Math 6410 • ODE

Traffic flow may be modeled by the equation \[ \frac {\partial }{\partial x}\bigl (G(\rho )\bigr )+\frac {\partial }{\partial t}\rho =0 \] where \(\rho (x,t)\) is the density of traffic flow along a highway. If \(\rho _{\max }\) is the maximum density of cars in a bumper to bumper situation, then \[ G(\rho )=c\rho \left (1-\frac {\rho }{\rho _{\max }}\right ) \] where \(c>0\) is the free speed constant. Suppose that the initial concentration is given by \[ \rho (x,o)= \begin {cases} \frac 12\rho _{\max }, & \text {if }x<0,\\ 0, & \text {if }x\ge 0. \end {cases} \] Describe the weak solution. Does a shock form? Interpret your result in terms of traffic flow.