%**start of header
\documentclass[12pt]{article}
%\usepackage{html}
\usepackage{pslatex}
\usepackage[T1]{fontenc}
\usepackage[latin1]{inputenc}
\usepackage{graphicx}
\usepackage{color}
\setlength{\oddsidemargin}{0pt}
\setlength{\evensidemargin}{0pt}
\setlength{\textwidth}{6.7in}
\setlength{\textheight}{9in}
\setlength{\topmargin}{0.0in}
\DeclareGraphicsRule{.png}{eps}{.png.bb}{`convert #1 'eps:-'}
\DeclareGraphicsRule{.jpg}{eps}{.jpg.bb}{`convert #1 'eps:-'}
\DeclareGraphicsRule{.gif}{eps}{.gif.bb}{`convert #1 'eps:-'}
\begin{document}
%**end of header

\centerline{Workshop \#5, outline for TF, not students}
\centerline{APM 115: mathematical modeling}

\bigskip
Traffic and delay equations:
\bigskip

Use the Taylor expansion method for approximately solving the traffic
delay ODEs.  some guidelines:

\begin{enumerate}

\item start from eqn 2.108 in mesterton-gibbons (scanned on course web
  page).  set number of cars to 2, where the velocity of the first car
  is specified and we only need to solve for the second car.

\item linearize the log function around one,

\item Taylor expand the delay term around t to eliminate delay,
keeping up to second derivative in time
  \begin{eqnarray*}
    \frac{d\phi_2}{dt}=\phi_2'(t)
    &=&v\ln\left[1+x_{max}\left\lbrace\phi_1(t-\tau)-\phi_2(t-\tau)\right\rbrace\right] \\
    &\approx&v\left[x_{max}\left\lbrace\phi_1(t-\tau)-\phi_2(t-\tau)\right\rbrace\right] \\
    &\approx&v\left[x_{max}\left\lbrace\phi_1(t-\tau)-
        (\phi_2(t)-\tau\phi_2'(t)+\tau^2\phi_2'')\right\rbrace\right] \\
  \end{eqnarray*}

\item change variables $\xi(t)=\phi_2'$ to obtain a set of two 1st order
  ODEs
  \begin{eqnarray*}
    \xi=vx_{max}\left\lbrace\phi_1(t-\tau)-(\phi_2-\tau\xi+\tau^2\xi')\right\rbrace
  \end{eqnarray*}
  so that we have 2 equations,
  \begin{eqnarray*} 
    \phi_2'&=&\xi(t) \\
    \xi'&=&\phi_1(t-\tau)/\tau^2-\phi_2/\tau^2+\xi(1/\tau-1/(vx_{max}\tau^2))
  \end{eqnarray*}
\item integrate using same specified scenario for the first car and
  same model coefficients as used in traffic.m on the course web page
  and compare the solutions.

\item vary both the delay $\tau$ and the mean velocity $v$ of the cars
  and solve for the perturbation displacement $phi_2(t)$ using both
  the delay equation and the approximate Taylor solution.  Based on
  your calculations, plot a line in the $\tau-v$ plane that separate
  stable traffic from unstable traffic.  Interpret your results.

\item time permitting: add another term in Taylor expansion, is
  accuracy relative to solution of delay equation improved?

\item time permitting: linearized stability analysis of ODEs by
solving for eigenvalues of linearized matrix operator.

\end{enumerate}

\end{document}
