Edit to the Traffic Differential Equation

Hey guys,

The system of differential equations I introduced last time was very hard for me to solve (If you have any ideas or suggestions, please comment below). To simplify it, I will use the very basic density-velocity model, that's preliminary in any traffic theory class: the Greenshield's model. It basically implies a negative linear relationship between the twos:

Rho i is the density in lane number i. Rho max is basically the density when there's traffic jam. That's verifiable as then speed is technically 0, then substitute v = 0 in the equation to get rho = rho max. We want the time derivative of rho i:

This is when there's only one lane. To make it for 2 lanes, coupled term is introduced just like last blog.

This is easily solvable. By adding them together, and also take the difference, we can get these two equations:

You can solve this for rho 1 + rho 2 and rho 1 - rho 2, to which from there, we're solving a simultaneous equation for each term rho 1 and 2, as a function of time. After some painful solving, they are:

After a very long time, the e exponentiated term vanishes. We actually need the two densities to be then equal, because that's the balanced position, to which no cars will switch anymore and we will then have no change in density. Doing so gives:

Voila.