$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$

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Problems to Chapter 4

• Problem 1
• Problem 3
• Problem 5
• Problem 7
• Problem 9
• Problem 11
• Problem 13

Part I. ODEs

Problem 1. Find up to $O(\varepsilon^2)$ solution $u$ to \begin{align*} &u''-\varepsilon u=\cos(x)&& 0< x< \pi\\ &u(0)=u(\pi)=0. \end{align*}

Problem 2. Find up to $O(\varepsilon^2)$ solution $u$ to \begin{align*} &u''-\varepsilon \cos(x)u=\cos(x)&& 0< x< \pi\\ &u(0)=u(\pi)=0. \end{align*}

Problem 3. Find up to $O(\varepsilon)$ solution $u$ to \begin{align*} -&\varepsilon^2u''+u=\cos(x)&& 0< x< \pi\\ &u(0)=u(\pi)=0. \end{align*}

Problem 4. Find up to $O(\varepsilon)$ solution $u$ to \begin{align*} -&\varepsilon^2u''+u=\sin(x)&& 0< x< \pi\\ &u'(0)=u'(\pi)=0. \end{align*}

Problem 5. Find up to $O(\varepsilon)$ solution $u$ to \begin{align*} -&\varepsilon u''+ u'=\sin(x)&& 0< x< \pi\\ &u'(0)=0, \quad u'(\pi)=1. \end{align*}

Problem 6. Find up to $O(\varepsilon)$ solution $u$ to \begin{align*} -&\varepsilon u''+ u'=\cos(x)&& 0< x< \pi\\ &u(0)=0, \quad u(\pi)=1. \end{align*}

Problem 7. Find up to $O(\varepsilon)$ solution $u$ to \begin{align*} &\varepsilon u^{IV}+ u=\cos(x)&& 0< x< \pi\\ &u(0)=u'(0)=u(\pi)=u'(\pi)=0. \end{align*}

Part II. PDEs

In the disk $D:-\{ (x,y)\colon x^2+y^2 <1\}$ with the boundary $\Gamma=\partial D = \{(x,y)\colon x^2+y^2=1\}$ find up to $O(\varepsilon)$ solutions $u$ to the following problems.

Hint. Use polar coordinates $(r,\theta)$.

Problem 8. \begin{align*} &-\varepsilon ^2 \Delta u+ u= 0,\\ &u|_\Gamma = 1. \end{align*}

Problem 9. \begin{align*} &-\varepsilon ^2 \Delta u+ u= 1,\\ &u|_\Gamma = 0. \end{align*}

Problem 10. \begin{align*} &-\varepsilon ^2 \Delta u+ u= x,\\ &u|_\Gamma = 0. \end{align*}

In the ring $D:-\{ (x,y)\colon a^2 < x^2+y^2 <b^2\}$ with the boundary $\Gamma=\partial D =\Gamma_1\cup \Gamma_2$, $\Gamma_1= \{(x,y)\colon x^2+y^2=a^2\}$, $\Gamma_2 = \{(x,y)\colon x^2+y^2=b^2\}$ find up to $O(\varepsilon)$ solutions $u$ to the following problems

Problem 11. \begin{align*} &-\varepsilon ^2 \Delta u+ u= 1,\\ &u|_{\Gamma_1}=0,\qquad u|_{\Gamma_2}=0 . \end{align*}

Problem 12. \begin{align*} &-\varepsilon ^2 \Delta u+ u= x,\\ &u|_{\Gamma_1}=0,\qquad u|_{\Gamma_2}=0 . \end{align*}

Problem 13. \begin{align*} &-\varepsilon \Delta u+ u_r= 1,\\ &u|_{\Gamma_1}=0,\qquad u|_{\Gamma_2}=0 . \end{align*}

Problem 14. \begin{align*} &-\varepsilon \Delta u- u_r= 1,\\ &u|_{\Gamma_1}=0,\qquad u|_{\Gamma_2}=0 . \end{align*}

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