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\pagestyle{fancy} \lhead{{\sf Homework 11 $|$ Due July 31 (Monday) }}
\rhead{\thepage} \cfoot{{\sf Math 4340/6340 $|$ Advanced Engineering
    Mathematics $|$ Summer II 2017 (online) $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-6mm}

\begin{center}
  \textsc{Topics: Harmonic functions and higher-dimensional PDEs}
\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 For each of the following functions, compute its
  \emph{Laplacian}, $\nabla^2f$, and determine if it is
  \emph{harmonic}. Feel free to use WolframAlpha to compute each
  individual second-partial derivative, as that is just a calculus
  exercise.
  \begin{multicols}{2}
    \begin{enumerate}
    \item $f(x)=10-3x$.
    \item $f(x,y)=x^2+y^2$.
    \item $f(x,y)=e^x\cos y$.
    \item $f(x,y)=x^3-3xy^2$.
    \item $f(x,y)=\ln(x^2+y^2)$.
    \item $f(x,y,z)=\ln(x^2+y^2+z^2)$.
    \item $f(x,y,z)=\frac{1}{\sqrt{x^2+y^2+z^2}}$. \medskip
    \end{enumerate}
  \end{multicols}


\item Let $u(x,y)$ be a function defined on a square region, $0\leq
  x,y\leq 1$. In this problem, we will solve \emph{Laplace's equation} in
  this region subject to different \emph{Dirichlet} boundary
  conditions.
  \begin{enumerate}
  \item Solve the following boundary value problem:
    \[
    \nabla^2u=0,\qquad u(0,y)=u(1,y)=u(x,0)=0,\quad u(x,1)=4\sin\pi
    x-3\sin 2\pi x+2\sin 3\pi x.
    \]
  \item Solve the following boundary value problem:
    \[
    \nabla^2u=0,\qquad u(x,0)=u(x,1)=u(0,y)=0,\quad u(1,y)=y(1-y).
    \]
  \item Solve the following boundary value problem:
    \begin{align*}
    \nabla^2u=0,\qquad u(x,0)=u(0,y)=0,\quad u(x,1)&=4\sin\pi
    x-3\sin 2\pi x+2\sin 3\pi x, \\ u(1,y)&=y(1-y).
    \end{align*}
    \emph{Hint}: Use superposition!
  \item Sketch the solutions to (a), (b), and (c). Feel free to use
    WolframAlpha to help. \bigskip
  \end{enumerate}



%\item Let $f(x,y)$ be a function defined on a square region, $0\leq
%  x,y\leq 1$. Find the general solution to the following PDE,
%  called the \emph{Helmholtz equation}, subject to the given
%  (Dirichlet) boundary conditions:
%  \[
%  \nabla^2 f=-\lambda f,\qquad f(x,0)=f(x,1)=f(0,y)=f(1,y)=0.
%  \]
%  The solutions to this PDE can be thought of as \emph{eigenfunctions}
%  of the Laplace operator (in rectangular coordinates) and the values
%  of $\lambda$ are the corresponding (Dirchlet) eigenvaules.



\item Let $u(x,y,t)$ be a function defined on a square region, $0\leq
  x,y\leq 1$ and $t\geq0$. Consider the following initial/boundary value
  problem for the \emph{two-dimensional heat equation}:
  \begin{align*}
    u_t=c^2\,\nabla^2u,\qquad
    &u(x,0,t)=u(x,1,t)=u(0,y,t)=u(1,y,t)=0, \\
    &u(x,y,0)=2\sin\pi x\sin\pi y+5\sin 2\pi x\sin\pi y.
  \end{align*}
  \begin{enumerate}
  \item Briefly describe, and sketch, a physical situation which this
    models. Be sure to explain the effect of the boundary conditions
    and the initial condition.
  \item Assume there is a solution of the form
    $u(x,y,t)=f(x,y)g(t)$. Find $u_{xx}$, $u_{yy}$, and $u_t$.
  \item Plug $u=fg$ back into the PDE and ``separate variables'' by
    dividing both sides by $c^2fg$.  Briefly justify why this quantity
    must be a constant.  Call this constant $-\lambda$. 
  \item Write down, and then solve, an ODE for $g(t)$, and a PDE for
    $f(x,y)$ with four boundary conditions.
  \item Find the general solution to this boundary value problem. 
  \item Find the particular solution that additionally satisfies the
    initial condition.
  \item What is the steady-state solution? Give a mathematical and
    intuitive (physical) justification. \bigskip
  \end{enumerate}
  
  
\item Consider the following initial/boundary value problem for the
  heat equation in a square region, and the funtion $u(x,y,t)$, where
  $0\leq x,y\leq 1$ and $t\geq0$.
  \begin{align*}
    u_t=c^2\,\nabla^2u,\qquad
    &u(x,0,t)=u(x,1,t)=u(0,y,t)=u(1,y,t)=0,\\
    &u(x,y,0)=(3\sin\pi x+5\sin 2\pi x)\,y(1-y).
  \end{align*}
  Since the only difference between this problem and the previous one
  is in the initial condition, steps (b)--(e) are the same and need
  not be repeated. Carry out steps (f) and (g), given this new initial
  condition. \bigskip



\item Consider the following initial boundary value problem for the heat
  equation in a square region, $0\leq x,y\leq 1$ and $t\geq0$:
  \begin{align*}
    u_t=c^2\,\nabla^2u,\qquad &u(x,0,t)=u(0,y,t)=0 \hspace{1.75in}
    u(x,y,0)=1+h(x,y) \\ &u(x,1,t)=4\sin\pi x-3\sin 2\pi x+2\sin 3\pi
    x \\ &u(1,y,t)=y(1-y).
  \end{align*}
  Here, $h(x,y)$ denotes the \emph{steady-state} solution.
  \begin{enumerate}
  \item What is $h(x,y)$? Justify your answer. [\emph{Hint}: This can
    be found by setting $u_t=0$. Use the results of a previous
    problem.]
  \item Make the substitution $v(x,y,t)=u(x,y,t)-h(x,y)$ and re-write
    the PDE above, including the boundary and initial conditions, in
    terms of $v$ instead of $u$. (The resulting PDE is
    \emph{homogeneous} -- you should recognize it from a previous problem.)
  \item Briefly describe, and sketch, a physical situation which this
    models. Be sure to explain the effect of the boundary conditions
    and the initial condition, as well as the difference between what
    $u$ and $v$ represent.
  \item Find the general solution for $u(x,y,t)$. [\emph{Hint}:
    Observe that $u$ is just the sum of the steady-state solution and
    the general solution of the related homogeneous boundary value
    problem.]
  \item Find the particular solution that additionally satisfies the
    initial condition.
  \end{enumerate}



\item Consider the following initial boundary value problem for the wave
  equation in a square region, where $0\leq x,y\leq 1$ and $t\geq 0$:
  \begin{align*}
    u_{tt}=c^2\,\nabla^2u,\qquad
    &u(x,0,t)=u(x,1,t)=u(0,y,t)=u(1,y,t)=0,\\
    &u(x,y,0)=x(1-x)y(1-y),\\ &u_t(x,y,0)=1.
  \end{align*}
  \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.
  \item Solve this PDE. The major difference between this and the heat
    equation is in the function $g(t)$. You do not need to repeat
    steps that are identical to previous problems.
  \item What is the long-term behavior of this system? Give a
    mathematical and intuitive (physical) justification.
  \end{enumerate}




\end{enumerate}

\end{document}
