Multiple-scale analysis

In mathematics and physics, multiple-scale analysis refers to 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 variable(s) for an independent variable, and subsequently treating these variables, fast and slow, as independent. In the solution process of the perturbation problem thereafter, the additional freedom – introduced by the new independent variables – is used to remove (unwanted) secular terms. This puts constraints on the approximate solution, which are called solvability conditions.

Example: undamped Duffing equation

As an example for the method of multiple-scale analysis, consider the undamped and unforced Duffing equation:^[1]

${\displaystyle {\frac {d^{2}y}{dt^{2}}}+y+\epsilon y^{3}=0,}$    ${\displaystyle y(0)=1,\qquad {\frac {dy}{dt}}(0)=0,}$ 

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 Duffing equation is known to be a Hamiltonian system:

${\displaystyle {\frac {dp}{dt}}=-{\frac {dH}{dq}},\qquad {\frac {dq}{dt}}=+{\frac {dH}{dp}},}$    with   ${\displaystyle H={\tfrac {1}{2}}p^{2}+{\tfrac {1}{2}}q^{2}+{\tfrac {1}{4}}\epsilon q^{4},}$ 

with q = y(t) and p = dy/dt. Consequently, the Hamiltonian H is a conserved quantity, a constant, equal to H = ½ + ¼ ε. This implies that both y and dy/dt have to be bounded:

${\displaystyle \left|y(t)\right|\leq {\sqrt {1+{\tfrac {1}{2}}\epsilon }}}$    and   ${\displaystyle \left|{\frac {dy}{dt}}\right|\leq {\sqrt {1+{\tfrac {1}{2}}\epsilon }}}$   for all t.

A regular perturbation-series approach to the problem gives the result:

${\displaystyle y(t)=\cos(t)+\epsilon \left[{\tfrac {1}{32}}\cos(3t)-{\tfrac {1}{32}}\cos(t)-\underbrace {{\tfrac {3}{8}}\,t\,\sin(t)} _{\text{secular}}\right]+{\mathcal {O}}(\epsilon ^{2}).}$ 

The last term between the square braces is secular: it grows without bound for large |t|, making the perturbation solution only valid for small values of the time t.

To construct a global valid solution, the method of multiple-scale analysis is used. Introduce the slow scale t₁:

${\displaystyle t_{1}=\epsilon t\,}$ 

and assume the solution y(t) is a perturbation-series solution dependent both on t and t₁, treated as:

${\displaystyle y(t)=Y_{0}(t,t_{1})+\epsilon Y_{1}(t,t_{1})+\cdots .}$ 

So:

${\displaystyle {\begin{aligned}{\frac {dy}{dt}}&=\left({\frac {\partial Y_{0}}{\partial t}}+{\frac {dt_{1}}{dt}}{\frac {\partial Y_{0}}{\partial t_{1}}}\right)+\epsilon \left({\frac {\partial Y_{1}}{\partial t}}+{\frac {dt_{1}}{dt}}{\frac {\partial Y_{1}}{\partial t_{1}}}\right)+\cdots \\&={\frac {\partial Y_{0}}{\partial t}}+\epsilon \left({\frac {\partial Y_{0}}{\partial t_{1}}}+{\frac {\partial Y_{1}}{\partial t}}\right)+{\mathcal {O}}(\epsilon ^{2}),\end{aligned}}}$ 

using dt₁/dt = ε. Similarly:

${\displaystyle {\frac {d^{2}y}{dt^{2}}}={\frac {\partial ^{2}Y_{0}}{\partial t^{2}}}+\epsilon \left(2{\frac {\partial ^{2}Y_{0}}{\partial t\,\partial t_{1}}}+{\frac {\partial ^{2}Y_{1}}{\partial t^{2}}}\right)+{\mathcal {O}}(\epsilon ^{2}).}$ 

Then the zeroth- and first-order problems of the multiple-scales perturbation series for the Duffing equation become:

${\displaystyle {\begin{aligned}{\frac {\partial ^{2}Y_{0}}{\partial t^{2}}}+Y_{0}&=0,\\{\frac {\partial ^{2}Y_{1}}{\partial t^{2}}}+Y_{1}&=-Y_{0}^{3}-2\,{\frac {\partial ^{2}Y_{0}}{\partial t\,\partial t_{1}}}.\end{aligned}}}$ 

The zeroth-order problem has the general solution:

${\displaystyle Y_{0}(t,t_{1})=A(t_{1})\,e^{+it}+A^{\ast }(t_{1})\,e^{-it},}$ 

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

${\displaystyle \left[-3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1}}}\right]\,e^{+it}-A^{3}\,e^{+3it}+c.c.}$ 

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(t₁) the solvability condition

${\displaystyle -3\,A^{2}\,A^{\ast }-2\,i\,{\frac {dA}{dt_{1}}}=0.}$ 

The solution to the solvability condition, also satisifying the initial conditions y(0) = 1 and dy/dt(0) = 0, is:

${\displaystyle A={\tfrac {1}{2}}\,\exp \left({\tfrac {3}{8}}\,t_{1}\right).}$ 

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

${\displaystyle y(t)=\cos \left[\left(1+{\tfrac {3}{8}}\,\epsilon \right)t\right]+{\mathcal {O}}(\epsilon ),}$ 

using t₁ = εt and valid for εt = 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.: t₂ = ε² t, t₃ = ε³ t, etc. However, this introduces possible ambiguities in the perturbation series solution which require a careful treatment.^[2]

See also

• WKB approximation

Notes

1. This example is treated in: Bender & Orszag (1999) pp. 545–551.
2. Bender & Orszag (1999) p. 551.

References

• Kevorkian, J.; Cole, J. D. (1996), Multiple scale and singular perturbation methods, Springer, ISBN 0-387-94202-5
• Bender, C.M.; Orszag, S.A. (1999), Advanced mathematical methods for scientists and engineers, Springer, pp. 544–568, ISBN 0-387-98931-5

External links

• Chow, Carson C., Multiple scale analysis, Scholarpedia, retrieved 2009-08-09