Solving the heat equation

Thanks! I hope you enjoy the next one :)

3blue1brown

Grant, this stuff is so good. These videos make me a little ashamed of how poorly I understood my PDE class. I think this is my favorite series you've done (and that's saying a lot). Glad you are taking your time and drawing it out.

Great job! The visual approach to solving the heat equation is very different than what is taught in school.

Thanks so much, that means a lot to me!

3blue1brown

This is the kind of video where I'm particularly thankful to you for allowing me, as a patreon of 3blue1brown, to express my support for this kind of videos, and I don't think I am too reckless if I say that this is also true for the majority of your patreons. Thanks again for belonging to this universe.

Ah, good tip!

3blue1brown

Great video. A super-minor thing that irked me was how the dashed lines jumped around at (e.g. at 8:20) Here's how I would've done those dashed lines: https://www.shadertoy.com/view/Wtj3zt

DomNomNom

Yes! This is a linear equation, and as you'll see, linear systems, in general, are intimately connected with generalizations of e^x, with solutions that can often be described purely with an exponential of some kind. So the fact that our sin(x)exp(-at) solution can be written more elegantly just with exp() is no coincidence!

3blue1brown

Great video as always Grant!

"sine curves are exponentials in disguise" -> so if I can decompose most functions to sinusoids, can I decompose further to exponentials? What a hook! And yes, take your time, get it right.

Max Goldstein

Hey. Thank you for this video, I'm looking forward to part two. One thing I wondered: Is there any significance to the fact that the solution is the real part of the form e^{k(x+\alpha k t)} with k = \omega i?

Yeah, I went back and forth on whether to go that route. I sort of agree that it always felt out of the blue. Why should I expect any solution to have that form? However, it seems slightly _less_ out of the blue is to play around with sine waves since you at least know something special might happen given their clean second derivatives.

3blue1brown

Thanks!

3blue1brown

Nice work. I see you decided to avoid the usual route of "assume u(x,t) = X(x)T(t). I recall that being very mysterious or overly-clever when I first read it, but I also found it satisfying to work through and solve the resulting ODE. It also adds to the sense that, by solving a PDE, we're solving infinitely many ODEs. That's not to say I disagree with your choice. It's kind of like solving the quadratic equation with geometry like an ancient Greek rather than algebraically completing the square. It fits the style of the channel, which is themed on intensely visual explanations. Anyway, this was a joy to watch, sorry for rambling, thanks for your work as always.

Jacob Mirra

Found this to be a fantastic breakdown of ideas I've had to use more and more in the last few years. Would love to see the spherical cases with Legendre polynomials broken down similarly, given their huge importance and relevance to this topic. Your physics audience would love you even more!

I really wish I had this video 15 years ago. All I learned was a few specific techniques for solving a few special cases of a few PDEs with no real insight into what the formulas were doing. Your videos are great for connecting all these dots I have into a coherent picture.

Kevin Davis

I have my exam in Differential equations on tuesday! Perfect timing, and great video as always <3 Thank you so much Grant :)

Great. Ps: it’s, not its (worth;-)