Math 6410 • ODE

Suppose that \(\rho(x,t)\) is the number density of cars evolving according to the
traffic model
\[
  \frac{\partial\rho}{\partial t}+\frac{\partial(\rho u)}{\partial x}=0.
\]
with \(u\) the car speed. Take \(u=1-\rho\). A queue is building up at a traffic light
\(x=1\) so that, when the light turns green at \(t=0\),
\[
  \rho(x,0)=
  \begin{cases}
    0, & x<0\text{ and }x>1,\\
    x, & 0<x<1.
  \end{cases}
\]
\begin{enumerate}
\item[(a)] Use the method of characteristics to solve the Cauchy problem. In particular,
construct the characteristic projections and identify any shocks or rarefaction waves.
\item[(b)] Hence, show that a collision first occurs at \(x=1/2\) when \(t=1/2\) and,
thereafter, there is a shock \(S(t)\). Use the Rankine-Hugoniot condition to derive
the following expression for the speed of the shock:
\[
  \frac{dS}{dt}=\frac{S+t-1}{2t}.
\]
\end{enumerate}