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\hyphenation{op-tical net-works semi-conduc-tor}
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\begin{document}
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% paper title
\title{Homework 8}

\markboth{CS 237C, Spring 2006} {Shell \MakeLowercase{\textit{et
al.}}: Bare Demo of IEEEtran.cls for Journals} \maketitle

\section{}
Consider the heat equation in $1D$,
\begin{displaymath}
u_t = bu_{xx}.
\end{displaymath}

\begin{enumerate}
\item Under which conditions are each of the schemes below stable?  Justify your answers.
\item For each scheme, state the order of accuracy in time and space (you don't need to prove this, just state it).
\end{enumerate}

\subsection{Forward Euler}
\begin{displaymath}
\frac{u^{n+1}_j-u^n_j}{\triangle t} = b\frac{u^n_{j+1} - 2u^n_j + u^n_{j-1}}{\triangle x^2}
\end{displaymath}

\subsection{Backward Euler}
\begin{displaymath}
\frac{u^{n+1}_j-u^n_j}{\triangle t} = b\frac{u^{n+1}_{j+1} - 2u^{n+1}_j + u^{n+1}_{j-1}}{\triangle x^2}
\end{displaymath}

\subsection{Crank-Nicholson}
\begin{displaymath}
\frac{u^{n+1}_j-u^n_j}{\triangle t} = \frac{b}{2}\left( \frac{u^n_{j+1} - 2u^n_j + u^n_{j-1}}{\triangle x^2} + \frac{u^{n+1}_{j+1} - 2u^{n+1}_j + u^{n+1}_{j-1}}{\triangle x^2}\right)
\end{displaymath}

% that's all folks
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