# Rigorous results on the method of multiple scales

The method of multiple scales (Scholarpedia) is a technique used to obtain approximate solutions to differential equations, most commonly when some of the more standard approaches to perturbation theory fail due to the appearance of secular terms, i.e., terms that grow unboundedly (over time, in the case of ordinary differential equations).

While this method seems to be useful and quite popular, all the sources I've seen describe the technique/algorithm and demonstrate its use (and observed accuracy) on various problems, but never mention any theorems guaranteeing accuracy. Are there any such theorems? Is there a good source on this topic in the mathematics literature (perhaps under a different title)?

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A number of results similar to what you are looking for were obtained in the literature on asymptotic homogenization. The classic monographs in these area are: Bensoussan, Lions & Papanicolaou (1978) "Asymptotic analysis for periodic structures", Sanchez-Palencia (1980) "Non-homogeneous media and vibration theory" and Bakhvalov & Panasenko (1989) "Homogenization: averaging processes in periodic media". They all use the method of multiple scales to get a formal expansion and then derive rigorous estimates on the accuracy of the resulting asymptotics. – Aleksey Pichugin Jun 4 '12 at 16:39
However, be warned: these monographs are mainly focussed on "classic" homogenization, in which the leading order problem is, essentially, one-scale. The extensions of these techniques to problems that are essentially two-scale even at the leading order were developed by G. Nguetseng and G. Allaire (the paper by G. Allaire called "Homogenization and two-scale convergence", SIAM J. Math. Anal., 23(1992), pp. 1482-1518 is a popular entry point to this area). – Aleksey Pichugin Jun 4 '12 at 16:45
When one turns to applied mathematics there is generally more interest on examples rather than to present a series of theorems to support the results. But in the book I cite there is the general theory described starting from page 360 and this should contain enough material to draw sound conclusions about accuracy. On the other hand, a two-scale approach is well supported by the adiabatic approximation and this is more mathematically studied. I think any book about ODE by Arnold should fit the bill. – Jon Jun 5 '12 at 19:17
You are right, my comment was aimed to give support to an eventual proof of a theorem. But, giving a look to the exterminate literature on this argument, I have found another reference providing an explicit theorem on accuracy for the two-scale approximation: epubs.siam.org/ebooks/siam/classics_in_applied_mathematics/cl27/…. You should check page 238 and following. This author claims the theory of multiple scale techniques well-understood. The proof of the theorem is also given. – Jon Jun 7 '12 at 7:28