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\begin{document}

\lhead{\large{\textbf{APPM 5470, Methods of Applied Mathematics:
      Partial Differential Equations and Integral Equations, HW 7, due
      12/8}}} \rhead{\textit{Fall 2017}} %\lfoot{} \cfoot{} \rfoot{}


\vspace{0.25in}

\textbf{Book Problems:}

Chapter 9: 3, 6, 11

\textbf{Additional Problems:}

\begin{enumerate}
\item [A1)] This problem will use methods of potential theory to find
  a boundary integral equation representation of the solution to the
  Dirichlet problem
  \begin{equation*}
    \begin{split}
      \Delta u(\mathbf{x}) &= 0, \quad \mathbf{x} \in \Omega \subset
      \R^3 \\
      u(\mathbf{x}) &= f(\mathbf{x}), \quad \mathbf{x} \in \partial
      \Omega ,
    \end{split}
  \end{equation*}
  where $\Omega$ is bounded and its boundary $\partial \Omega$ is
  smooth.  Recall the fundamental solution in $\R^3$:
  $\Phi(\mathbf{x},\mathbf{y}) = (4\pi |\mathbf{x}-\mathbf{y}|)^{-1}$.
  \begin{enumerate}
  \item Show that for any $\mu \in C(\partial \Omega)$, the double
    layer potential
    \begin{equation}
      \label{eq:1}
      u(\mathbf{x}) = \int_{\partial \Omega} \left [ \frac{\partial}{\partial
        n_y} \Phi(\mathbf{x},\mathbf{y}) \right ] \mu(\mathbf{y})\,
    \mathrm{d}S_y
    \end{equation}
    is harmonic in $\R^3 \setminus \partial \Omega$.
  \item Prove 
    \begin{equation*}
      \int_{\partial \Omega} \left [ \frac{\partial}{\partial
        n_y} \Phi(\mathbf{x},\mathbf{y}) \right ] 
    \mathrm{d}S_y =
    \begin{cases}
      -1 , \quad \mathbf{x} \in \Omega, \\
      -\frac{1}{2}, \quad \mathbf{x} \in \partial \Omega, \\
      0, \quad \mathbf{x} \in \overline{\Omega}^c = \R^3 \setminus
      \overline{\Omega} .
    \end{cases}
    \end{equation*}
  \item For the double layer potential in Eq.~\eqref{eq:1} to solve
    the Dirichlet problem, we require
    \begin{equation*}
      \lim_{\substack{\mathbf{z} \to \mathbf{x} \in \partial \Omega \\
        \mathbf{z} \in \Omega}} u(\mathbf{z}) = f(\mathbf{x}) .
    \end{equation*}
    Assuming this and using the relations you proved in (b), show that
    the density $\mu$ in \eqref{eq:1} satisfies the singular boundary
    integral equation
    \begin{equation*}
      \int_{\partial \Omega} \left [ \frac{\partial}{\partial
        n_y} \Phi(\mathbf{x},\mathbf{y}) \right ] \mu(\mathbf{y})\,
    \mathrm{d}S_y - \frac{1}{2} \mu(\mathbf{x}) = f(\mathbf{x}), \quad
    \mathbf{x} \in \partial \Omega .
    \end{equation*}
    Integral equations like this one are at the heart of modern
    boundary integral numerical methods.
  \end{enumerate}
\item [A2)] In this problem, you will show that the Fourier transform
  of a constant is a delta distribution.  The general strategy is to
  find the Fourier transform of $f_L(x) = H(L-|x|) =
  \begin{cases}
    1, \quad |x| < L \\ 0, \quad \mathrm{else}
  \end{cases}$ and then show that $\lim_{L \to \infty} \hat{f}_L(k)$
  is a delta distribution.  Recall the Fourier transform
  \begin{equation}
    \label{eq:2}
    \hat{f}(k) = \int_{\R} f(x) e^{ikx}\, \mathrm{d}x .
  \end{equation}
  \begin{enumerate}
  \item Compute $\hat{f}_L(k)$.
  \item Show that $\lim_{R \to \infty} \int_{-R}^R \hat{f}_L(k) \,
    \mathrm{d} k = 2 \pi$.  \textit{Hint:  some basic complex
      variables can be handy here.}
  \item Now prove that $\hat{f}_L(k) \to 2\pi \delta(k)$ in the sense
    of distributions.
  \end{enumerate}
\item [A3)] 
  \begin{enumerate}
  \item Explain mathematically the concept of finite or infinite
    signal propagation speed for the heat $u_t = k u_{xx}$ and wave
    $u_{tt} - c^2 u_{xx} = 0$ equations.  Assume compactly supported
    initial conditions on $\R$ and justify your answer explicitly by
    using the solution to each problem.
  \item Consider the ``initial value problem'' for Laplace's equation
    \begin{equation*}
      \begin{split}
        u_{xx} + u_{yy} &= 0, \quad x \in \R, \quad y > 0, \\
        u(x,0) &= f(x), \quad \lim_{y \to \infty} u(x,y) = 0, \quad x
        \in \R ,
      \end{split}
    \end{equation*}
    where $y$ plays the role of ``time''.  Assume $f$ is smooth and
    compactly supported.  
    \begin{enumerate}
    \item Solve this boundary value problem.
    \item Does the solution exhibit finite or infinite propagation
      speed?  Justify your answer mathematically and explicitly as in
      part (a).
    \end{enumerate}
  \end{enumerate}
\item [A4)] Let $\Omega = \{(x,y) \in \R^2 ~ | ~ x \in (0,\pi), ~ y > 0
  \}$.
  \begin{enumerate}
  \item Solve
    \begin{equation}
      \label{eq:8}
      \begin{split}
        u_{xx} + u_{yy} &= 0, \quad x \in \Omega, \\
        u(x,0) &= V, \quad \lim_{y \to \infty} u(x,y) = 0, \quad  x \in (0,\pi), \\
        u(0,y) &= u(\pi,y) = 0, \quad y > 0 ,
      \end{split}
    \end{equation}
    using separation of variables.
  \item Why did we require $\lim_{y \to \infty} u(x,y) = 0$?
  \item Show that the solution can be written as
    \begin{equation}
      \label{eq:9}
      u(x,y) = \frac{4 V}{\pi} \textrm{Im} \sum_{n ~\textrm{odd}} \frac{Z^n}{n} 
      , \quad Z = e^{i(x + i y)} .
    \end{equation}
  \item Now sum the infinite series (the Taylor expansion $\log (1+Z)
    = Z - \frac{1}{2} Z^2 + \frac{1}{3} Z^3 - \frac{1}{4} Z^4 +
    \cdots$ may be helpful).
  \item Manipulate the summed series and recal $\textrm{Im} \log z =
    \arg z$ on the principal branch of $\log z$ to show that the
    solution in closed form is
    \begin{equation}
      \label{eq:10}
      u(x,y) = \frac{2 V}{\pi} \tan^{-1} \left ( \frac{\sin x}{\sinh y}
      \right ) .
    \end{equation}
  \end{enumerate}
\end{enumerate}

\end{document}

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