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.