Multiple-scale analysis: Difference between revisions

Revision as of 02:56, 11 August 2009

Revision as of 03:00, 11 August 2009

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:<math>Y_0(t,t_1) = A(t_1)\, e^{+it} + A^\ast(t_1)\, e^{-it},</math>

with ''A''(''t''₁'') a [[complex number|complex-valued]] [[amplitude]] to the zeroth-order solution ''Y''₀''(''t'',&nbsp;''t''₁'') and ''i''²''&nbsp;=&nbsp;−1. Now, in the first-order problem the forcing in the [[right hand side]] of the differential equation is

:<math>\left[ -3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} \right]\, e^{+it} - A^3\, e^{+3it} + c.c.</math>

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''₁'') the ''solvability condition''

:<math>-3\, A^2\, A^\ast - 2\, i\, \frac{dA}{dt_1} = 0.</math>

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As a result, the approximate solution by the multiple-scales analysis is

:<math>y(t) = \cos \left[ \left( 1 + \tfrac38\, \epsilonvarepsilon \right) t \right] + \mathcal{O}(\epsilonvarepsilon),</math>

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]].

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>

==See also==