$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$ ###[Problems to Section 5.3](id:sect-3.2.P) > 1. [Problem 1](#problem-5.3.P.1) > 2. [Problem 2](#problem-5.3.P.2) > 3. [Problem 3](#problem-5.3.P.3) > 4. [Problem 4](#problem-5.3.P.4) **[Problem 1.](id:problem-5.3.P.1)** a. Consider Dirichlet problem \begin{align} &u\_{xx}+u\_{yy}=0,\qquad -\infty0, \\\\ &u|\_{y=0}=f(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. b. Consider Neumann problem \begin{align} &u\_{xx}+u\_{yy}=0,\qquad -\infty0, \\\\ &u\_y|\_{y=0}=f(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition must satisfy $f$? **[Problem 2.](id:problem-5.3.P.2)** a. Consider Dirichlet problem \begin{align} &u\_{xx}+u\_{yy}=0,\qquad -\infty **[Problem 3.](id:problem-5.3.P.3)** Consider Robin problem \begin{align} &u\_{xx}+u\_{yy}=0,\qquad -\infty0, \\\\ &(u\_y+\alpha u)|\_{y=0}=f(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition (if any) must satisfy $f$? *Hint.* Consider separately $\alpha>0$ and $\alpha<0$. **[Problem 4.](id:problem-5.3.P.4)** a. Consider problem \begin{align} &\Delta^2u=0,\qquad -\infty0, \\\\ &u|\_{y=0}=f(x),\quad &u\_y|\_{y=0}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. b. Consider problem \begin{align} &\Delta^2u=0,\qquad -\infty0, \\\\ &u\_{yy}|\_{y=0}=f(x),\quad &\Delta u\_{y}|\_{y=0}=g(x). \end{align} Make Fourier transform by $x$, solve problem for ODE for $\hat{u}(k,y)$ which you get as a result and write $u(x,y)$ as a Fourier integral. What condition must satisfy $f,g$? ________________ [$\Uparrow$](../contents.html)  [$\uparrow$](./S5.3.html)  [$\Rightarrow$](../Chapter6/S6.1.html)