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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.