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\begin{document}

\title{Introduction to Partial Differential Equations}
\author{Midterm Exam}
\date{April 12, 2011}
\maketitle

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\begin{enumerate}

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

\item Draw the characteristic curves, then state a set of boundary
and/or initial conditions that specify the solution uniquely in the
first quadrant of the $(x,t)$ plane.
\end{enumerate}
\hfill (5+5)


\item
\begin{enumerate}
\item Show that 
\begin{equation*}
  \Delta u = 2n \lim_{\varepsilon\to 0} 
  \frac{1}{\varepsilon^2} \, 
  \PVint_{\partial B(x,\varepsilon)} (u(y)-u(x)) \, dS(y) \,.
\end{equation*}

\item Let 
\[
  U^+ = \{ x \in \R^n \colon 0 < x_1 < 1, \lvert x_2 \rvert < 1,
           \dots, \lvert x_n \rvert < 1
\]
with $n \geq 2$ denote an open half-cube.  Suppose $u \in C(\bar U^+)$
is harmonic in $U^+$ with $u(0,x_2, \dots, x_n)=0$ for $\lvert x_2
\rvert \leq 1, \dots, \lvert x_n
\rvert \leq 1$.

Show, by referring to the result from part (a) or otherwise, that
\[
  v(x) = 
  \begin{cases}
    u(x) & \text{for } x_1 \geq 0 \\
    - u(-x_1, x_2, \dots, x_n) & \text{for } x_1 < 0
  \end{cases}
\]
is harmonic in the open cube $\lvert x_1 \rvert < 1, \dots, \lvert x_n
\rvert < 1$.
\end{enumerate}
\hfill (5+5)




\item Recall that the solution to the heat equation
\begin{gather*}
  u_t - \Delta u = 0
  \quad \text{in } \mathbb{R}^n \times (0,\infty) \,, \\
  u = g
  \quad \text{on } \mathbb{R}^n \times \{t=0\} 
\end{gather*}
is given by
\[
  u(x,t) = \int_{\mathbb{R}^n} \Phi(x-y,t) \, g(y) \, dy \,,
\]
where, for $t>0$,
\[
  \Phi(z,t) = \frac1{(4\pi t)^{n/2}} \,
              e^{-\tfrac{\lvert z \rvert^2}{4t}} \,.
\]
Suppose that $g \in L^1(\mathbb R^n)$.

\begin{enumerate}
\item Show that $\lVert u \rVert_{L^\infty} \to 0$ as $t \to \infty$.
\item Show that, for all $t \geq 0$,
\[
  \int_{\mathbb R^n} u(x,t) \, d x = \text{const} \,.
\]
\item Give a physical interpretation of (a) vs.\ (b).

\end{enumerate}
\hfill (3+3+4)



\item Consider the wave equation 
\begin{gather*}
  u_{tt} - u_{xx} = 0 \qquad \text{in } \R \times (0, \infty) \,, \\
  u = g \qquad \text{on } \R \times \{t =0 \} \,, \\
  u_t = h \qquad \text{on } \R \times \{t =0 \} 
\end{gather*}
for $g \in C^2$ and $h\in C^1$.  Derive d'Alembert's solution formula
\[
  u(x,t) = \frac{g(x+t) + g(x-t)}2 
  + \frac12 \int_{x-t}^{x+t} h(y) \, dy \,.
\]

\emph{Note:} A constructive derivation is required for full credit.

\emph{Hint:} Factorize the wave equation as $(\partial_t
+ \partial_x)(\partial_t - \partial_x) u = 0$.

\hfill (10)




\item 
\begin{enumerate}
\item Let $u \in C^3_1(\R \times [0,\infty))$ solve the \emph{Airy
equation} 
\begin{gather*}
  u_t + u_{xxx} = 0 
  \quad \text{in } \R \times (0, \infty) \,, \\
  u(x,0) = g(x) 
  \quad \text{on } \R \times \{ t = 0 \} \,,
\end{gather*}
and suppose that $u,u_x \to 0$ as $x \to \pm \infty$.

Prove that $u$ is the unique solution in this class.

\emph{Hint:} Energy methods.

\item Extend your uniqueness proof to the \emph{Korteweg--de Vries
equation}
\begin{gather*}
  u_t + u \, u_x + u_{xxx} = 0 
  \quad \text{in } \R \times (0, \infty) \,, \\
  u(x,0) = g(x) 
  \quad \text{on } \R \times \{ t = 0 \} \,.
\end{gather*}
\end{enumerate}
\hfill (5+5)

\end{enumerate}

\end{document}
