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PDEs you should know

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Re: PDEs you should know

#91

Post by dan-robertson »

aidenn0 wrote: post #81
dan-robertson wrote: post #50

Earlier quoted context omitted.

To give Maxwell a little more credit (not that you aren’t), the wave equations and PDEs of today are much nicer thanks to modern knowledge and computational techniques. Maxwell didn’t have div, grad or curl and so he had dozens of equations to look at instead of just a few, and I think the terms and patterns weren’t as well known as they are today.

Do you know if the surface/line integral forms taught to those without vector-calculus under their belts an actual stepping stone to the modern ones? The number of equations are the same, they are just a lot more hairy.

I don’t know enough about the history and I don’t know what forms you are talking about. When I learned ‘vector calculus’ at university, we were introduced to surface/line integrals and div/grad/curl at around the same time.

Re: PDEs you should know

#93

Post by infotropy »

_carbyau_ wrote: post #69
valbaca wrote: post #6

Partial Differential Equations you should know but with no explanation...I mean...shouldn't you...just know? /s

Hell, I didn't know what PDE meant. Thanks!

Same here. I had similar issues with the other TLAs in the comments.

(TLA - Three Letter Abbreviation)

Re: PDEs you should know

#94

Post by aaaaaaaaaaab »

aidenn0 wrote: post #82
aaaaaaaaaaab wrote: post #51

To me the most baffling thing about differential equations is the fact that somehow the Universe is able to solve them in real time. I mean, of course there are PDEs like the Navier-Stokes equation that describe phenomena emerging from the simple interactions of an immense number of particles, so you could say that the Universe doesn’t “solve” them per se, rather, it runs the discretized simulation on an extremely fi…

Maybe we are just simulations living in a computer that is designed to calculate solutions to PDEs?

Sure, but you’ve just moved the problem of solving PDEs one layer higher to the simulating universe.

Re: PDEs you should know

#96

Post by mindcrime »

Kwpolska wrote: post #36
travisporter wrote: post #9

Why do i need to enable javascript to see the equations tex-style

This website is using MathJax [0] to render math. MathJax and its faster and leaner competitor KaTeX are the only ways to display beautiful, human-friendly math on the Web. They can be run server-side, but many sites do it client-side. The alternative, MathML [2] is a pain for humans to write [3] — it’s a late-90s XML format — and is only supported by Firefox and Safari [4]. [0] https://www.mathjax.org/ [1] https://k…

and is only supported by Firefox and Safari [4].

That's no longer (entirely) true. Chrome are re-adding MathML support (thanks to Igalia) and a significant degree of MathML support is already there. It's hidden behind a feature toggle, but if you turn it on, it does work. It's not complete as far as handling every detail of the spec, but what's there is not insignificant from what I've seen.

https://chromestatus.com/feature/5240822173794304

https://mathml.igalia.com/news/2021/02/15/mathml-plans-for-2...

Re: PDEs you should know

#98

Post by docfort »

agumonkey wrote: post #79
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 ?

The fun thing is that a PDE’s solution is emergent by definition: it is an interplay of the dynamics (specified by the the PDE), the boundary conditions, initial conditions, and maybe a forcing function (energy or information pump). Change one of those things and you’ll get a different solution. You might be reminded of fractals, which are a special case of iterated function systems, roughly a discrete version of iterated function systems. Or maybe you might think of cellular automata, all based on local update rules and some initial conditions.

The answer to your actual question is literally called the wave equation (first in the list on the linked webpage). The left side talks about some variable u and how it changes over time. The right side is how u changes over space. And the two sides are linked via a constant c. By observing the solution or by working out the units, we can understand c to be the phase velocity, or roughly, the wave speed. So the way that u evolves over space is limited by how it evolves over time (and vice versa)! u cannot react to all things in space instantaneously. Therefore, a wave emerges, carrying updates from one part of space to another.

Re: PDEs you should know

#99

Post by agumonkey »

docfort wrote: post #98
agumonkey wrote: post #79

Earlier quoted context omitted.

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

The fun thing is that a PDE’s solution is emergent by definition: it is an interplay of the dynamics (specified by the the PDE), the boundary conditions, initial conditions, and maybe a forcing function (energy or information pump). Change one of those things and you’ll get a different solution. You might be reminded of fractals, which are a special case of iterated function systems, roughly a discrete version of ite…

Ok but the wave equation doesn't have a term to represent the area of information you have in the system right ?

Re: PDEs you should know

#100

Post by blueplanet200 »

marginalia_nu wrote: post #16
medo-bear wrote: post #12

Earlier quoted context omitted.

Most useful equations don't have a closed form solution. This is why things like machine learning exist. More specifically PDEs are a hot topic in DNNs at the moment. See physics informed neural networks: https://en.m.wikipedia.org/wiki/Physics-informed_neural_netw...

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.

Depends on what you're doing, but you can numerically simulate solutions to PDEs. To me that makes them useful even if we struggle to conjure up closed form solutions.
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