\documentclass[11pt]{article} \usepackage{amssymb,amsmath} \usepackage{enumerate} \usepackage{graphicx} \setlength{\textwidth}{6.6in} \setlength{\textheight}{8.5in} \setlength{\evensidemargin}{-0.2in} \setlength{\oddsidemargin}{-0.2in} \setlength{\topmargin}{-0.3in} \pagestyle{empty} \newcommand{\R}{\mathbb{R}} \newcommand{\x}{\mathbf{x}} \newcommand{\y}{\mathbf{y}} \newcommand{\z}{\mathbf{z}} \newcommand{\vv}{\mathbf{v}} \newcommand{\rd}{\mathrm{d}} \newcommand{\pp}{\mathbf{p}} \newcommand{\0}{\mathbf{0}} \newcommand{\nnu}{\boldsymbol{\nu}} \newcommand{\divg}{\mathrm{div}} \begin{document} \begin{center} {\sc Fall 2020 \\ [1ex] Partial Differential Equation Comprehensive Exam} \end{center} \vskip 1.5in Please read and sign the integrity statement below, and attach it to the answer that you submit to dropbox. \\ \vskip .5in {\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.} \vskip 2in 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 00,\\ u(x,0)& =& f(x), \quad 00, \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}