from Hacker News

PDEs you should know

by lucaspauker on 2/8/22, 7:32 PM with 94 comments

• by docfort on 2/8/22, 8:43 PM

  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 speed limit on information passing from time into space!

  Don’t like traffic waves? Well, why is there some limit on spatial information connected to temporal information? It’s because I cannot see through the cars in front of me. The “fog of war” creates the waves. The denser the fog (e.g. I’m surrounded by semitrucks), the greater the likelihood of waves developing.

  This intuition is formed by being able to recognize the form of the PDE with general knowledge of the solutions, without needing to actually solve the PDE. Sure, additional insights are possible if you solve it, but knowing that traffic is like springs gives you leverage to use your ordinary intuition to understand unfamiliar things.

  Point of fact, James Maxwell of E&M fame saw the wave equation and the separate electric and magnetic field PDEs and came up with a detailed spring model to give himself a more familiar analog to play with.

• by dls2016 on 2/8/22, 9:38 PM

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

• by marginalia_nu on 2/8/22, 8:04 PM

  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 should know these.

• by valbaca on 2/8/22, 8:03 PM

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

• by prof-dr-ir on 2/8/22, 8:50 PM

  The author is an undergraduate student, and judging from this list it appears that he has yet to encounter non-linear PDEs?

  Besides the Navier-Stokes equations, which are already frequently mentioned, I would have very much liked to see Einstein's equations added as well.

• by rudiger on 2/8/22, 8:16 PM

  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.

• by aaaaaaaaaaab on 2/8/22, 10:23 PM

  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 fine scale, and the whole continuous PDE is our “simplification” of the problem.

  However, there are equations like the Einstein field equations that operate on a seemingly continuous domain, and whose solutions are impossibly complex in nontrivial cases… So how does the Universe do it?

  One can say that this question is beyond what science should be concerned with; the Universe evolves according to these equations, because this is what the Universe is. Yet, from a computational point of view it irks me…

• by frakt0x90 on 2/8/22, 7:43 PM

  I know I could look it up but having an explanation of what any of the variables mean would make this not useless.

• by Orangeair on 2/8/22, 8:11 PM

  It's been awhile since I've had a Diff EQ class, but isn't the harmonic motion one an ODE?

• by brummm on 2/8/22, 10:10 PM

  This seems like a random collection of equations from a 1st/2nd year undergrad physics class + Black Scholes.

• by sampo on 2/8/22, 10:24 PM

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

• by travisporter on 2/8/22, 8:05 PM

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

• by tgsovlerkhgsel on 2/9/22, 7:16 AM

  I've read the linked page and skimmed the comments and I still don't know what "PDE" is supposed to mean.

  From a quick search, my best guess is https://en.wikipedia.org/wiki/Partial_differential_equation?

• by Extigy on 2/8/22, 8:45 PM

  I too would have liked to have seen Navier-Stokes included, or least an inviscid Euler equation for modelling fluid flow.

• by hoseja on 2/9/22, 7:41 AM

  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.

• by jvanderbot on 2/8/22, 10:37 PM

  Pet peeve: Define your constants (at least units!). If I know the constants by heart, I probably remember the equation.

• by divbzero on 2/9/22, 8:56 AM

  Navier–Stokes is a notable omission. Perhaps because it comes in so many forms [1]?

  [1]: https://en.wikipedia.org/wiki/Navier–Stokes_equations

• by pc86 on 2/8/22, 8:38 PM

  Who is "you" in this scenario? Who actually needs to know these by heart day to day?

• by ccosm on 2/8/22, 8:01 PM

  No Navier-Stokes? Elasticity?

• by elil17 on 2/8/22, 9:13 PM

  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.

• by eternityforest on 2/9/22, 2:54 PM

  I love how the title makes it seem like they're just ordinary general knowledge things that average people will totally get.

• by fithisux on 2/8/22, 8:14 PM

  Boltzmann equation?

• by actusual on 2/8/22, 7:47 PM

  Cool, why?

• by user-the-name on 2/9/22, 11:57 AM

  The only people who will understand this page are those who already know those equations, so what's the point?

• by rq1 on 2/8/22, 10:46 PM

  Black Scholes and Heat Equation are the same, up to a change of variable.

• by ramshanker on 2/9/22, 2:12 AM

  No navier-Stocks equation? or is it in there with a different name.

• by quantum_state on 2/9/22, 12:25 AM

  Klein Gordon equation and Dirac equation should be included.

• by bally0241 on 2/8/22, 10:07 PM

  Helmholtz equation?

• by mjfl on 2/8/22, 8:10 PM

  what's the best black scholes tutorial?

• by JabavuAdams on 2/8/22, 10:13 PM

  One of these is not like the others.

• by groos on 2/8/22, 9:02 PM

  One of them is not like the rest.