Review of Pertubation Methods

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American Mathematical Society





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Differential equations can be divided into those that can be solved and those that cannot. The first class is nearly exhausted by a sophomore "cookbook" differential equations course. The study of the second class is roughly divided into three approaches: qualitative, numerical, and asymptotic. The qualitative approach (which, at least for initial value problems, more or less coincides with "dynamical systems theory") gives up the attempt to find solutions and instead seeks to describe the behavior of the solutions. (In many cases this is what one wants the solutions for anyway.) The numerical approach obtains approximate solutions in the form of tables or graphs. The asymptotic approach, otherwise known as perturbation theory, also looks for approximate solutions but obtains them as formulas. When these formulas are simple enough to comprehend, they can reveal a great deal about the solution. At the simplest level, for instance, an approximate formula for a periodic solution can immediately show the influence of each variable upon the period and amplitude. At a deeper level, it is in the very nature of an asymptotic solution that its terms are sorted into orders of importance. This forces the mathematician into a style of thinking that is reminiscent of pragmatic common sense: when faced with a complicated problem, one asks which features of the problem are most important and attempts to incorporate them into the solution first. In this way, guesswork (more politely known as heuristics) comes to play an important role in the construction of solutions, and proofs of validity (that is, proof of error bounds) get pushed to the end. Often, because the mind-set needed for heuristics differs from that needed for proofs, the proofs get ignored altogether. For many authors, if a solution "looks" asymptotic (that is, if it is "uniformly ordered", as defined below) and agrees well enough with numerical solutions, then it is good enough.


This review is published as Review of Pertubation Methods by E. J. Hinch, BAMS 28, 1993, 418-421. Posted with permission.

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American Mathematical Society



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