Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

PDEs you should know

lucaspauker.com

Post reply on HN
Print view View original

81–90 of 102 posts

Re: PDEs you should know

#81

Post by aidenn0 »

dan-robertson wrote: post #50
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…

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.

Re: PDEs you should know

#82

Post by aidenn0 »

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?

Re: PDEs you should know

#83

Post by throwaway81523 »

pvarangot wrote: post #20
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.

Isn't Hooke's law a solution to the harmonic motion PDE in that page?

Harmonic motion on that page is an ODE rather than a PDE. The whole page is kind of strange since it doesn't explain much.

Re: PDEs you should know

#84

Post by salty_biscuits »

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…

Navier stokes are incredibly important pdes for science and engineering, yet proof of even the existence of solutions is not currently available. Numerical methods are far more important than analytical solutions.

Re: PDEs you should know

#85

Post by civilized »

sampo wrote: post #53

The harmonic motion equation is an ordinary differential equation (ODE), not partial differential equation (PDE).

Technically, ordinary derivatives and partial derivatives are the same thing on a function of a single variable, so ODEs and PDEs are the same thing in this case.

On the other hand, the author should have included many other ODEs if they wanted to go down this path.

Re: PDEs you should know

#86

Post by kergonath »

tagoregrtst wrote: post #43
pvarangot wrote: post #20

Earlier quoted context omitted.

Isn't Hooke's law a solution to the harmonic motion PDE in that page?

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

Yes, it is. Hooke’s law tells basically that the force is proportional to the deformation, or in standard textbook physics, f = - k x, the classical harmonic oscillator.

It’s not surprising if you see a solid material that way:

- it is in stable equilibrium when no force is applied, i.e., it is at an energy minimum

- therefore either extending it or compressing it increases its energy

- therefore the second derivative of its energy is positive.

On a high level, this sounds very much like a simple parabola, because it is one.

For more technical details, we can always locally approximate a function using a Taylor’s series. In this case, the constant term and the linear term are zero if we place the frame of reference correctly (not necessary but it simplifies the equations). So the leading term is quadratic. If we are close enough to the energy minimum (i.e. if the deformations are small), we can ignore the other terms. Therefore, a solid is, to a very good degree of approximation for most of them, a harmonic oscillator.

Alternatively, to a physicist, almost everything looks like a parabola.

If the energy is a parabola, then the force is a linear function, or, as it was written in the linked document, -dE/dx = m d^2 E/dt^2 = - k x . (The first step being Newton’s second law).

The analogy can be pushed a bit: a solid deformed quickly enough will have periodic deformations, periodically contracting and expanding, i.e. vibrating. This is strictly equivalent to a harmonic oscillator oscillating in its energy well.

I suspect (please tell me if I am wrong) the sine(x) solution you mention refers to elastic waves, which appear if we take the next step and consider the solid as a bunch of coupled harmonic oscillators. Long story short, if we do that, we end up with the wave equation, from which you get sines (or more accurately, complex exponential settings).

Re: PDEs you should know

#89

Post by hoseja »

I see these as very nice, very elegant, mostly useless tautologies.

Each one needs a couple pages of explanation to be useful and if you know the explanation you don't need the ~four symbol equation.

Post reply on HN