Problems to Chapter 6

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

Problems to Chapter 6

Laplace in the polar coordinates

Problems in the disk

Problem 1.

\begin{align} & \Delta u:=u_{rr}+r^{-1}u_r +r^{-2}u_{\theta\theta}=0 && \text{for }\ r < a, -\pi\le\theta\le \pi, \label{a}\\[3pt] & u|_{r=a}=f(\theta). \label{b} \end{align}

Problem 2. \begin{align} & \Delta u=0&& \text{for } r< a, -\pi\le\theta\le \pi, \label{c}\\[3pt] & u_r|_{r=a}=f(\theta), \label{d}\\[3pt] &u|_{r=0}=0.\label{e} \end{align}

1. What condition should satisfy $f(\theta)$?
2. Explain why we need condition (\ref{e}).
Problems in the exterior of the disk

Problem 3. \begin{align} & \Delta u=0&& \text{for } r>a, -\pi\le\theta\le \pi , \label{f}\\[3pt] & u|_{r=a}=f(\theta),\label{g}\\[3pt] & \max |u| <\infty.\label{h} \end{align}

Problem 4. \begin{align} & \Delta u =0&& \text{for } r >a, -\pi\le\theta\le \pi , \label{i}\\[3pt] & u_r|_{r=a}=f(\theta),\label{j}\\[3pt] & u\to 0 \quad \text{as } r\to\infty.\label{k} \end{align}

1. What condition should satisfy $f(\theta)$?
2. Explain why we need condition (\ref{k}) rather than (\ref{h}).
Problems in the ring

Problem 5.

\begin{align} & \Delta u =0&& \text{for } a<r<b , -\pi\le\theta\le \pi , \label{l}\\[3pt] & u|_{r=a}=f(\theta),\label{m}\\[3pt] & u|_{r=b}=g(\theta).\label{n}\\[3pt] \end{align}

Problem 6. \begin{align} & \Delta u =0&& \text{for } a<r<b , -\pi\le\theta\le \pi , \label{p}\\[3pt] & u_r|_{r=a}=f(\theta),\label{q}\\[3pt] & u|_{r=b}=g(\theta).\label{r}\\[3pt] \end{align}

Problem 7. \begin{align} & \Delta u =0&& \text{for } a<r<b , -\pi\le\theta\le \pi , \label{s}\\[3pt] & u_r|_{r=a}=f(\theta),\label{t}\\[3pt] & u_r|_{r=b}=g(\theta),\label{u}\\[3pt] & \int_{-\pi}^\pi u(r^*,\theta)\,d\theta=0.\label{v} \end{align}

1. What condition should satisfy $f(\theta)$ and $g(\theta)$?
2. Explain why we need condition (\ref{v}). Does the choice of $r^*\in [a,b]$ matter?
Problems in the disk sector

Problem 8.

\begin{align} & \Delta u =0&& \text{for } r< a , 0<\theta < \alpha , \label{aa}\\[3pt] & u|_{r=a}=f(\theta),\label{ab}\\[3pt] & u|_{\theta=0}=u|_{\theta=\alpha}=0.\label{ac}\\[3pt] \end{align}

Problem 9.

\begin{align} & \Delta u =0&& \text{for } r< a , 0<\theta < \alpha , \label{ad}\\[3pt] & u|_{r=a}=f(\theta),\label{ae}\\[3pt] & u_\theta|_{\theta=0}=u_\theta|_{\theta=\alpha}=0.\label{af}\\[3pt] \end{align}

Problem 10.

\begin{align} & \Delta u =0&& \text{for } r< a , 0< \theta < \alpha , \label{ag}\\[3pt] & u_r|_{r=a}=f(\theta),\label{ah}\\[3pt] & u|_{\theta=0}=u|_{\theta=\alpha}=0.\label{ai}\\[3pt] \end{align}

Problem 11.

\begin{align} & \Delta u =0&& \text{for } r< a , 0< \theta< \alpha , \label{aj}\\[3pt] & u_r|_{r=a}=f(\theta),\label{ak}\\[3pt] & u_\theta|_{\theta=0}=u_\theta|_{\theta=\alpha}=0,\label{al}\\[3pt] & u|_{r=0}=0.\label{am} \end{align}

1. What condition should satisfy $f(\theta)$?
2. Explain why we need condition (\ref{am}).

Remark. One can consider other domains: $\{r >a, 0<\theta<\alpha\}$, $\{a< r< b, 0<\theta<\alpha\}$, and other boundary condition (Robin).

Miscellaneous

Problem 12. Describe all real-valued solutions of biharmonic equation $$\Delta^2u:=u_{xxxx}+2u_{xxyy}+u_{yyyy}=0 \label{ba}$$ which one can obtain by a method of separation $u(x,y)=X(x)Y(y)$.

1. Try to do it directly; find the term preventing us.
2. However if we set $X(x)=e^{ikx}$ (or $X=\cos (kx)$, or $X(x)=\sin (kx)$), we will be able to get ODE for $Y(y)$. Solve it.
3. In the rectangle $\{0 < x < a, 0< y < b\}$ we can find solutions, $a$-periodic with respect to $x$ and satisfying boundary conditions with respect to $y$: at $\{y=0\}$ and $\{y=b\}$ are given either $u, u_y$, or $u, u_{yy}$, or $u_{yy}, u_{yyy}$ (conditions do not necessary match).
4. Write the general solution in the form of series.

Problem 13. Describe all real-valued solutions of biharmonic equation $$\Delta^2u:= (\partial_r^2 +r^{-1}\partial_r + r^{-2}\partial_\theta^2)^2u=0 \label{bb}$$ which one can obtain by a method of separation $u(r,\theta)=R(r)\Theta(\theta)$.

1. Like in the previous problem, consider solution $2\pi$-periodic with respect to $\theta$, take $\Theta(\theta)=e^{in\theta}$ and find equation to $R(r)$.
2. Since it will be Euler equation, solve it.
3. Write the general solution in the form of series.

Problem 14.

1. Find the solutions that depend only on $r$ of the equation $$\Delta u:=u_{xx}+u_{yy}+u_{zz}=k^2u,$$ where $k$ is a positive constant.
Hint. Substitute $u=v/r$.

2. Find the solutions that depend only on $r$ of the equation $$\Delta u:=u_{xx}+u_{yy}+u_{zz}=-k^2u,$$ where $k$ is a positive constant.
Hint. Substitute $u=v/r$.

Problem 15.

1. Try to find the solutions that depend only on $r$ of the equation $$\Delta u:=u_{xx}+u_{yy}=k^2u,$$ where $k$ is a positive constant. What ODE should satisfy $u(r)$?

2. Try to find the solutions that depend only on $r$ of the equation $$\Delta u:=u_{xx}+u_{yy}=-k^2u,$$ where $k$ is a positive constant. What ODE should satisftfy $u(r)$?

Bessel functions