  a greater or a lesser extent. Two  texts, in particular,  give a good overview of the
          subject: Nayfeh (1973), in which a number of problems are investigated in many dif-
          ferent ways  (including variants  of the  method of multiple scales),  and Kevorkian  &
          Cole (1996)  which provides an  up-to-date and wide-ranging discussion. The  appli-
          cations to  ordinary differential equations are  nicely presented in both  Smith  (1985)
          and O’Malley (1991),  where a lot of technical detail is included, as well as a careful
          discussion of asymptotic correctness. Holmes (1995) provides an excellent account of
          both the method of multiple scales and the WKB method. This latter is also discussed
          in Wasow (1965) and in Eckhaus (1979). Finally, an excellent introduction to the rôle
          of asymptotic methods in the analysis of oscillations (mainly those that are nonlinear)
          can be found in Bogoliubov & Mitropolsky  (1961)—an older text, but a classic that
          can be highly recommended (even though it does not possess an index!).

          EXERCISES
           Q4.1 Nearly linear oscillator I. A weakly nonlinear oscillator is described by the equation




               where a (> 0) is a constant (independent of  and   the initial conditions

               are

               Use the method  of multiple  scales to  find, completely, the  first  term in  a
               uniformly valid asymptotic expansion.  (It is sufficient to  use the time scales
                T =  t and     Explain why your solution fails if a <  0.
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