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\noindent PCMI UFP Program  \hfill  Andrew J. Bernoff \\
July 3,  2003 \hfill Jon Jacobsen
\begin{center}
\Large{  PDE's \& Maple Lab 1 }
\end{center}
\section{Easy}
You can download a MAPLE worksheet that does these problems from:
\begin{verbatim}
     http://www.math.hmc.edu/~ajb/PCMI/PDE_Lab1.mws
\end{verbatim}
\begin{enumerate}
\item Consider the convection equation
\begin{equation}
u_t + c u_x = 0,  \qquad x \in \R, t > 0,
\label{ce}
\end{equation}
and let $F(x)=e^{-x^2}$.
\begin{enumerate}
\item Show $u(x,t)=F(x-ct)$ solves (\ref{ce}).
\item Let $c=1$.  Plot $u(x,t)$ at time $t=0,1,2$ on the same axes.
\item Plot the solution surface $u(x,t)$ for $-6 < x < 6$ and $0 \leq t < 2$.
\item Create an animation of the solution $F(x-t)$ for $0 < t < 2$.
\end{enumerate}

\item Consider Laplace's equation in $\R^2$:
\begin{equation}
\Phi_{xx} + \Phi_{yy} = 0, \qquad (x,y) \in \R^2,
\label{lape}
\end{equation}
and let $F(x,y) = e^{-x} \cos y$.
\begin{enumerate}
\item Show $F$ solves (\ref{lape}).
\item Plot $F(x,y)$ for $0 < x < 1$ and $0 < y < 2 \pi$.
\end{enumerate}

\item Consider the function $f(x)=x$ on the interval $[0,\pi]$.   The Fourier sine series
for $f$ is given by
\begin{equation}
 f(x) = 2 \sum_{m=1}^{\infty} \frac{(-1)^{m+1}}{m} \, \sin m x.  
 \label{fss}
\end{equation}
Plot the function $f$ together with its Fourier approximation taking
$2$,  $4$, $8$, $16$ and then $32$ terms of the series.
\end{enumerate}
\section{Medium}
\begin{enumerate}

\item Consider the wave equation
\begin{equation}
u_{tt} =  c^2 u_{xx},  \qquad x \in \R, \, t > 0,
\label{we}
\end{equation}
and let $F(x)=e^{-x^2}$.
\begin{enumerate}
\item Show $u(x,t)=\frac{1}{2}\left(F(x-ct) + F(x+ct)  \right)$ solves (\ref{we}).
\item Let $c=1$.  Plot $u(x,t)$ at time $t=0,1,2$ on the same axes.
\item Plot the solution surface $u(x,t)$ for $-6 < x < 6$ and $0 \leq t < 2$.
\item Create an animation of the solution $u(x,t)$ for $0 < t < 2$.
\end{enumerate}

\item Consider the heat equation
\begin{equation}
u_t =  u_{xx},  \qquad x \in \R, \, t > 0,
\label{he}
\end{equation}
and let $u(x,t)=\frac{1}{\sqrt{4 \pi (t+1)}}e^{-\frac{x^2}{4 (t+1) }}$.
\begin{enumerate}
\item Show $u(x,t)$ solves (\ref{he}).
\item Plot $u(x,t)$ at time $t=0,1,2$ on the same axes.
\item Plot the solution surface $u(x,t)$ for $-6 < x < 6$ and $0 \leq t < 2$.
\item Create an animation of the solution $u(x,t)$ for $0 < t < 2$.
\item Show that $\int_{-\infty}^{\infty} u(x,t) \, dx = 1$ for each $t > 0$.
\end{enumerate}

\item Consider the following 
initial boundary value problem for the heat equation: 
\begin{equation}
\begin{cases}
u_t =  u_{xx},  & x \in (0,\pi), \, t > 0, \\
u(0,t)=u(\pi,t)=0,  & t > 0,  \\
u(x,0) = x,   & x \in [0,\pi].
\end{cases}
\label{he2}
\end{equation}
The Fourier series solution to this is given by
\[     u(x,t) = \sum_{m=1}^{\infty} \frac{2 (-1)^{m+1} }{m} e^{-m^2 t} \sin m x.  \]
Plot the 16 term approximation to the solution $u(x,t)$ at time $t=0,1,2$. 
Create an animation of the solution $u(x,t)$ for $0 < t < 4$.
\end{enumerate}
\section{Challenge}
\begin{enumerate}
\item \textbf{Harmonic Polynomials.}
\begin{enumerate}
\item Show $F(x,y) = x^3 - 3 x y^2$ is harmonic on $\R^2$.  Plot $F$.
\item Find all cubic harmonic polynomials, i.e., all harmonic polynomials
of the form $H(x,y)=a x^3 + b x^2 y + c x y^2 + d y^3$.
\end{enumerate}

\item \textbf{d'Alembert's Solution.}  Let $u(x,t)$ be defined by
\begin{equation}
u(x,t) = \frac{1}{2} \int_{x-t}^{x+t} e^{-s^2} \, ds.
\label{wes}
\end{equation}
\begin{enumerate}
\item Show $u(x,t)$ solves the wave equation $u_{tt} = u_{xx}$.   What is
the initial displacement and velocity? 
\item Plot $u(x,t)$ at time $t=0,1,2$ on the same axes.
\item Plot the solution surface $u(x,t)$ for $-6 < x < 6$ and $0 \leq t < 2$.
\item Create an animation of the solution $u(x,t)$ for $0 < t < 2$.
\end{enumerate}

\item \textbf{The Erf Function.} Let $v(x)$ be defined by
\begin{equation}
v(x) = \int_0^x e^{-s^2} \, ds.
\label{erf}
\end{equation}
\begin{enumerate}
\item Show $u(x,t)=\frac{1}{2} + \frac{1}{\sqrt{\pi}} v \left(\frac{x}{\sqrt{4 t}} \right)$ solves the 
heat equation $u_{t} = u_{xx}$.
\item Plot $u(x,t)$ at time $t=.5, 1, 2$ on the same axes.  What is the initial temperature
distribution?
\item Plot the solution surface $u(x,t)$ for $-6 < x < 6$ and $0 \leq t < 2$.
\item Create an animation of the solution $u(x,t)$ for $0 < t < 2$.
\item Show $u_x(x,t)$ also solves the heat equation.  Find and plot this solution.
What is its initial temperature distribution?  
\end{enumerate}


\end{enumerate}


\end{document}