## traffic flow - strange problem!

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,