Method Of Multiple Scales

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

Differential equation and energy conservation

As an example for the method of multiple-scale analysis, consider the undamped and unforced Duffing equation:
$\frac + y + \varepsilon y^3 = 0,$ $y(0)=1, \qquad \frac(0)=0,$
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:
$\frac=-\frac, \qquad \frac=+\frac, \quad \text \quad H = \tfrac12 p^2 + \tfrac12 q^2 + \tfrac14 \varepsilon q^4,$
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:
$\left, y(t) \ \le \sqrt \quad \text \quad \left, \frac \ \le \sqrt \qquad \text t.$

Straightforward perturbation-series solution

A regular perturbation-series approach to the problem proceeds by writing $y(t) = y_0(t) + \varepsilon y_1(t) + \mathcal(\varepsilon^2)$ and substituting this into the undamped Duffing equation. Matching powers of $\varepsilon$ gives the system of equations
$\begin \frac + y_0 &= 0,\\ \frac + y_1 &= - y_0^3. \end$

Solving these subject to the initial conditions yields

y(t) = \cos(t)
+ \varepsilon \left \tfrac \cos(3t) - \tfrac \cos(t) - \underbrace_\text \right
+ \mathcal(\varepsilon^2).

Note that the last term between the square braces is secular: it grows without bound for large , ''t'', . In particular, for $t = O(\varepsilon^)$ this term is ''O''(1) and has the same order of magnitude as the leading-order term. Because the terms have become disordered, the series is no longer an asymptotic expansion of the solution.

Method of multiple scales

To construct a solution that is valid beyond $t = O(\epsilon^)$, the method of ''multiple-scale analysis'' is used. Introduce the slow scale ''t''₁:
$t_1 = \varepsilon t$
and assume the solution ''y''(''t'') is a perturbation-series solution dependent both on ''t'' and ''t''₁, treated as:
$y(t) = Y_0(t,t_1) + \varepsilon Y_1(t,t_1) + \cdots.$

So:
$\begin \frac &= \left( \frac + \frac \frac \right) + \varepsilon \left( \frac + \frac \frac \right) + \cdots \\ &= \frac + \varepsilon \left( \frac + \frac \right) + \mathcal(\varepsilon^2), \end$
using ''dt''₁/''dt'' = ''ε''. Similarly:
$\frac = \frac + \varepsilon \left( 2 \frac + \frac \right) + \mathcal(\varepsilon^2).$

Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become:
$\begin \frac + Y_0 &= 0, \\ \frac + Y_1 &= - Y_0^3 - 2\, \frac. \end$

Solution

The zeroth-order problem has the general solution:
$Y_0(t,t_1) = A(t_1)\, e^ + A^\ast(t_1)\, e^,$
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

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