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PDEs you should know

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Re: PDEs you should know

#51

Post by aaaaaaaaaaab »

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…

Re: PDEs you should know

#54

Post by jcla1 »

dls2016 wrote: post #42
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.

The Schrodinger equation is the heat equation with complex time. Although qualitatively it’s dispersive, not dissipative.

The difference is that in the Schrödinger case you're effectively 'turning' the solution (in the complex plane) which leads to the uncomfortable question of whether the solution to the heat equation you'd start with is still defined. When going from heat to Black-Scholes you're just rescaling in 'existing' dimensions which doesn't change the character of the PDE.

Re: PDEs you should know

#55

Post by [deleted] »

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…

[deleted]

Re: PDEs you should know

#56

Post by Koshkin »

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.

Re: PDEs you should know

#58

Post by dan-robertson »

jvanderbot wrote: post #57

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

Most of the units can be inferred, e.g. for the wave equation, say the units of u are A (for amplitude, but you can guess whatever), x is L (for length) and t is T for time (both are extremely conventional dimensions). You convert the pde to units and get:

  A/T^2 = units(c)^2 A/L^2,
and can therefore say:

  units(c) = L/T
And guess that c is the speed of the wave or something proportional to it (it is, in fact, the speed).

For the simple harmonic oscillator you get:

  units(m) L/T^2 = units(k) L
which is insufficiently determined but gives units(k/m) = 1/T^2, and you might guess m is mass (in kg say) and then k is kgs^-2, or force per distance, a reasonable set of units for a spring constant (the ode is just Hook’s law: F=kl, but F=ma)

For the other equations it becomes harder but the point of the website isn’t really to teach you what the pde is. It’s extremely easy to search for the equation on Wikipedia (as the site gives their names) and look up the units and a bit about the equations there.

Re: PDEs you should know

#60

Post by [deleted] »

aqme28 wrote: post #26
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.

The math isn't that hairy. I took numerical methods and computational physics classes in undergrad.

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