Multiple-scale analysis: Difference between revisions

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In [[mathematics]] and [[physics]], '''multiple-scale analysis''' (also called the '''method of multiple scales''') comprises techniques used to construct uniformly valid [[approximation]]s to the solutions of [[perturbation theory|perturbation problems]], both for small as well as large values of the [[independent variable]]s. This is done by introducing fast-scale and slow-scale variables for an independent variable, and subsequently treating these variables, fast and slow, as if they are independent. In the solution process of the perturbation problem thereafter, the resulting additional freedom – introduced by the new independent variables – is used to remove (unwanted) [[secular variation|secular terms]]. The latter puts constraints on the approximate solution, which are called '''solvability conditions'''.

Mathematics research from about the 1980s proposes that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling (for example, see [[center manifold]] and [[slow manifold]]).

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:<math>\left| y(t) \right| \le \sqrt{1 + \tfrac12 \varepsilon} \quad \text{ and } \quad \left| \frac{dy}{dt} \right| \le \sqrt{1 + \tfrac12 \varepsilon} \qquad \text{ for all } t.</math>{{pad|3em}}

===Straightforward perturbation-series solution===

A regular [[perturbation theory|perturbation-series approach]] to the problem gives the result:

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</math>

The last term between the square braces is secular: it grows without bound for large |''t''|. In particular, for <math>t = O(\epsilon^{-1})</math> this term is ''O''(1) and has the same order of magnitude as the leading-order term. Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution.

===Method of multiple scales===

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+ \mathcal{O}(\varepsilon^2),

\end{align}

</math>

using ''dt''₁/''dt''&nbsp;=&nbsp;''ε''. Similarly:

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using ''t''₁&nbsp;=&nbsp;''εt'' and valid for ''εt''&nbsp;=&nbsp;O(1). This agrees with the nonlinear [[frequency]] changes found by employing the [[Lindstedt–Poincaré method]].

This new solution is valid until <math>t = O(\epsilon^{-2})</math>. Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, ''i.e.'': ''t''₂&nbsp;=&nbsp;''ε''²&nbsp;''t'', ''t''₃&nbsp;=&nbsp;''ε''³&nbsp;''t'', etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see {{harvnb|Kevorkian|Cole|1996}}; {{harvnb|Bender|Orszag|1999}}).<ref>Bender & Orszag (1999) p. 551.</ref>

===Coordinate transform to amplitude/phase variables===

Alternatively, modern sound approaches derive these sorts of models using coordinate transforms,<ref>{{citation| first1=C.-H. |last1=Lamarque |first2=C. |last2=Touze |first3=O. |last3=Thomas |title=An upper bound for validity limits of asymptotic analytical approaches based on normal form theory |journal=[[Nonlinear Dynamics (journal)|Nonlinear Dynamics]] |pages=1931–1919 1931–1949|year=2012 |volume=70 |issue=3 |doi=10.1007/s11071-012-0584-y }}</ref> as described next.

A solution <math>y\approx r\cos\theta</math> is sought in new coordinates <math>(r,\theta)</math> where the amplitude <math>r(t)</math> varies slowly and the phase <math>\theta(t)</math> varies at an almost constant rate, namely <math>d\theta/dt\approx 1.</math> Straightforward algebra finds the coordinate transform{{citation needed|date=June 2015}}