Multiple Scales
When one uses matched asymptotic expansions, the solution is constructed in different regions that are then patched together to form a composite expansion. The method of multiple scales differs from this approach in that it essentially starts with a gener
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Multiple Scales
3.1 Introduction When one uses matched asymptotic expansions, the solution is constructed in different regions that are then patched together to form a composite expansion. The method of multiple scales differs from this approach in that it essentially starts with a generalized version of a composite expansion. In doing this, one introduces coordinates for each region (or layer), and these new variables are considered to be independent of one another. A consequence of this is that what may start out as an ordinary differential equation is transformed into a partial differential equation. Exactly why this helps to solve the problem, rather than make it harder, will be discussed as the method is developed in this chapter. The history of multiple scales is more difficult to delineate than, say, boundary-layer theory. This is because the method is so general that many apparently unrelated approximation procedures are special cases of it. One might argue that the procedure got its start in the first half of the nineteenth century. For example, Stokes (1843) used a type of coordinate expansion in his calculations of fluid flow around an elliptic cylinder. Most of these early efforts were limited, and it was not until the latter half of the nineteenth century that Poincar´e (1886), based on the work of Lindstedt (1882), made more extensive use of the ideas underlying multiple scales in his investigations into the periodic motion of planets. He found that the approximation obtained from a regular expansion accurately described the motion for only a few revolutions of a planet, after which the approximation becomes progressively worse. The error was due, in part, to the contributions of the second term of the expansion. He referred to this difficulty as the presence of a secular term. To remedy the situation, he expanded the independent variable with the intention of making the approximation uniformly valid by removing the secular term. This idea is also at the heart of the modern version of the method. What Poincar´e was missing was the introduction of multiple
M.H. Holmes, Introduction to Perturbation Methods, Texts in Applied Mathematics 20, DOI 10.1007/978-1-4614-5477-9 3, © Springer Science+Business Media New York 2013
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3 Multiple Scales
independent variables based on the expansion parameter. This step came much later; the most influential work in this regard is by Kuzmak (1959) and Cole and Kevorkian (1963).
3.2 Introductory Example As in the last chapter, we will introduce the ideas underlying the method by going through a relatively simple example. The problem to be considered is to find the function y(t) that satisfies
where
y + εy + y = 0 for 0 < t,
(3.1)
y(0) = 0 and y (0) = 1.
(3.2)
This equation corresponds to an oscillator (i.e., a mass–spring–dashpot) with weak damping, where the time variable has been scaled by the period of the undamped system (Exercise 1.42). This is the classic example used to illustrate multiple scales. Our approach to constructing the approximation comes fro
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