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\pagestyle{fancy} \lhead{{\sf Homework 13 $|$ Due August 2 (Thursday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2018 (online) $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: PDEs in other coordinate systems}
\end{center}

%$\;$

%\noindent Read the following:
%\begin{itemize}
%\item Chapters 1 \& 2 of \emph{Visual Group Theory} (VGT). \vspace{-2mm}
%\item VGT Exercises 1.1--1.4, 1.8--1.12, 2.13--2.17. \vspace{-2mm}
%\item The article \emph{Group Think} by Steven Strogatz, which
%  appeared in the NY Times in 2010. \vspace{-2mm}
%\item Francis Su's essay on good mathematical writing.
%\end{itemize} 

\noindent In the process of solving these problems, you will encounter
several ODEs, Sturm-Liouville problems, PDEs, and Fourier series, many
of which you have encountered before. You do not need to re-derive the
solutions of anything you have previously solved.

\begin{enumerate}


\item Let $u(r,\theta)$ be a function defined on the disk of radius
  $R$. Consider the following boundary value problem for
  \emph{Laplace's equation} in polar coordinates:
  \[
  u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\qquad
  u(r,\theta+2\pi)=u(r,\theta),\qquad u(R,\theta)=2-3\cos\theta+5\sin
  2\theta.
  \]
  \begin{enumerate}
    \item Assume that a solution has the form
      $u(r,\theta)=R(r)T(\theta)$. Plug this back in and seperate
      variables to get an equation for $R$ and $T$, including boundary
      conditions.
    \item Solve the ODEs for $R(r)$ and $T(\theta)$, and determine all
      possible eigenvalues $\lambda_n$. Make sure to impose the additional
      requirement that $R(0)$ exists. 
    \item Find the general solution to Laplace's equation in polar
      coordinates.
    \item Plug in $r=R$ to find the particular solution to this
      boundary value problem. \medskip
  \end{enumerate}



\item Let $u(r,\theta,t)$ be a function defined on the disk of radius
  $R=1$, and for all $t\geq 0$. Consider the following initial/boundary
  value problem for the \emph{heat equation} in polar coordinates:
  \begin{align*}
    u_t=c^2\,\Delta u,\qquad &u(r,\theta+2\pi)=u(r,\theta),\qquad\qquad\qquad 
    u(r,\theta,0)=1-r^2+h(r,\theta)\\
    &u(1,\theta,t)=2-3\cos\theta+5\sin 2\theta.
  \end{align*}
  Here, $h(r,\theta)$ denotes the \emph{steady-state} solution.
  \begin{enumerate}
  \item What is $h(r,\theta)$?
  \item Make the change of varibles
    $v(r,\theta,t)=u(r,\theta,t)-h(r,\theta)$, and re-write the PDE
    above, including the boundary and initial condtions, in terms of
    $v$ instead of $u$. 
  \item Find the general solution for this \emph{homogeneous} PDE using
    separation of variables. Assume that
    $v(r,\theta,t)=f(r,\theta)g(t)$.
  \item Find the particular solution that satisfies the initial
    condition. \medskip
  \end{enumerate}
  


\item Let $u(r,\theta,t)$ be a function defined on the disk of radius
  $R=1$, and for all $t\geq 0$. Consider the following initial/boundary
  value problem for the \emph{wave equation} in polar coordinates:
  \[
  \begin{array}{lll}
    u_{tt}=c^2\,\Delta u,\qquad & u(r,\theta+2\pi)=u(r,\theta)\qquad &
    u(r,\theta,0)=1-r^2 \\
    & u(1,\theta,t)=0,\qquad & u_t(r,\theta,0)=0.
  \end{array}
  \]
  \begin{enumerate}
  \item Briefly describe, and sketch, a physical situation which this
    models. Be sure to explain the effect of the boundary conditions
    and both initial conditions. [\emph{Hint}: What does the function
      $u(r,\theta,0)=1-r^2$ look like?]
  \item Assume that there is a solution of the form
    $u(r,\theta,t)=f(r,\theta)g(t)$. Plug this back in and separate
    variables to get an ODE for $g$ and a PDE for $f$. Include
    boundary conditions for $f$, and one initial condition for $g$.
  \item Find the general solution to this BVP. 
  \item Find the particular solution that additionally satisfies the
    initial conditions.

    \bigskip
  \end{enumerate}



\item Let $u(r,\theta,\phi,t)$ be the temperature of a sphere of
  radius $R=\pi$. Assume that the initial temperature is constant, and
  that temperature does not depend on the latitude or longitude. In
  this case, $u(r,\theta,\phi,t)=u(r,t)$, and the heat equation
  reduces down to the following:
  \[
  u_t=c^2(u_{rr}+\frac{2}{r}u_r),\qquad u(\pi,t)=0,\qquad u(r,0)=T_0.
  \]
  There is an implied boundary condition at $r=0$, that $u(0,t)$ is
  finite.
  \begin{enumerate}
    \item Assume that there is a solution of the form
      $u(r,t)=f(r)g(t)$. Separate variables to get two equation, and
      ODE for $g$, and a (singular) Sturm-Liouville problem for $f$.
    \item Solve the Sturm-Liouville problem for $f$. [\emph{Hint}: One
      way would be to use the power series method. However, a much easier
      way is to define $y(r)=rf(r)$, and re-write the problem in terms of
      $y$.] 
    \item Write the general solution to this PDE. 
    \item Find the particular solution that additionally satisfies the
      initial condition. Leave the formulas for coefficients in terms
      of integrals; no need to solve them. \bigskip
  \end{enumerate}


  % Logan, Exercise 4.4.4
\item Consider a sphere of radius $R=1$, and suppose that
  $u(r,\phi)$ represents a potential the depends only on the radius
  $r\in[0,1]$ and latitude $\phi\in[0,\pi]$. In this problem, we will
  solve Laplace's equation under these conditions. The boundary value
  problem is
  \[
  u_{rr}+\frac{2}{r}u_r+\frac{1}{r^2\sin\phi}(\sin\phi
  u_\phi)_\phi=0,\qquad u(1,\phi)=f(\phi).
  \]
  \begin{enumerate}
    \item Assume $u(r,\phi)=R(r)Y(\phi)$, and derive the following two
      equations
      \[
      1((1-x^2)y')'=\lambda y,\qquad (r^2R')'=\lambda R,
      \]
      where $x=\cos\phi$ for $-1<x<1$ and $y(x)=Y(\cos^{-1}(x))$.
    \item The equation for $y(x)$ may look familiar -- it is
      \emph{Legendre's equation} (see HW 4, 6, and 9). Recall that it
      has bounded, continuous solutions on $[-1,1]$ when 
      \[
      \lambda_n=n(n+1),\qquad y_n(x)=P_n(x),\quad n=0,1,2,\dots.
      \]
      These are the eigenvalues and eigenfunctions of the related
      (singular) Sturm-Liouville problem (see HW 9). Carry out the
      details of deriving the general solution to Laplace's equation,
      which will be
      \[
      u(r,\phi)=\sum_{n=0}^\infty c_nr^nP_n(\cos\phi),\qquad\text{where}\qquad
      c_n=\frac{1}{||P_n||^2}\int_0^\pi f(\phi)P_n(\cos\phi)\sin\phi\;d\phi.
      \]
    \item If $f(\phi)=\sin\phi$, find an approximate,
      \emph{fourth-order} solution. That is, truncate it after the
      $n=4$ term. \emph{Hint}: The Legendre
      polynomials can be derived from the following formula:
      \[
      P_n(x)=\frac{1}{n!2^n}\frac{d^n}{dx^n}\left[(x^2-1)^n\right].
      \]
      
  \end{enumerate}
  
  

%  % Logan, Exercise 4.4.5
%\item In the mid 1860s, Lord Kelvin used the heat equation to estimate
%  the age of the earth. He assumed that it was a perfect sphere, that
%  its initial temperature was $7000^\circ$, and the temperature of the
%  surrounding atmosphere has always been $0$ degrees, the diffusivity
%  is $c^2=0.007$ cm$^2/$sec, and that the \emph{geothermal gradient}
%  is $3.7\times 10^{-4}$ deg/cm. 


\end{enumerate}

\end{document}
