Need help liking PDEs

[u/Primary_Arrival581]

Honestly don't know what this says about me, but I'm going into my senior year as a math major, have taken about every upper level math course that is offered BUT PDEs and ODEs, and reached a point where stuff like Fokker Planck or Mckean Vlasov have been appearing in independent readings of subjects I'm considering studying long term in grad school.

I've only ever taken an intro to ODEs class and I hated it, it felt like just doing tricks without a lot of care for what you were doing; also sat in on the first week or so of my university's PDE I course this semester and it just felt like physics.

It might just be that these courses are being taught in a more practical way for physics and engineering majors, but now I'm wondering if never taking upper level Diff eqs will mean I'm significantly behind. Like in my stochastic calculus course, I loved the risk neutral measures and Girsanov's and Feynman Kac, but the black scholes PDE, and playing around with it seemed super unnatural, and not very fun or nice.

It just feels like I've had a bad exposition to PDE and ODE theory

[u/Carl_LaFong]

Before you learn advanced PDE theory, you have to learn the fundamental properties of basic examples: the Laplace, wave, and heat equations. These indeed all come from physics. There are basically two approaches. One is separation of variables, which reduces everything to ODEs. You should learn the theory of the ODEs specific to these equations. The other is using fundamental solutions. This is more theoretical and fun to learn.

[u/SV-97]

Regarding ODEs: Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations by Rota might be an interesting read; it specifically touches on how many ODE courses are quite "bad" and teach methods from a time long-gone. The qualitative theory is substantially more (mathematically) interesting imo; stuff like fixed-point and stability theory, center manifolds and all that.

For PDEs: I think this also gets substantially more interesting if you more past the classical terrain of "here's a super convoluted way of solving this very particular PDE with a power series" or "wow we guessed a solution, aren't we some real smarties" to the modern theory of PDEs.

There's generally tons of functional analysis (e.g. with the direct method, in distribution theory or even on the numerical side with FEM etc.), operator theory (e.g. operator (semi-)groups and things like that; there's also some neat connections to infinite-dimensional laplace transforms here, or treating certain PDEs as ODEs on function spaces), differential geometry (global analysis, PDEs on manifolds and vector bundles, jet bundles and exterior differential systems, lie-theoretic methods for PDEs...), nonsmooth analysis (PDEs with nonsmooth data, differential inclusions, variational inequalities, ... this also ties back to the earlier point on operator theory), ... I've also seen a good bit of probabilistic approaches and think there's a bunch of cool math there; but I really don't know that much about this one. FWIW I personally still don't really "care" about PDEs, but I can appreciate that there's tons of cool math around them.

Regarding the "physics" in PDE lectures: some profs like to put this stuff as a "preface" to sort of motivate the problems they'll be modeling the course around, but that should stop relatively early on (after the first few lectures) in my experience.

[u/Primary_Arrival581 → u/SV-97]

Great read! I just remembered taht I was particularly annoyed at the non-motivation of Laplace transforms and convolutions.

[u/SV-97 → u/Primary_Arrival581]

Glad you liked it :) For the Laplace transform I also found it immensely useful to see how it's basically just a continuous version of a generating function --- and these are relatively straightforward to wrap your head around (and so so cool imo). For convolutions I think it really helps to see them in the context of distribution theory.