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PDEs you should know

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Re: PDEs you should know

#61

Post by blablabla123 »

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.

I think the most generic approach is to prove existence and uniqueness and then go with numerical methods.

Re: PDEs you should know

#62

Post by sampo »

aqme28 wrote: post #24
marginalia_nu wrote: post #17

Earlier quoted context omitted.

Well, to be useful, wouldn't you need to be able to use it? Most people outside of physicists and physics-adjacent fields are very far away from the mathematical tools to deal with these equations.

Numerical methods exist. There's a whole field to simulate PDEs that you can't solve exactly.

> There's a whole field to simulate PDEs that you can't solve exactly.

There are whole fields to simulate one particular PDE: Computational fluid mechanics for the Navier-Stokes equation. Computational electromagnetics for Maxwell's equations. Computational chemistry for the Schrödinger equation. Mathematical finance ...probably does also other things than just simulates the Black-Scholes equation.

Re: PDEs you should know

#63

Post by bernulli »

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…

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!

Re: PDEs you should know

#64

Post by nautilius »

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.

You really should read up on numerical math, we really don't need closed form solutions, and this makes PDE far from useless.

Re: PDEs you should know

#65

Post by aaaaaaaaaaab »

Koshkin wrote: post #56
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…

> the Universe is able to solve them in real time Not just the Universe - analog computers can do that, too.

Analog computers are part of the Universe, so IMO they give no insight into how the Universe solves PDEs.

In contrast, digital computing can be modeled on a purely logical basis.

Re: PDEs you should know

#66

Post by pfdietz »

bernulli wrote: post #63
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…

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%93Pasta%E2%80%93Ul...

https://en.wikipedia.org/wiki/Korteweg%E2%80%93De_Vries_equa...

Re: PDEs you should know

#67

Post by dan-robertson »

rudiger wrote: post #15

The Black–Scholes equation is basically identical to the heat equation. Divide through by σ^2 and let n = σ^2 * (T - t) if you want to derive it.

Black–Scholes is the heat equation backwards, which has pretty different behaviour to the heat equation as the latter smooths things out over time and the former makes them less smooth over time. But this does make some sense: when an option reaches expiry you know exactly how much it’s worth (as a function of strike price) but the further you are before, the less well you can predict the strike price and the smoother the price function should be. Indeed your substitution reverses the direction of time but intuitions about the heat equation aren’t so applicable to Black–Scholes because intuitions are often directional.

Re: PDEs you should know

#68

Post by dan-robertson »

marginalia_nu wrote: post #8

PDEs are really useful if you are in the rare domains where they are useful. But most PDEs don't have even have closed form solutions for non-trivial boundary conditions. So unless you are a physicist or something adjacent, no, no you really don't need to know these. Hard to say what's the intended audience for the page though. Could be a message aimed at physics undergrads or something. If so, then indeed, you shoul…

I was once asked a (programming) interview question and suggested solving it with differential equations (it was a system of odes but could be made into pdes too). But that wasn’t the solution they were looking for.

Re: PDEs you should know

#70

Post by dang »

dls2016 wrote: post #41

The Unfinished PDE Coffee Table Book https://people.maths.ox.ac.uk/trefethen/pdectb.html

You should definitely submit that someday—just leave enough time for the hivemind caches to clear (ideally a few months). If you do, email us at hn@ycombinator.com so we can put it in the second-chance pool (https://news.ycombinator.com/pool, explained at https://news.ycombinator.com/item?id=26998308), so it will get a random placement on HN's front page.
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