Multiple-scale analysis
Introduction In mathematics and physics, '''multiple-scale analysis''' (also called the '''method of multiple scales''') comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. 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 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).
Example: undamped Duffing equation Here the differences between  approaches for both regular perturbation theory and multiple-scale analysis can be seen, and how they compare to the exact solution for ===Differential equation and energy conservation===As an example for the method of multiple-scale analysis, consider the undamped and unforced Duffing equation: which is a second-order ordinary differential equation describing a nonlinear oscillator. A solution ''y''(''t'') is sought for small values of the (positive) nonlinearity parameter 0 1:and assume the solution ''y''(''t'') is a perturbation-series solution dependent both on ''t'' and ''t''1, treated as:So:using ''dt''1/''dt'' = ''ε''. Similarly:Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become:===Solution===The zeroth-order problem has the general solution:with ''A''(''t''1) a complex-valued amplitude to the zeroth-order solution ''Y''0(''t'', ''t''1) and ''i''2 = −1.Now, in the first-order problem the forcing in the right hand side of the differential equation iswhere ''c.c.'' denotes the complex conjugate of the preceding terms. The occurrence of ''secular terms'' can be prevented by imposing on the – yet unknown – amplitude ''A''(''t''1) the ''solvability condition''The solution to the solvability condition, also satisfying the initial conditions  and , is:As a result, the approximate solution by the multiple-scales analysis isusing  and valid for . This agrees with the nonlinear frequency changes found by employing the Lindstedt–Poincaré method.This new solution is valid until . Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, i.e., , , etc. However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see ; ).===Coordinate transform to amplitude/phase variables===Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms, as described next.A solution  is sought in new coordinates  where the amplitude  varies slowly and the phase  varies at an almost constant rate, namely  Straightforward algebra finds the coordinate transformtransforms Duffing's equation into the pair that the radius is constant  and the phase evolves according toThat is, Duffing's oscillations are of constant amplitude  but have different frequencies  depending upon the amplitude.More difficult examples are better treated using a time-dependent coordinate transform involving complex exponentials (as also invoked in the previous multiple time-scale approach). A web service will perform the analysis for a wide range of examples.
See also * Method of matched asymptotic expansions* WKB approximation*Method of averaging*Krylov–Bogoliubov averaging method
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