are nonlinear partial differential equations the hardest form of mathematics to study and break down? it seems like there are a shitload of them with no correlation. why is that so? is there an inherent powergrap to understanding just even one equation that it would be considered worthy of a phd thesis and an entire sub-field modelling nature? conceptually there are some relationships but is it just a struggle beyond notions of solving methods?
i see them as pretty important structures that define methods of connecting different areas of mathematics, so i was just curious if this was worthy of further inquiry or if i should just stop and focus on what i currently study which is nowhere near this scope.
i never really thought about it before but PDEs does seem like a really "disjoint" field of study, more like a blanket term for a bunch of crazy-ass mathematical magic tricks
Easton Howard
it really does seem that way. is there just no way to look at it as an underlying theory like how functions and their derivatives/integrals have been slapped together under the fundamental theorem of calculus? do we need a fundamental theorem of ᚠᚢᚦᚨᚱᚲ that will change the way we live forever?
Isaac Long
lookup numerical analysis, its usually the last class in an engineering degree and it pretty much says "heres a computer that will solve all those de/pde youve been memorizing loool" it teaches how to 'find' the equation you are looking for solution to solve (the computer solves it with numbers plugged in, since there is no "general" solution ever )
Isaac Rivera
aren't those just approximations like newtons method? i mean perelman used NPDEs to solve the poincaire conjecture so it can't be just as easy as plugging it in, otherwise why didn't someone do that and make a million bucks?
Carson Edwards
Yeah it is shit like Newton's method or 4th order Runge-Kutta. Problem is that they don't always converge. Even simple shit like Euler flow equations can shit their pants with non-trivial geometry of walls.
Gabriel Miller
oh wow really? I didn't think that would actually be the case i was just spitting hot air. would understanding scalar fields help solve them?
Ryder Miller
When it comes to nonlinear PDEs, people do in fact spend their entire career studying one equation (with appropriate generalizations). It can certainly compose an entire thesis.
They are solved using a wide variety of techniques. I'd have to say that bifurcation theory comes up quite often, and can be a very powerful tool for showing nontrivial solutions to a wide variety of nonlinear PDE problems.
This is my field. Nonlinear PDEs are great. I like seeing all the crazy shit that can happen, and I like reaching for a wide variety of tools (including functional analysis, bifurcation theory) to try to tackle them.
Ethan Morgan
oh i've seen bifurcation theory pop up before, i have a few books on it. what would you reccomend as essential reading for someone who wants to delve into the world of NPDEs? Spectral Theory? inverse scattering method? Toda Lattices? it seems so massive of a field that doesn't really give you a guide on how to approach it.
Leo Long
PDEs = god tier math
Luis Price
I like Kielhöfer's bifurcation theory book. As for spectral theory; I don't really have any recommendations - I just had it from previous background. I'm not familiar with the other techniques you listed. As you said, it's a huge field, and there are so many tools out there to try. I wouldn't say there is an over-arching guide.
Another "technique" is a very brute-force perturbation series method that's used a lot. I think Stokes used it himself for his water waves, and it can be made rigorous. It might be interesting to read up on these techniques. I don't know where it can be found in books (I'm sure it can - just search "perturbation theory") - but it can be learned by reading others' papers and seeing the techniques in action.
Christopher Davis
Source of image tho ?
Blake Campbell
Yes. I studied linear PDE, it was hard as fuck, and now the non linear...
>To elaborate on Steve Huntsman's comment, I remember reading the following on Terence Tao's blog: there exist PDE that can simulate Newtonian mechanics, and using such a PDE and the correct initial conditions it is possible, in principle, to simulate an arbitrary analog Turing machine. So a general-purpose algorithm to determine even the qualitative behavior of an arbitrary PDE cannot exist because such an algorithm could be used to solve the halting problem. kek
Justin Gutierrez
Please no there are thousands of pdes surely they cam be reduced right?
Adam Allen
>is there an inherent powergrap to understanding just even one equation that it would be considered worthy of a phd thesis and an entire sub-field modelling nature? Yes. You really shouldn't think of PDE as a single field. It has a variety of domains dealing with widely different objects and using widely different techniques
Jacob Turner
SOURCE OF IMAGE ? ? ? ?
Kevin Davis
There is underlying theory. Have you studied any analysis, modern algebra or linear algebra?
Ian Young
Just use numerical methods to solve them and make tons of money like the engineers do.
Andrew Gutierrez
So what do I need to study starting now to solve the 7 Millennium Problems?
