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\begin{document}

\title{Introduction to Partial Differential Equations}
\author{Final Exam}
\date{May 25, 2011}
\maketitle

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\begin{enumerate}
\item
\begin{enumerate}
\item
Solve the partial differential equation
\begin{gather*}
  t \, u_t + x \, u_x = 0 \,, \\
  u(x,1) = g(x) \,,
\end{gather*}
where $u=u(x,t)$ using the method of characteristics for $t \geq 1$. 

\item Draw the characteristic curves.  Discuss how you could solve the
equation in the exterior of the unit disk.

\end{enumerate}
\hfill (5+5)



\item Suppose $u$ is a radial solution of the \emph{Helmholtz
equation}
\[
  u - \Delta u = 0 \qquad \text{in } \R^n \,.
\]
Set $u(x) = v(r)$ with $r = \lvert x \rvert$.  Show that $v$ must
satisfy the \emph{modified Bessel equation}
\[
  v - (n-1) \, \frac{v'}r - v'' = 0 \,.
\]
(Do not attempt to solve it.)
\hfill (10)



\item Let $U \subset \R^n$ be open, bounded, and connected.  Let $u
\colon U \to \R$ be a nonnegative harmonic function.  Show that if
$u(x)=0$ for some $x\in U$, then $u=0$ everywhere in $U$.
\hfill (10)



\item Show that if the initial datum to the heat equation is even, the
solution will be even for any fixed $t \geq 0$.
\hfill (5)



\item \label{i} 
Suppose $u=u(x,t)$ is a smooth function on $\R \times [0,\infty)$ with
compact support in the $x$-direction for every fixed $t \geq 0$.
Suppose further that $u$ solves Burgers' equation
\[
  u_t + F(u)_x = 0 \qquad
  \text{with} \qquad F(u) = \tfrac12 \, u^2 \,.
\]

\begin{enumerate}
\item Show that
\[
  M(t) = \int_\R u(x,t) \, dx
  \qquad \text{and} \qquad
  E(t) = \int_\R u(x,t)^2 \, d x
\]
are constants of the motion.

\item Show that 
\[
  \int_0^\infty \int_\R (u \, v_t + F(u) \, v_x) \, dx \, dt
  + \int_\R u(x,0) \, v(x,0) \, dx = 0
\]
for every $v \in C^1(\R \times [0,\infty))$ with compact support.
\end{enumerate}
\hfill (5+5)





\item \emph{Note:} This problem continues Question~\ref{i} which
should be attempted beforehand.

Suppose that $u$ is an integral solution of Burgers' equation (i.e.,
it satisfies the condition stated in \ref{i}b) which is smooth
everywhere except on a curve $C$.  Suppose further that $C$ has a
smooth parametrization of the form $(s(t),t)$.  Let $u_l$ denote the
left-hand limit of $u$ on $C$ and let $u_r$ denote the right-hand
limit of $u$ on $C$.

\begin{enumerate}
\item Show that, on $C$,
\[ 
  [[F(u)]] = \dot s \, [[u]]
\]
where $[[F(u)]] = F(u_l)-F(u_r)$ and $[[u]]=u_l-u_r$ denote the jump
in $F(u)$ and the jump in $u$ across $C$, respectively.

\emph{Hint:} Use the divergence theorem.

\item Show that $M(t)$ is a constant of the motion.

\emph{Hint:} Use the divergence theorem on $V_l$ and $V_r$ separately
and notice that the contribution on $C$ cancels.

\input{burgers.pdf_t}

\item Suppose $u$ satisfies the \emph{entropy condition}
\[ 
  u(x+h,t) - u(x,t) \leq \frac{c}t \, h
\]
for some constant $c>0$.  Show that this implies $u_l \geq u_r$.  

\item Show that, when $u$ satisfies the entropy condition, $E(t)$ is a
decreasing function of time.

\emph{Hint:} Show that in the interior of $V_l$ and $V_r$, 
\[
  \tfrac12 \, \partial_t u^2 + \tfrac13 \, \partial_x u^3 = 0 \,.
\]
Now use the divergence theorem as in (b) and discuss the sign of the
contribution on $C$.

\item Give a physical interpretation of (d) vs.\ (b).



\end{enumerate}
\hfill (5+5+5+5+5)







\end{enumerate}

\end{document}
