Multiple-scale Anlysis. 4

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Problems to Chapter 8

Reminder. In Section 8.2 was constructed solution $u_\varepsilon (t)$ to the problem \begin{align} &u'' +2\varepsilon u' +u=0, \label{eq-8.P.1}\\ &u(0)=1, \qquad u'(0)=0\label{eq-8.P.2} \end{align} for $\varepsilon ^3t\ll 1$ modulo $O(\varepsilon^3)$ as \begin{gather} u_\varepsilon (t) \sim w _0(t, \varepsilon t, \varepsilon_2 t) + w _1(t, \varepsilon t)\varepsilon + w _2(t)\varepsilon^2. \label{eq-8.P.3} \end{gather}

Problem 1.

  1. Write explicit solution for problem \begin{align} &u'' +(1-\varepsilon )u =0,\label{eq-8.P.4}\\ &u(0)=1, \qquad u'(0)=0.\label{eq-8.P.5} \end{align}
  2. Repeat construction of Section Section 8.2 for this problem.

Problem 2.

For problem (\ref{eq-8.P.4})-(\ref{eq-8.P.5}) write solution modulo $O(\varepsilon^4)$ for $\varepsilon ^2t \ll 1$ (lesser error but under stronger restriction). What should be instead of (\ref{eq-8.P.3})?

Problem 3.

For problem (\ref{eq-8.P.1})-(\ref{eq-8.P.2}) write solution modulo $O(\varepsilon^4)$ for $\varepsilon ^2t \ll 1$ (lesser error but under stronger restriction). What should be instead of (\ref{eq-8.P.3})?

Problem 4. Write what form we would look for solution $u_\varepsilon (t)$ for $\varepsilon ^Kt\ll 1$ modulo $O(\varepsilon^M)$.

Reminder. In Section 8.3 was constructed solution $u_\varepsilon (t)$ to the problem (\ref{eq-8.P.1})-(\ref{eq-8.P.2}) modulo $O(\varepsilon ^3)$ in the form \begin{gather} u _\varepsilon = w_0 (T,\tau) + w_1 (T, \tau)\varepsilon + w_2(T,\tau)\varepsilon^2 \label{eq-8.P.6} \end{gather} with \begin{gather} T= (1+\nu_2 \varepsilon^2)t , \qquad \tau =\varepsilon t. \label{eq-8.P.7} \end{gather}

Problem 5.

Repeat construction of Section Section 8.3 for problem (\ref{eq-8.P.4})-(\ref{eq-8.P.5}).

Problem 6.

For problem (\ref{eq-8.P.4})-(\ref{eq-8.P.5}) write solution modulo $O(\varepsilon^4)$ for $\varepsilon ^2t \ll 1$ (lesser error but under stronger restriction). What should be instead of (\ref{eq-8.P.6})?

Problem 7.

For problem (\ref{eq-8.P.1})-(\ref{eq-8.P.2}) write solution modulo $O(\varepsilon^4)$ for $\varepsilon ^2t \ll 1$ (lesser error but under stronger restriction). What should be instead of (\ref{eq-8.P.6})?

Problem 8.

Write what form we would look for solution $u_\varepsilon (t)$ for $\varepsilon ^Kt\ll 1$ modulo $O(\varepsilon^M)$.

Problem 9.

Repeat construction of Section Section 8.3 for problem (\ref{eq-8.P.4})-(\ref{eq-8.P.5}).

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