Why do PDE's seem so unnatural? [closed]

First let me preface by saying that I'm highly aware of the fact that plenty a math topic seems unnatural upon first learning. But PDEs seem to have a special place in my "unnatural" category of mathematics. Specifically because I'm comfortabe with pretty much everything else, even when I'm lost. In PDE texts I frequently encounter statements by authors such as "but if we demand system X has Y property", "if we assume the solution is of the form TX", etc. All backwards statements compared to say, real analysis.

It seems to me like one of the Bernoulli's came up with separation of variables for a way to get a solution and nobody ever tried anything else, they just kept pushing eigenfunction expansion until there's no way to back out. It's like we found an alien spacecraft and found some parts that do stuff but at the end of the day we don't know what the inspiration for the design was.

Is there another method of handling PDEs that seems more "natural" (in the sense that number theory seems natural) as opposed to separation of variables, eigenfunction expansion, green's functions? There's no way I'm the first person to question the current trajectory of PDEs.

Note: I do find some of the math to be pretty cool in PDE's, in a problem solving sense. And eigenfunction expansions and fourier series are cool in their own right. I like Green's functions too. I just can't get over the fact that we're forcing the math to work and then back-justifying everything. Like the Dirac Delta, for example.

Rant over. Can anybody lead me to the light, or atleast tell me to stop whining in a motivational way? Thank you.

• Do you find eigenfunctions unnatural in general, or just in the context of PDEs? Like if someone said "I want to compute $A^n$, so let me first compute its eigendecomposition $A = PDP^{-1}$", would you find that unnatural? If yes, then the problem isn't really PDEs here. But if no, then what's so unnatural at generalizing eigenvectors to eigenfunctions when you look at PDEs? I'm not a PDE expert, but I feel like when you see a linear operator whose long-term behavior you want to understand, you try to look at its eigenstuff... you did that in linear algebra; isn't it the same here? Apr 8, 2018 at 22:50
• Regarding separation of variables, to me, trying $u(x,t) = X(x) T(t)$ is just trying to see if the system is in fact decoupled... which is how you solve problems in everyday life, right? Like if you want to eat pizza but your spouse wants pasta, you first try to see if it's possible to solve those problems independently (you buy yourself pizza and you buy your spouse pasta), and only if you encounter difficulties do you try to find a coupled solution (maybe you don't have enough money so you split half and half or something). (Sorry if this sounds dumb, I couldn't think of a better example...) Apr 8, 2018 at 23:05

The big problem with PDEs that makes them so difficult is geometry. ODEs are fairly natural because we only have to consider a few cases when it comes to the geometry of the set and the known information about it. We can find general solutions to many (linear) ODEs because of this, and so there's usually a natural progression to get there.

PDEs on the other hand, have at least 2-dimensional independent variables, so the variety in the kinds of domains is increased from just intervals to any reasonable connected domain. This means that initial and boundary values contain a lot more information about the solution, so a general solution would have to take into account all possible geometries. This is not really possible in any meaningful way, so there usually aren't general solutions.

When we do pick a geometry, it often simplifies the problem significantly. One nice domain is $\mathbb{R}^n$. Many simple PDEs have invariance properties, which means that if we're given enough space to "shift" and "scale" parts of the equation, we can probably come to reason about what the solution should look like. For these situations, there may be general solutions (see PDEs on unbounded domains). These solutions are also more of the straightforward kinds of solutions we see in ODEs.

Many PDEs and ODEs simply don't have closed form solutions, and so usually rely on series methods and other roundabout ways to write solutions which don't really "look" like solutions.

Separation of variables is a kind of reasonable guess that the effect of each independent variable should be independent in some way. We can try writing the solution as a sum or a product or some other combination of independent functions of each independent variable, and this often reduces the problem in some way which allows up to separate the PDE into a series of ODEs. We don't know that this will work in every case, but if we can show uniqueness of a solution, then finding any kind of solution means we found the solution to the problem.

The last main reason is that the theory of PDEs is way harder than the theory of ODEs. So, when you're first learning to solve ODEs, you can be introduced to these methods with a bit of theory and some background on why each of the guesses and techniques makes some sense. When first learning to solve PDEs, however, you probably will not have anywhere near the amount of background you need to fully understand the problems. You can be taught the methods, but they will always seem like a random guess or just a technique that happens to work, until you learn about the theory behind it. As Eric Towers mentions, some Lie algebra would be a good place to start, and I would also recommend PDE books with a more theoretical slant to them, such as Lawrence Evans' text. Since you seem to have some background in real analysis (and so presumably some basic modern/abstract algebra), I think both of these paths should be achievable at your level.

I would strongly recommend learning about Lie symmetry analysis of differential equations.

• Even in ODEs, the sentence structure you describe is common. "If $N_x = M_y$, then the equation is exact and we can..."
• All the techniques from ODEs and all the techniques you mention for PDEs can be expressed in this one framework.
• Computer technology is up to the task of computing the (algebraically horrendous) prolongations needed to calculate anything. This was definitely not so in Lie's time.

Starting places (Links are to the publisher, you can find these other places):

Additionally, if you get your head wrapped around these ideas, you will have a head start understanding Galois theory. (Groups of symmetries holding the set of solutions of a differential frame invariant are remarkably analogous to groups of symmetries holding the set of roots of a polynomial invariant.)

I'd tend to agree that "random" PDE's can seem un-natural, or, at least ... random. But, as with other parts of mathematics, in my experience, even though we might like to prove the most general results possible without more effort, we don't really care nearly as much about random items as we do about things that arise in the course of relatively serious (mathematical and otherwise) enterprises.

Also, technical possibilities constrain what we can do, which, as it happens oftens leads to consideration of linear PDEs, at least as linearized versions of possibly more "genuine" PDEs that may better describe physical, geometric, or other phenomena.

Then, if we throw in natural symmetries, on Euclidean spaces we quickly find the three canonical partial differential operators: Laplacian $\Delta=\sum_i \partial^2/\partial x^2$, the heat operator $\Delta-\partial/\partial t$, and the wave operator $\Delta-\partial^2/\partial t^2$.

The constant-coefficient aspect is due to being translation-invariant, which is a common/reasonable feature. And so on.

And, as we know, to systematically describe the process of "solve differential equation with boundary conditions..." we often can use, construct, or prove existence of a "Green's function", whose description and properties are probably best understood in terms of distributions, a.k.a., generalized functions. And the latter are usefully discussed using Fourier series, Fourier transforms, and other eigenfunctions expansions when available.

So, in my opinion, these basic things are just more calculus, and are nearly universally useful.