\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts}

\newcommand\PVint{\mathop{\setbox0\hbox{$\displaystyle\intop$}%
    \hskip0.2\wd0%
    \vcenter{\hrule width0.6\wd0height0.5pt depth0.5pt}%
    \hskip-0.8\wd0%
    }\mskip-\thinmuskip\intop\nolimits}

\begin{document}

\title{Introduction to Partial Differential Equations}
\author{Homework 6}
\date{due March 29, 2011}
\maketitle

\begin{enumerate}
\item Evans, p.~88 problem 14

\item Prove a maximum principle for the following semilinear PDE, called
Burger's equation,
\begin{gather*}
  u_t + u \, u_x = u_{xx} \,, \\
  u(x,0) = g(x)
\end{gather*}
where $u=u(x,t)$ and $(x,t) \in \mathbb{R} \times [0,\infty)$.

\item Find the Fourier transforms for the following functions on
$\mathbb{R}$:
\begin{enumerate}
\item $f(x) = e^{-t |x|}$,
\item $\displaystyle f(x) =
\begin{cases}
1 & \text{for }|x| \leq 1 \\
0 & \text{for }|x| > 1
\end{cases}$.
\end{enumerate}

\item The so-called \emph{Helmholtz equation} on $\mathbb{R}$,
\[
  u - u_{xx} = f \,,
\]
is similar to the Poisson equation: Its solution can be written in the
form
\[
  u(x) = \int_{\mathbb{R}} \Psi (x-y) \, f(y) \, dy \,.
\]
Use the Fourier transform and the result from (3a) to find $\Psi$.

\end{enumerate}


\end{document}
