Multiple-scale Anlysis. 1

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Chapter 8. Multiple-scale Analysis

Secular terms

We already saw two-scales in the singular perturbations theory. But this is different.

We follow Chapter 7 of Bowman. Consider the Cauchy problem for ODE: \begin{align} &u''+ 2\varepsilon u'+ u=0,\qquad t>0 \label{eq-8.1.1}\\ &u(0)=1,\qquad u'(0)=0. \label{eq-8.1.2} \end{align} The exact solution is \begin{equation} u=u(t,\varepsilon)= e^{-\varepsilon t} \Bigl[ \cos\bigl(\sqrt{1-\varepsilon^2}t\bigr) + \frac{\varepsilon}{\sqrt{1-\varepsilon^2}} \sin\bigl(\sqrt{1-\varepsilon^2}t\bigr) \Bigr] \label{eq-8.1.3} \end{equation} which is bounded: $|u(t,\varepsilon)|\le 1/\sqrt{1-\varepsilon^2}$ for all $t\ge 0$.

On the other hand. using standard perturbation method \begin{equation} u(t,\varepsilon)\sim \sum_{n\ge 0} u_n(t)\varepsilon^n \label{eq-8.1.4} \end{equation} we get \begin{align} &u_0''+ u_0=0,\qquad t>0 \label{eq-8.1.5}\\ &u_0(0)=1,\qquad u_0'(0)=0. \label{eq-8.1.6} \end{align} and \begin{align} &u_n''+ u_n=-2u_{n-1}',\qquad t>0 \label{eq-8.1.7}\\ &u_n(0)=1,\qquad u_n'(0)=0. \label{eq-8.1.8} \end{align} Then \begin{equation} u_0=\cos(t),\quad u_1=-t\cos(t)+\sin(t),\quad u_n= (-1)^n t^n\cos(t)+ O(t^{n-1}) \label{eq-8.1.9} \end{equation} so these terms are unbounded and approximation (\ref{eq-8.1.4}) works only as $\varepsilon t\ll 1$. We want an approximation valid for larger $t$.

The amplitudes of the perturbations are unbounded despite the fact that the exact solution is bounded. This is known as a secularity. The perturbation expansion is invalid since it attempts to separate the true dependence of $u$ on $t$ and $\varepsilon$ into a series containing products of functions of $t$ and functions of $\varepsilon$; the exact solution evidently cannot be written in this form. Instead, we see that for small $\varepsilon$ there are really two time scales, normal $t$ and slow $\varepsilon t$.. The method of multiple scales provides a means of dealing with such problems.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$