# How do differential equations relate to traffic flow?

1. Jan 1, 2016

### cmkluza

Hello, I'm trying to learn about the role that differential equations play in traffic flow, and how I can use them to model/predict/whatever you do with them with traffic flow. Do you guys know of any good and in-depth (preferably online and free) resources I can use to learn about this?

2. Jan 2, 2016

### pasmith

The [US] Federal Highway Administration has published a monograph on the subject, which may be found http://www.fhwa.dot.gov/publications/research/operations/tft/index.cfm[/url [Broken]. Chapters 4, 5 and 6 seem to be the most relevant for your purpose.

Last edited by a moderator: May 7, 2017
3. Jan 2, 2016

### lavinia

An old book is Whitham's Linear and Non-linear Waves. Chapter 3 begins with a discussion of traffic flow.

Last edited: Jan 2, 2016
4. Jan 2, 2016

### cmkluza

Thanks, I'll have to read over that when I get a chance, but it seems promising!

Alright, I'll have to see if I can find that in my library or online somewhere to check it out. Thanks for the suggestion!

Last edited by a moderator: May 7, 2017
5. Jan 2, 2016

### lavinia

Whitham was one the the first to model traffic flow with PDE's.

6. Jan 4, 2016

### lavinia

BTW; Whitham, in his book, describes how to model a simple traffic flow as a quasi-linear hyperbolic PDE. This is the case of a one lane road with no exit or entrance ramps. If the traffic is sufficiently dense, one can define quantities such as the density of cars per unit of road and the flow rate(flux) of cars past any point on the road. Since there are no exits or entrances cars are preserved so one can write down the standard conservation equation,

$d/dt ∫_a^bρdx + q(b,t) - q(a,t) = 0$

Dividing by $b-a$ and letting $b$ approach $a$ one gets the limiting equation $∂ρ/∂t + ∂q/∂x = 0$ If one assumes that the flux, $q$, depends on the local density of cars (which is not completely unreasonable) then there is some function, $φ(ρ) = q$, so $∂q/∂x = φ^{'}(ρ)∂ρ/∂x$ and the conservation equation becomes

$∂ρ/∂t + φ^{'}(ρ)∂ρ/∂x = 0$

and this is a 1 dimensional quasi-linear PDE. This is a simple case, but shows some ideas on how to do this modeling. Whitham also illustrates how to model traffic lights and other traffic phenomena.

Last edited: Jan 5, 2016