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Comment by aqme28 | Hacker News Reader

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aqme28·

Why does a PDE need an analytical solution?

Are you arguing that e.g. Navier-Stokes isn't useful?

edit: Just noticed that Navier-Stokes isn't even on here. This is frankly a weird list.

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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.

cs is a dominant topic on hn. cs is definitely physics and math adjacent

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.

I think what the other commenters are getting at is that PDEs can be used without having a closed form solution (and mostly are used that way as closed form solutions usually only come up in special artificial cases). You start your system in a real known state and then propagate it forward in time using (for example) the finite difference method on the equations to figure out the state at a later time. https://en.wikipedia.org/wiki/Numerical_methods_for_partial_...

>> Well, to be useful, wouldn't you need to be able to use it?

That's what we have computers for, numerical solutions to PDEs ;-)

Do you have to solve it to use it?

NS is a case in point. No general solution, but thousands of special cases that are solved and many more that can be understood using numerical methods

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