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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 much sounder support for multiscale modelling (for example, see [[center manifold]] and [[slow manifold]]).
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| ==Example: undamped Duffing equation==
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| ===Differential equation and energy conservation===
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| As an example for the method of multiple-scale analysis, consider the undamped and unforced [[Duffing equation]]:<ref>This example is treated in: Bender & Orszag (1999) pp. 545–551.</ref>
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| :<math>\frac{d^2 y}{d t^2} + y + \varepsilon y^3 = 0,</math> {{pad|3em}} <math>y(0)=1, \qquad \frac{dy}{dt}(0)=0,</math>
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| 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 < ''ε'' ≪ 1. The undamped Duffing equation is known to be a [[Hamiltonian system]]:
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| :<math>\frac{dp}{dt}=-\frac{\partial H}{\partial q}, \qquad \frac{dq}{dt}=+\frac{\partial H}{\partial p}, \quad \text{ with } \quad H = \tfrac12 p^2 + \tfrac12 q^2 + \tfrac14 \varepsilon q^4,</math>
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| with ''q'' = ''y''(''t'') and ''p'' = ''dy''/''dt''. Consequently, the Hamiltonian ''H''(''p'', ''q'') is a conserved quantity, a constant, equal to ''H'' = ½ + ¼ ''ε'' for the given [[initial conditions]]. This implies that both ''y'' and ''dy''/''dt'' have to be bounded:
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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}}
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| ===Straightforward perturbation-series solution===
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| A regular [[perturbation theory|perturbation-series approach]] to the problem gives the result:
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| :<math>
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| y(t) = \cos(t)
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| + \varepsilon \left[ \tfrac{1}{32} \cos(3t) - \tfrac{1}{32} \cos(t) - \underbrace{\tfrac38\, t\, \sin(t)}_\text{secular} \right]
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| + \mathcal{O}(\varepsilon^2).
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| </math>
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| The last term between the square braces is secular: it grows without bound for large |''t''|, making the perturbation solution valid for only small values of the time ''t''.
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| ===Method of multiple scales===
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| To construct a global valid solution, the method of ''multiple-scale analysis'' is used. Introduce the slow scale ''t''<sub>1</sub>:
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| :<math>t_1 = \varepsilon t\,</math>
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| and assume the solution ''y''(''t'') is a perturbation-series solution dependent both on ''t'' and ''t''<sub>1</sub>, treated as:
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| :<math>y(t) = Y_0(t,t_1) + \varepsilon Y_1(t,t_1) + \cdots.</math>
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| So:
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| :<math>
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| \begin{align}
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| \frac{dy}{dt}
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| &= \left( \frac{\partial Y_0}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_0}{\partial t_1} \right)
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| + \varepsilon \left( \frac{\partial Y_1}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_1}{\partial t_1} \right)
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| + \cdots
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| \\
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| &= \frac{\partial Y_0}{\partial t}
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| + \varepsilon \left( \frac{\partial Y_0}{\partial t_1} + \frac{\partial Y_1}{\partial t} \right)
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| + \mathcal{O}(\varepsilon^2),
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| \end{align}
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| </math>
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| using ''dt''<sub>1</sub>/''dt'' = ''ε''. Similarly:
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| :<math>
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| \frac{d^2 y}{d t^2}
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| = \frac{\partial^2 Y_0}{\partial t^2}
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| + \varepsilon \left( 2 \frac{\partial^2 Y_0}{\partial t\, \partial t_1} + \frac{\partial^2 Y_1}{\partial t^2} \right)
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| + \mathcal{O}(\varepsilon^2).
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| </math>
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| Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become:
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| :<math>
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| \begin{align}
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| \frac{\partial^2 Y_0}{\partial t^2} + Y_0 &= 0,
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| \\
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| \frac{\partial^2 Y_1}{\partial t^2} + Y_1 &= - Y_0^3 - 2\, \frac{\partial^2 Y_0}{\partial t\, \partial t_1}.
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| \end{align}
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| </math>
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| ===Solution===
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| The zeroth-order problem has the general solution:
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| :<math>Y_0(t,t_1) = A(t_1)\, e^{+it} + A^\ast(t_1)\, e^{-it},</math>
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| with ''A''(''t''<sub>1</sub>) a [[complex number|complex-valued]] [[amplitude]] to the zeroth-order solution ''Y''<sub>0</sub>(''t'', ''t''<sub>1</sub>) and ''i''<sup>2</sup> = −1. Now, in the first-order problem the forcing in the [[right hand side]] of the differential equation is
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| :<math>\left[ -3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} \right]\, e^{+it} - A^3\, e^{+3it} + c.c.</math>
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| where ''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''<sub>1</sub>) the ''solvability condition''
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| :<math>-3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} = 0.</math>
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| The solution to the solvability condition, also satisfying the initial conditions ''y''(0) = 1 and ''dy''/''dt''(0) = 0, is:
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| :<math>A = \tfrac12\, \exp \left(\tfrac38\, i \, t_1 \right).</math>
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|
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| As a result, the approximate solution by the multiple-scales analysis is
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| :<math>y(t) = \cos \left[ \left( 1 + \tfrac38\, \varepsilon \right) t \right] + \mathcal{O}(\varepsilon),</math>
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| using ''t''<sub>1</sub> = ''εt'' and valid for ''εt'' = O(1). This agrees with the nonlinear [[frequency]] changes found by employing the [[Lindstedt–Poincaré method]].
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| Higher-order solutions – using the method of multiple scales – require the introduction of additional slow scales, ''i.e.'': ''t''<sub>2</sub> = ''ε''<sup>2</sup> ''t'', ''t''<sub>3</sub> = ''ε''<sup>3</sup> ''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>
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| 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 |year=2012 |volume=70 |issue=3 |doi=10.1007/s11071-012-0584-y }}</ref> as also described next.
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| ===Coordinate transform to amplitude/phase variables===
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| We seek a solution <math>y\approx r\cos\theta</math> 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>.
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| Straightforward algebra finds the coordinate transform
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| :<math>y=r\cos\theta +\frac1{32}\varepsilon r^3\cos3\theta +\frac1{1024}\varepsilon^2r^5(-21\cos3\theta+\cos5\theta)+\mathcal O(\varepsilon^3)</math>
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| transforms Duffing's equation into the pair that the radius is constant <math>dr/dt=0</math> and the phase evolves according to
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| :<math>\frac{d\theta}{dt}=1 +\frac38\varepsilon r^2 -\frac{15}{256}\varepsilon^2r^4 +\mathcal O(\varepsilon^3).</math>
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| That is, Duffing's oscillations are constant amplitude but a different frequencies depending upon the amplitude.<ref>{{citation |first=A.J. |last=Roberts |title=Modelling emergent dynamics in complex systems |url=http://www.maths.adelaide.edu.au/anthony.roberts/modelling.php |accessdate=2013-10-03 }}</ref>
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| 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.<ref>{{citation |first=A.J. |last=Roberts |title=Construct centre manifolds of ordinary or delay differential equations (autonomous) |url=http://www.maths.adelaide.edu.au/anthony.roberts/gencm.php |accessdate=2013-10-03 }}</ref>
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| ==See also==
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| * [[Method of matched asymptotic expansions]]
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| * [[WKB approximation]]
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| ==Notes==
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| {{reflist}}
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| A good reference is Nayfeh's perturbation method book
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| ==References==
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| *{{citation
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| | last1=Kevorkian | first1=J.
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| | last2=Cole | first2=J. D.
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| | title=Multiple scale and singular perturbation methods
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| | year=1996
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| | publisher=Springer
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| | isbn=0-387-94202-5
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| }}
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| *{{citation
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| | first1=C.M. | last1=Bender | authorlink1=Carl M. Bender
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| | first2=S.A. | last2=Orszag | authorlink2=Steven A. Orszag
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| | title=Advanced mathematical methods for scientists and engineers
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| | publisher=Springer
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| | year=1999
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| | isbn=0-387-98931-5
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| | pages=544–568
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| }}
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| ==External links==
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| *{{citation
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| | url=http://www.scholarpedia.org/article/Multiple_scale_analysis
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| | publisher=[[Scholarpedia]]
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| | title=Multiple scale analysis
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| | first=Carson C. | last=Chow
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| | accessdate=2009-08-09
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| | doi=10.4249/scholarpedia.1617
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| | year=2007
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| | journal=Scholarpedia
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| | volume=2
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| | issue=10
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| | pages=1617
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| }}
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| [[Category:Mathematical physics]]
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| [[Category:Asymptotic analysis]]
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| [[Category:Perturbation theory]]
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