The Unfinished PDE Coffee Table Book
https://people.maths.ox.ac.uk/trefethen/pdectb.html
PDEs you should know
41–50 of 102 posts
Re: PDEs you should know
#42The 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.
Re: PDEs you should know
#43Earlier 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?
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
Re: PDEs you should know
#44Earlier quoted context omitted.
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.
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.
>> 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 ;-)
Re: PDEs you should know
#45Earlier quoted context omitted.
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.
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.
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
Re: PDEs you should know
#46Earlier 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
ASin(wx) is a solution for certain initial conditions. In general there's no single "solution" to a PDE.
Re: PDEs you should know
#47Helmholtz equation?
Re: PDEs you should know
#48This seems like a random collection of equations from a 1st/2nd year undergrad physics class + Black Scholes.
Re: PDEs you should know
#49One of these is not like the others.
Re: PDEs you should know
#50I 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.