Math 391 Lecture 23 - Separation of Variables for PDEs and the Heat Equation

In this lecture we briefly look at an equation (a partial differential equation to be precise) that describes the conduction of heat through a rod. This is called the heat equation. We state the equation without proof, and then quickly look at an outline of how it is solved using separation of variables for the PDE.We obtain a general formula for the solution, and content ourselves with memorizing it and applying it from hence forth. As it turns out, the solution is a series solutions, having arbitrary constant coefficients given by the coefficients of a Fourier sine series.

This is the last official class of this semester. There may be a review session.

[ctelwardt]

Dude, you rock. Thanks for taking the time to put your lectures online. Ive spent hours taking notes from your videos and they help me far more than my current professor ever could.

[pinarcokugras]

I am studying industrial enginnering and taking differatial equations second time, but with your lectures my first exam was very high. And I hope my second exam will be too. I am very grateful to you. When I was in highschool all I want to do is to teach math to my friends and childreen. Now, in universtiy, I still like to study with my freinds and show them so different techniquies. ANyways, I am sorry for my bad bad English . It is just, I am really really gratefull to your work! Be safe!

[nishabisht]

Thanks a lot for these videos. Tbh you make up for all the not so good math professors in CCNY.

[FCBgamalARAB]

Thank you very much i been watching all of your videos and you been helping me all yer long.
Tahnk you

[mariiachernova]

This is my third course "with you", Prof. Smith, thank you so much for that!
Do you know if you will teach any 300 math courses like 30800 Bridge to Advanced Math in the next academic year?

[UnforsakenXII]

47:30 wew lad.

I'm actually a bit happy you mentioned eigenvalues and eigenfunctions because that sorta leads into Linear Algebra which is why I'm going to try to tackle right after your 392 class.

Oh speaking of the future.

Halfway through 392 you guys switch-over into Linear Algebra. Would it better to just do half of 392 and start on 346? I'm asking because it seems as though the linear algebra in 392 specifically for engineering applications.


Overall, thank you so much for posting this online. It's helped a great deal.

[Jumpertj]

You are the best. Thank You very much today i have a final exam.
A+ for sure im ready.
I just have a little question. On the finals solution they always derived the solution for X''-(lambda)*X=0 and T'=(alpha^2)*(lambda)*T
however, your method really fast, are they gonna deduct points for not deriving those equations?

[jonnybrates]

At 26:41, why can you multiply the integral by 2, and go from 0 to L if the function is odd. I thought that was only applicable if the thing being integrated is even. Thanks.