Fourier transform, Fourier integral

$\renewcommand{\Re}{\operatorname{Re}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ Deadline Thursday, March 5.

APM346 (2015) Home Assignment 7

This assignment is based on Lecture 19--Lecture 21.

  1. Problem 1
  2. Problem 2
  3. Problem 3
  4. Problem 4

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

  2. Solve \begin{align*} & \Delta :=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 $f(\theta)=\left\{\begin{aligned} &\sin(\theta) &&0<\theta<\pi\\ &0 &&\pi<\theta<2\pi. \end{aligned}\right.$

Problem 4

  1. Solve \begin{align*} & \Delta :=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 $f(\theta)=\left\{\begin{aligned} &\sin(\theta) &&0<\theta<\pi\\ &0 &&\pi<\theta<2\pi. \end{aligned}\right.$

  2. Solve \begin{align*} & \Delta :=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 $f(\theta)=\left\{\begin{aligned} &\sin(\theta) &&0<\theta<\pi\\ &0 &&\pi<\theta<2\pi. \end{aligned}\right.$