Multiple-scale analysis: Difference between revisions

Revision as of 03:00, 11 August 2009

Revision as of 03:05, 11 August 2009

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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 constraints are called '''solvability conditions'''.

==Example: undamped Duffing equation==

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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>

:<math>\frac{d^2 y}{d t^2} + y + \epsilonvarepsilon y^3 = 0,</math> {{pad|3em}} <math>y(0)=1, \qquad \frac{dy}{dt}(0)=0,</math>

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&nbsp;<&nbsp;''ε''&nbsp;≪&nbsp;1. The undamped Duffing equation is known to be a [[Hamiltonian system]]:

:<math>\frac{dp}{dt}=-\frac{dH}{dq}, \qquad \frac{dq}{dt}=+\frac{dH}{dp},</math> \text{{pad|2em}} with {{pad|2em}} <math>H = \tfrac12 p^2 + \tfrac12 q^2 + \tfrac14 \epsilonvarepsilon q^4,</math>

with ''q''&nbsp;=&nbsp;''y''(''t'') and ''p''&nbsp;=&nbsp;''dy''/''dt''. Consequently, the Hamiltonian ''H''(''p'',&nbsp;''q'') is a conserved quantity, a constant, equal to ''H''&nbsp;=&nbsp;½&nbsp;+&nbsp;¼&nbsp;''ε'' for the given [[initial conditions]]. This implies that both ''y'' and ''dy''/''dt'' have to be bounded:

:<math>\left| y(t) \right| \le \sqrt{1 + \tfrac12 \epsilonvarepsilon}</math> \text{{pad|2em}} and {{pad|2em}} <math>\left| \frac{dy}{dt} \right| \le \sqrt{1 + \tfrac12 \epsilonvarepsilon}\text{ for all } t.</math>{{pad|3em}} for all ''t''.

===Straightforward perturbation-series solution===

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:<math>

y(t) = \cos(t)

+ \epsilonvarepsilon \left[ \tfrac{1}{32} \cos(3t) - \tfrac{1}{32} \cos(t) - \underbrace{\tfrac38\, t\, \sin(t)}_\text{secular} \right]

+ \mathcal{O}(\epsilonvarepsilon^2).

</math>

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===Method of multiple scales===

To construct a global valid solution, the method of ''multiple-scale analysis'' is used. Introduce the slow scale ''t''₁'':

:<math>t_1 = \epsilonvarepsilon t\,</math>

and assume the solution ''y''(''t'') is a perturbation-series solution dependent both on ''t'' and ''t''₁'', treated as:

:<math>y(t) = Y_0(t,t_1) + \epsilonvarepsilon Y_1(t,t_1) + \cdots.</math>

So:

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\frac{dy}{dt}

&= \left( \frac{\partial Y_0}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_0}{\partial t_1} \right)

+ \epsilonvarepsilon \left( \frac{\partial Y_1}{\partial t} + \frac{dt_1}{dt} \frac{\partial Y_1}{\partial t_1} \right)

+ \cdots

\\

&= \frac{\partial Y_0}{\partial t}

+ \epsilonvarepsilon \left( \frac{\partial Y_0}{\partial t_1} + \frac{\partial Y_1}{\partial t} \right)

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

\end{align}

</math>

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

:<math>

\frac{d^2 y}{d t^2}

= \frac{\partial^2 Y_0}{\partial t^2}

+ \epsilonvarepsilon \left( 2 \frac{\partial^2 Y_0}{\partial t\, \partial t_1} + \frac{\partial^2 Y_1}{\partial t^2} \right)

+ \mathcal{O}(\epsilonvarepsilon^2).

</math>