%  outline_cj.tex
%  04 July 2001
%
\documentclass{article}

\title{REU Report Outline}
\author{Chandra Butterfield and Joel Jorgensen}
\date{\today}

\begin{document}

\maketitle

\begin{abstract}
This is an outline of what your report should look like.  Your abstract
should summarize your work in two or three sentences.
\end{abstract}

\tableofcontents

\section{Introduction}

Your introduction should be an overview of the whole project, and a summary
of the report itself.  Roughly, you might try to write a paragraph 
here corresponding to each section that follows.

\section{The Biological Setting}

Explain the biology of the problem.  Your readers may never have heard of
\textit{angiogenesis}, which you should define.  They may not have a clear
idea of the properties of a cancerous tumor, of how vital a blood supply is,
and of how small blood vessels can grow in the body.  

\section{The Mathematical Model}

You should explain the relationship between the biological process and
the mathematical model you are setting up.  The first thing would be
to establish the \textit{geometry}; a three dimensional problem has
become a one dimensional problem.  Then you could discuss what biological
quantities you are going to be keeping track of.  And then you can
consider how you are going to simulate the changes in these quantities 
over time, using mathematical relationships that we hope work
more or less like the biological quantities do.

\section{Algorithm}

Now it's time to talk about solving your mathematical problem.  You
have a set of coupled differential equations, whose solutions are really
continuous functions of time.  

Talk about the discretization you have to do.  Mention that this introduces
a set of errors into your problem, which presumably depend on the
coarseness of your discretization.  Now discuss how a derivative in the
continous problem is approximated by a finite difference in the discrete
problem.

Since the equations are coupled, they are very large.  The decision was
made to lag the equations, and solve for one variable at a time.  Explain
that this is done to reduce the size of the system, and that, presumably,
the result will be approximately the same, if we repeat the solution
process several times.

\section{Implementation}

Here you should have a brief discussion of the programming.  Point out that
you begin using MATLAB on a Macintosh, but that the code was so slow you 
converted the numerical processing to FORTRAN on a DEC Alphastation.  Mention 
that your graphics were done by transferring data from the Alphastation 
to the Macintosh, and using MATLAB.

Discuss a few interesting features of the program.  On a typical run, what
was the time step?  How many time steps did you take?  How many iterations 
did you have to carry out on each time step to get convergence?  How many
spatial points did you use?

\section{Results}

Show some results.  Pictures would be good.  

Discuss how the numerical output of your program is similar to the biological 
process.

Look for numerical work that others have done, and see if your program 
agrees in any way.

\section{Discussion}

We'll talk about what goes here later.

\begin{thebibliography}{99}

\bibitem{Cooke},
Robert Cooke,
\textit{Dr Folkman's War},
Random House, 2001.

\end{thebibliography}

\end{document}