Multiple-scale analysis
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.
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[citation needed] that coordinate transforms and invariant manifolds provide a sounder support for multiscale modelling (for example, see center manifold and slow manifold).
As an example for the method of multiple-scale analysis, consider the undamped and unforced Duffing equation:[1]
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:
Consequently, the Hamiltonian H(p, q) is a conserved quantity, a constant, equal to H = 1/2 + 1/4 ε for the given initial conditions.
A regular perturbation-series approach to the problem proceeds by writing
and substituting this into the undamped Duffing equation.
Solving these subject to the initial conditions yields
Note that the last term between the square braces is secular: it grows without bound for large |t|.
Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution.
Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become:
The zeroth-order problem has the general solution:
with A(t1) a complex-valued amplitude to the zeroth-order solution Y0(t, t1) and i2 = −1.
Now, in the first-order problem the forcing in the right hand side of the differential equation is
denotes the complex conjugate of the preceding terms.
The occurrence of secular terms can be prevented by imposing on the – yet unknown – amplitude A(t1) the solvability condition
As a result, the approximate solution by the multiple-scales analysis is
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., t2 = ε2 t, t3 = ε3 t, etc.
However, this introduces possible ambiguities in the perturbation series solution, which require a careful treatment (see Kevorkian & Cole 1996; Bender & Orszag 1999).
[2] Alternatively, modern approaches derive these sorts of models using coordinate transforms, like in the method of normal forms,[3] as described next.
Straightforward algebra finds the coordinate transform[citation needed]
transforms Duffing's equation into the pair that the radius is constant
That is, Duffing's oscillations are of constant amplitude
[4] 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.[when?
