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PDEs you should know

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Re: PDEs you should know

#71

Post by FabHK »

pfdietz wrote: post #66
bernulli wrote: post #63

Earlier quoted context omitted.

It's really cool how a Mach number emerges from traffic flow, with speed of cars vs speed of information, completely with shock waves and everything!

The equation you want for this is the Burgers Equation. https://en.wikipedia.org/wiki/Burgers%27_equation This equation was initially thought of as the appropriate continuous version of the discrete problem investigated by Fermi-Pasta-Ulam-Tsingou (on a very early digital computer shortly before Fermi's death), but then it was realized the KdV equation was better for that. https://en.wikipedia.org/wiki/Fermi%E2%80%93…

Nice short course on edX, if you want to see it in action:

Intro to Traffic Flow Modeling and Intelligent Transport Systems

(Note that thanks to aggressive monetisation of MOOCs they shut off access to the course a few weeks after you enrol in it.)

https://www.edx.org/course/intro-to-traffic-flow-modeling-an...

Re: PDEs you should know

#73

Post by tagoregrtst »

capn_duck wrote: post #46
tagoregrtst wrote: post #43

Earlier quoted context omitted.

No? Hooks law is an approximation to a material (and spring) property that sets the PDE up. Sin(x) is the solution. But I could be wrong :S

A Sin(w x) is a solution for certain initial conditions. In general there's no single "solution" to a PDE.

I meant its a solution to a specific problem set up using Hooke’s law

Re: PDEs you should know

#76

Post by madengr »

elil17 wrote: post #38

I don’t really get the point of this. Who should know these? Why should they know them? What should they know about them? As a mechanical engineer, for instance, it’s usually a bad idea for me to think about these equations - it’s to “in the weeds”, so to speak.

No post body was provided.

Re: PDEs you should know

#79

Post by agumonkey »

docfort wrote: post #29

I think it’s better to know that sometimes we only know how to describe something by relating rates of change to other states. And that’s ok. Maybe it has a closed form equation, or maybe can only be solved numerically. But if I see that a differential equation looks like a wave equation, then I get intuition that it’s describing waves. And why do the waves appear? Because the physical process the PDE describes has a…

emergent oscillations from partial information seems interesting, does this have a name ?

Re: PDEs you should know

#80

Post by squeaky-clean »

aqme28 wrote: post #25
marginalia_nu wrote: post #16

Earlier quoted context omitted.

The point is that equations without closed form solutions are pretty useless if you aren't into pretty hairy maths, Green's functions and the like. An equation everyone should know is Hooke's law. That's useful at a high school level.

I think we need a better mutual definition of useful. IMO an equation is "useful" if someone out there is doing practical things with it. You are taking a sort of definition where anyone can do math with it.

Well the page title is "PDEs You Should Know", not "PDEs Someone Should Know"
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