Math 6410 • ODE

Suppose that \(\rho(x,t)\) is the number density of cars per unit length along a road,
\(x\) being distance along the road, such that
\[
  \frac{\partial\rho}{\partial t}
  +\frac{\partial[\rho(1-\rho)]}{\partial x}=0.
\]
\begin{enumerate}
\item[(a)] Show that \(\rho\) is constant along the characteristics
\[
  \frac{dx}{dt}=1-2\rho,
\]
and derive the following Rankine-Hugoniot condition for the speed of a shock
\(x=S(t)\):
\[
  \frac{dS}{dt}=\frac{[\rho(1-\rho)]_-^+}{[\rho]_-^+}.
\]
\item[(b)] A queue is building up at a traffic light \(x=1\) so that, when the light
turns to green at \(t=0\),
\[
  \rho(x,0)=
  \begin{cases}
    0, & \text{if }x<0\text{ and }x>1,\\
    x, & \text{if }0<x<1.
  \end{cases}
\]
Solve the corresponding characteristic equations, and sketch the resulting
characteristic curves. Deduce that a collision first occurs at \(x=1/2\) when
\(t=1/2\), and that thereafter there is a shock such that
\[
  \frac{dS}{dt}=\frac{S+t-1}{2t}.
\]
\end{enumerate}