Traffic density,  \rho(x,t) satisfies

 \frac{\partial\rho}{\partial t} + u_{max} (1 - \frac{2\rho}{p_{max}}\frac{\partial\rho}{\partial x}) = 0

Before a traffic light turns red, we assume a simple situation where traffic density is a constant,  \rho = \rho_0 , so that all cars are moving at the same velocity,  u = u_{max}(1 - 2\rho_0 / \rho_{max}). We assume the green light (at x = 0) turns red at t = 0. Behind the light (x < 0), the traffic desnity is initially uniform,  \rho = \rho_0 , but the density is maximum,  \rho = \rho_{max}, at the light itself since the cars are stopped there. What is the density of cars at x = -1, when t = 1?


I tried this question, and since it says "behind the light (x<0), the traffic density is initially uniform,  \rho = \rho_0 and hence wrote

 \rho(x,0) = \rho_o

I then used this condition to solve it and came up with  \rho(-1,1) = \rho_0

However my friend (we have been comparing answers) looked at "the density is maximum,  \rho = \rho_{max} at the light and wrote

 \rho(0,t) = \rho_{max}

And he solved this to get  \rho(-1,1) = \rho_{max}

We do not understand why the answers are different, and can't see where either of us has gone wrong? Can anyone see what's going on here?!

Any help much appreciated,

thanks in advance!