I've recently been making some posts around the web and on this forum attempting to figure out how to use a PDE that models traffic flow in concrete examples. I realize that I have to solve this PDE in order to use it, but I'm sort of lost on how exactly one solves it. The PDE is as follows:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \frac{\partial \rho}{\partial t} + \frac{\partial v_{max}(\rho - \frac{\rho^2}{\rho_{max}})}{\partial x} = 0[/tex]

where ##\rho## is density of vehicles, ##v## is velocity, ##\rho_{max}## and ##v_{max}## are max density and velocity respectively, ##t## is time, and ##x## is distance.

That large bit on the top of the second partial is just velocity as a function of density in ##q(\rho) = \rho v(\rho)##, where ##q## is flow (number of vehicles per unit time).

From what I've been told, solving this is going to have to be solved numerically, or at least an analytical solution would be way over my head. I've been told that Euler's method could be used here, but I'm not seeing how. I feel like I need to define ##\rho(x, t)## first in order to do anything as far as solving this goes, but no one else has hinted that that is necessary, nor have I ever seen an expression for ##\rho(x, t)## in the different sources I've read through.

Can anyone explain (as simply as possible) how I can go about solving this numerically? Or perhaps there are mathematical utilities I can use to solve this? I do have access to Wolfram's Mathematica program at the moment, which I'm lead to believe can be helpful in solving this, though I'm not overly familiar in Wolfram's language so even with this I don't know how to solve this PDE.

Any help will be greatly appreciated!

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Can someone walk me through solving a PDE numerically?

Loading...

Similar Threads - someone walk through | Date |
---|---|

Derive gen sol of non-homogeneous DEs through linear algebra | Apr 22, 2015 |

Can someone help me ho to diferentiate this equation to get the | Mar 14, 2012 |

Can someone explain this to me please? | Nov 2, 2011 |

Can someone name this method testing DE's to be homogeneous | Sep 24, 2011 |

Can someone explain this step in the proof of the convolution theorem? | Jun 17, 2011 |

**Physics Forums - The Fusion of Science and Community**