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

Problem 1.

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

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

Problem 2.

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

Problem 3.

1. Solve \begin{align*} & \Delta u:=u_{xx}+u_{yy}=0&& \text{in } r<a\\[3pt] & u|_{r=a}=f(\theta). \end{align*} where we use polar coordinates $(r,\theta)$ and f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.

2. Solve \begin{align*} & \Delta u:=u_{xx}+u_{yy}=0&& \text{in } r>a\\[3pt] & u|_{r=a}=f(\theta),\\[3pt] & \max |u| <\infty. \end{align*} where we use polar coordinates $(r,\theta)$ and f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.

Problem 4.

1. Solve \begin{align*} & \Delta u:=u_{xx}+u_{yy}=0&& \text{in } r<a\\[3pt] & u_r|_{r=a}=f(\theta) \end{align*} where we use polar coordinates $(r,\theta)$ and f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.

2. Solve \begin{align*} & \Delta u :=u_{xx}+u_{yy}=0&& \text{in } r>a\\[3pt] & u_r|_{r=a}=f(\theta),\\[3pt] & \max |u| <\infty. \end{align*} where we use polar coordinates $(r,\theta)$ and f(\theta)=\left\{\begin{aligned} &1 &&0<\theta<\pi\\ -&1 &&\pi<\theta<2\pi. \end{aligned}\right.

Problem 5. Describe all real-valued solutions of biharmonic equation $$\Delta^2u:=u_{xxxx}+2u_{xxyy}+u_{yyyy}=0$$ 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 $X(x)=\cos (kx)$), 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 6. Describe all real-valued solutions of biharmonic equation $$\Delta^2u:= (\partial_r^2 +r^{-1}\partial_r + r^{-2}\partial_\theta^2)^2u=0$$ 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.