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\begin{center}
{\sc Spring 2022 \\ [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}}
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\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 Solve the following initial value problem using characteristics. 
$$u_x^2+u_y^2=u$$
with the initial condition $u(0,y)=ay^2$. For what positive $a$ are there solutions? Is the solution unique? 

\item Consider the second order linear equation $x^2u_{xx}-y^2u_{yy}=0$
\begin{enumerate}[{\rm (a)}]
\item Classify the equation as hyperbolic, parabolic, or elliptic.
\item Rewrite this equation in its canonical form. 
\end{enumerate}

\item Let $\Omega \subset \R^{n}$ denote a bounded, connected domain with smooth boundary. Use Green's identity and the energy method to show that $u(\x,t) = 0$ is the unique solution to the following parabolic PDE with bi-harmonic diffusion:
\begin{align*}
u_{t} &= -\Delta( \Delta u ) \qquad \, \x \in \Omega, \; t > 0, \\
\Delta u(\x,t) &= 0 \quad \qquad \qquad \;\, \x \in \partial \Omega, \; t > 0, \\
u(\x,t)  &= 0 \quad \qquad \qquad \;\, \x \in \partial \Omega, \; t > 0, \\
u(\x,0) &= 0 \quad \qquad \qquad \;\, \x \in \Omega, \; t = 0.
\end{align*}

\item 
\begin{enumerate}[{\rm (a)}]
\item Green's identity is given by 
$$\int_{\Omega} (g\Delta f- f\Delta g){\rm d}x =\int_{\partial\Omega}(g\partial_nf-f\partial_ng)$$
where $\partial_n$ is the normal derivative. Prove this by applying the divergence theorem. 
\item Let $K(x)$ denote the fundamental solution of the Laplace operator $\Delta$ in $\mathbb{R}^3$, and let $v(x)$ be an infinitely differentiable function which equals zero for $|x|>R$. Apply Green's identity to prove the following identity: 
$$\int_{\mathbb{R}^3}K(x)v(x){\rm d}x=v(0)$$
\end{enumerate}

\item If $\Omega$ is a bounded open set in $\mathbb{R}^2$ with smooth boundary $\partial\Omega$. Show that if $u$ satisfies $$\Delta u=0\quad\text{in}\quad \Omega$$ then, using the mean value property for harmonic functions to show $$\max_{\Omega}u=\max_{\partial\Omega}u$$

\item 
\begin{enumerate}[{\rm (a)}]
\item Verify that $u(x,t) = F(x+ct)+G(x-ct)$, $F$ and $G$ twice differentiable, is a solution of the wave equation $$u_{tt}=c^2u_{xx}$$ Use this to solve the initial value problem for the wave equation with initial conditions 
\begin{align*}
u(x,0) = f(x), \quad \text{for}\quad x \in\mathbb{R},\\
u_t(x,0) = g(x), \quad \text{for}\quad x \in\mathbb{R}.
\end{align*}
Verify your solution. 
\item Solve the initial boundary problem for the wave equation on the quarter plane $\{(x,t): x>0, t>0\}$ with general initial conditions, as above, but for $x>0$, and boundary condition $u(0,t)=0$, for $t>0$. 
\end{enumerate}

\item Consider the wave equation in the first quadrant $x>0,t>0$
\begin{align*}
u_{tt}  &= u_{xx}, \quad \;\,  0<x<\infty, \ t >0,\\
u(x,0) &=  f(x), \quad 0<x<\infty,\\
u_t(x,0)  &=g(x), \quad 0<x<\infty,\\
u(0,t)  &= 0,\qquad\;\, t>0,
\end{align*}
where $f \in C^2([0,\infty))$ and $g \in C^1([0,\infty))$ satisfy $f(0)=f'(0)=g(0)=0$.
\begin{enumerate}
\item
Solve the problem using the odd extensions of $f$ and $g$.
\item
Sketch the domain of dependence of a point $(x_0,t_0)$ where $0<x_0<\infty$ and $t_0>0$.
\item
Sketch the region of influence of a point $x_0$ where $0 < x_0 < \infty$.
\end{enumerate}

\item
Let $\Omega = B_{1}(\0)$ denote the unit ball in $\R^{2}$ centered at the origin. Show the solution to \begin{align}\label{eq:par}
u_{t}(\x,t) &= \Delta u(\x,t) \quad \, \text{in} \quad \Omega_{T} := \{ (\x,t) : \x \in \Omega , \; 0 < t < T\}\\
u(\x,t) &= h(\x,t) \quad \quad \x \in \partial \Omega, \; t > 0 \nonumber\\
u(\x,0) &= g(\x) \quad \quad \;\;\;\x \in \Omega, \; t=0 \nonumber
\end{align}
satisfies the inequality
$$
\mathrm{e}^{-8t}\left(1 - |\x|^2\right)^{2} \leq u(\x,t) \leq \mathrm{e}^{-4t}(1-|\x|^2)  
$$
if $ g(\x) = 1-|\x|^2$ and $h(\x,t) = 0$. You may use the identities $\Delta |\x|^2 = 4$ and $\Delta |\x|^4 = 16|\x|^2$ (valid in two dimensions) without proof.

\end{enumerate}

\end{document}
