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\begin{center}
{\sc Fall 2020 \\ [1ex] Partial Differential Equation Comprehensive Exam} 
\end{center}
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Please read and sign the integrity statement below, and attach it to the answer that you submit to dropbox. \\
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{\bf I will not share the contents of this comprehensive exam with any person or site. I have only used allowable resources for this comprehensive exam. I have neither given nor received help during this comprehensive exam.}
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Name \underline{\hspace{3cm}}\hskip 1in
Signature \underline{\hspace{3cm}}
\newpage


\vspace{0.1in}
 {\sl Do any six problems. Clearly indicate in the table below which problems you want to be graded. If you do not select any problems we will grade the first 6 problems. Good luck!}

\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
Problems& 1 & 2& 3& 4 & 5& 6& 7& 8\\
\hline
 & & & & & & & &  \\
 \hline
\end{tabular}

\vspace{0.2in}
\begin{enumerate}

\item Use the method of characteristics to solve the Cauchy problem $u = u^2_x - 3 u^2_y $ with $u(x,0)=x^2$. Is the solution uniquely defined? If so, justify. If not, produce two solutions.

\item Assume that $u \in C^{2}(\Omega) \cap C(\overline{\Omega})$ is sub-harmonic,
$$
\Delta u(\x) \geq  0 \quad \text{for all} \quad \x = (x_1,\ldots,x_n) \in \Omega
$$
with $\Omega \subset \R^n$ a bounded, connected domain. Show that any such $u \in C^{2}(\Omega) \cap C(\overline{\Omega})$ satisfies the weak maximum principle
$$
\max_{ \x \in \overline{\Omega} } u(\x) = \max_{ \z \in \partial \Omega} u(\z).
$$

\item Consider the initial value problem for a conservation law
\begin{align}\label{eq:cl}
&u_{t}(x,t) + q^{\prime}\big( u(x,t) \big) u_{x}(x,t) = 0 \nonumber \\
&u(x,0) = g(x)
\end{align}
\begin{enumerate}[(a)]
\item Use the Leibniz rule
$$
\frac{\rd}{\rd t} \left( \int^{b(t)}_{a(t)} u(x,t) \, \rd x \right) = u( b(t) , t ) b^{\prime}(t) - u( a(t) , t ) a^{\prime}(t) + \int^{b(t)}_{a(t)} u_{t}(x,t) \, \rd x
$$
to derive the Rankine-Hugoniot jump condition for the speed $s^{\prime}(t)$ of a shock from the following conservation law property --- the solution $u(x,t)$ of \eqref{eq:cl} must obey
$$
\frac{\rd}{\rd t} \int^{b}_{a} u(x,t) \, \rd x = q\big( u(a,t) \big) - q \big( u(b,t) \big)
$$
for any interval $(a,b) \subset \R$.
\item Consider following equation
\begin{equation}\label{eq:burg}
u_t + \frac{1}{2}uu_x = 0 \quad x \in \R , t > 0 \qquad u(x,0) = \begin{cases}
2 & \text{if} \quad x < 0 \\
1 & \text{if} \quad x > 0.
\end{cases}
\end{equation}
Find the entropy solution to \eqref{eq:burg}, and justify that your solution is the entropy solution.
\end{enumerate}
\newpage
\item Consider the hyperbolic equation
\begin{align}\label{eq:hyper}
u_{tt} - 2 \lambda u_{tx} - u_{xx} &= 0 \quad \qquad  \;\;\, x \in \R, t>0 \nonumber \\
u(x,0) &= g(x) \qquad  \; x \in \R, t = 0, \nonumber \\
u_{t}(x,0) &= h(x) \qquad  \; x \in \R, t = 0,
\end{align}
for $\lambda \in \R$ any real number. Use an ansatz of the form
$$
u(x,t) = F(x+\lambda_{+} t) + G(x+\lambda_{-}t) \qquad \lambda_{\pm} := \lambda \pm \sqrt{ 1 + \lambda^{2}}
$$
to derive the d'Alembert formula
$$
u(x,t) = \frac{\lambda_{+}g(x + \lambda_{-} t ) - \lambda_{-}g(x+\lambda_{+}t)}{\lambda_{+} - \lambda_{-}} + \frac1{\lambda_{+} - \lambda_{-}} \int^{x + \lambda_{+} t }_{x + \lambda_{-} t} h(z) \, \rd z
$$
for the solution of \eqref{eq:hyper}.

\item Solve the following problem ---
\begin{eqnarray*}
u_{tt}  -  u_{xx} &=& 0, \quad  t > \max\{ -x, x\}, \ t \ge0,\\
u(x,t)& =& \phi(t), \quad  x=t, \ t \ge 0\\
u(x,t) & =& \psi(t), \quad  x=-t, \ t\ge0,
\end{eqnarray*}
where $\phi, \psi \in C^2([0,\infty))$ and $\phi(0)=\psi(0)$.

\item
Use the odd extension to find the solution to the following problem
\begin{eqnarray*}
u_{t}  -  k u_{xx} &=& 0, \quad  0<x<\infty, \ t >0,\\
u(x,0)& =& f(x), \quad 0<x<\infty,\\
u(0,t)&=&0, \quad t>0,
\end{eqnarray*}
where $f \in C([0,\infty))$.

\newpage
\item Let $\Omega \subset{\mathbb{ R}}^n$ denote a smooth, bounded domain. Suppose that a smooth function $u(\x,t)$ satisfies the heat equation
$$
u_{t}(\x,t) = \Delta u(\x,t)
$$
in $\Omega \times \{t>0\},$ and that either $u(\x,t) = 0$ or $(\partial_{\nnu} u) (\x,t) = 0$ on $\partial \Omega$. Use the energy method to prove that
$$
E(t) := \frac1{2} \int_{\Omega} u^{2}(\x,t) \; \rd \x + \int^{t}_{0} \int_{\Omega} |\nabla u|^{2}(\x,s) \; \rd \x \rd s 
$$
is constant in time, then prove uniqueness for smooth solutions to non-homogeneous Dirichlet 
$$u(\x,0) = g(\x)\quad \text{and} \quad u(\x,t) = h(\x,t) \quad \text{on}\;\partial \Omega$$
and non-homogeneous Neumann 
$$u(\x,0) = g(\x)\quad \text{and} \quad (\partial_{\nnu} u)(\x,t) = h(\x,t) \quad \text{on}\;\partial \Omega$$
initial/boundary value problems for the heat equation.

%\item
%Consider the pure initial value problem for the damped wave equation,
%\begin{equation}\label{eq:damp}
%u_{tt} + \alpha u_{t} = u_{xx} \qquad u(x,0) = g(x) \;\; u_{t}(x,0) = h(x),
%\end{equation}
%for $\alpha > 0$ a drag coefficient.
%\begin{enumerate}
%\item Fix a point $x_0 \in \R$ and a time $t_0 > 0$. For $0 \leq t \leq t_0$ let 
%$$
%E(t) := \frac1{2} \int^{x_0 + (t_0-t)}_{x_0 - (t_0-t)} u^{2}_{t}(x,t) + u^{2}_{x}(x,t) \, \rd x
%$$
%denote the total energy (kinetic plus potential) in the interval $(x_0-(t_0-t),x_0+t_0-t)$. Use Leibniz rule and the PDE \eqref{eq:damp} to show that
%$$
%E^{\prime}(t) \leq 0
%$$
%for $0 < t < t_0,$ and so $E(t)$ is non-increasing.
%\item Suppose $g(x) = h(x) = 0$ in the interval $(x_0 - t_0, x_0 + t_0)$. Show that 
%$$
%u(x,t) = 0
%$$
%in the entire triangular region $T := \{ (x,t) : x_0 - (t_0-t) \leq x \leq x_0 + (t_0-t), \; 0 \leq t \leq t_0\}$.
%\item Show that solutions of the initial value problem \eqref{eq:damp} are unique.
%\end{enumerate}

\item Let $\Omega \subset \R^{n}$ denote a bounded, connected domain with smooth boundary. Let $u(\x)$ denote the solution to Poission's equation
$$
\Delta u(\x) = f(\x) \quad (\x \in \Omega) \qquad \text{and} \qquad u(\x) = g(\x) \quad (\x \in \partial \Omega),
$$
and let $G(\x,\y)$ denote the Green's function for $\Omega$. Prove Green's representation
$$
u(\x) = \int_{\partial \Omega} (\partial_{\nnu} G)(\x,\y) g(\y) \, \rd \sigma_{\y} + \int_{\Omega} G(\x,\y) f(\y) \, \rd \y
$$
for the solution to Poisson's equation. (Here $(\partial_{\nnu} G)(\x,\y) := \nabla_{\y} G(\x,\y) \cdot \nnu(\y)$ means normal derivative).

%\item Consider the following heat equation on the half-line with zero-Dirichlet boundary conditions
%\begin{align}\label{eq:thisthing}
%u_{t}(x,t) &= u_{xx}(x,t) \quad x>0,\;t>0 \\
%u(0,t) &= 0 \quad \;\;t>0, \nonumber \\
%u(x,0) &= g(x) \quad x>0,\; t=0 \nonumber
%\end{align}
%Use the odd extension of $g(x)$ to find the solution. Show that the solution $u(x,t)$ is positive \emph{everywhere} on $(0,\infty)$ if $g(x) \geq 0$ and $g(x_*) > 0$ at some point $x_* > 0$.

\end{enumerate}

\end{document}
