Regarding traffic flow and position

Question

I need help starting this problem I don't understand how to utilize the equation for q, traffic flow. I understand that this problem wishes to have one utilize the method of characteristics but how exactly? Thanks for all the help!

Answer 1

The density is constant along the characteristic curves, which are straight lines in the $x$-$t$ plane, satisfying $$ \frac{dx}{dt} = q'(\rho) = a \left[ \ln(\rho_\max) - \ln(\rho) - 1\right] . $$ One has several cases to examine:

• if $\rho_\max/\mathrm{e}\leq\rho(0)\leq\rho_\max$, then the slope $dx/dt = q'(\rho(0))$ is negative. The characteristics are decreasing linear functions in the $x$-$t$ plane.

• if $0<\rho(0)<\rho_\max/\mathrm{e}$, then the slope $dx/dt = q'(\rho(0))$ is strictly positive. The characteristics are increasing linear functions in the $x$-$t$ plane.

Therefore, with the initial conditions of the OP, a sketch of the characteristics in the $x$-$t$ plane shows that they cannot cross. The density is constant along the characteristic curve with initial condition $\rho(0) = \rho_\max/2$, which satisfies $$ x(t) = a\left[\ln 2 -1\right] t + \rho_\max/2 \, . $$