Separation of Variables

In this video I use the technique of separation of variables to solve the heat equation, by effectively turning a pde into two odes. This is a very classical problem at the end of a linear algebra and differential equations-course, and is a very nice blend of the two subjects. There is also a little bit of Fourier series involved. Enjoy!

[parinpatel]

I could've taken the time to write you a personal email, but I feel it would be best if the public read this:



Sir, you are an academic hero. Since failing an engineering math course for the first time early this year due to me not having an understanding of differential equations, especially this topic, I vowed that I will get back up and try again. I have watched countless videos on how to successfully solve a heat equation, and NONE of them by far taught it as simple as you did here. As of Fall 2022, I am retaking my failed course and am doing MUCH better than what I had before. This video seriously proved to be a game changer in my understanding of the heat equation and on how to solve them. Thank you so much once again and keep doing what you're doing!

[donnymcjonny]

Your enthusiasm is infectious which makes you a great teacher.

[daalfredLP]

6:03 "...and here it's the same Spiel"
German student approves!

[AnotherZoruaAmongUs]

If I get accepted to UC Irvine and decide to transfer there, I seriously need to sit in on some of your lectures or pray that you teach at least one of my upper division courses! You seem like such an amazing professor, and it's people like you that make me want to teach math!

[SirGamestop]

These videos are going to be immense help to a lot of people now that college classes are online for the rest of the semester.

[Jullan-M]

I'm glad I subscribed as this is exactly what I needed!

[tomatrix]

This is really useful. Great video Dr. P!

[othmaniarij]

Does he know how cute and funny he is ... I wish all professors can be this amazing. I didn't just understand this part in my physics lecture but I also had a great time watching and following this video.
A HUGE THANK YOU to you Dr. Peyam
PS: all my education is in french and I still find this pretty understandable

[carlavn]

Thank you for the awesome videos <3 I'm currently taking PDE class and this truly helps a lot!

[kasrashahidinamegh]

He went through all of that jumbo just for the joy of teaching, you watched it for free, learned, and had fun.The one thing he is asking you is to subscribe, so just do it <3

[LegendOfMurray]

who doesn't love heat equations? :D
super cool, Dr. Peyam!

[sushantdahal]

The reason we choose non-trivial soln is that since our PDE is one-dimensional heat eqn. So, temperature u(x, t) must satisfy newton's law of cooling i.e. decrease in temperature of the body within increase in time. So, the trivial soln.u(x, t)=0 looks absurd.

Is this correct dr.payam sir ?

[MrAssassins]

Simply amazing, now that I'm curse differential equations all this math tricks help me a lot, PM videos are god maths jajaja

[loicetienne]

Great video again. But something escapes me.
What is the value of U(1, 0) ?

Is a continuous solution possible under the given initial conditions ?
U(1, t) = 0 for t > 0
U(x, 0) = x for 0 < x < 1

[riccardoagazzi]

That was simply wonderful. I loved it.

[MrRyanroberson]

I wonder if this tactic finds any use in the three body problem? though the whole problem is not knowing the function in question, which certainly isn't a simple product.

[mht]

Absolutely amazing! (It's hard to find real maths in YouTube). It would be great if you could do something with tensors :D

[sansamman]

you are a teacher in Berkley, isn't that the place where the legend Cliff Stoll lived :0 ?!

[yuvalpaz]

pi'm>peyam!

but regardless, great video! I love the heat equation, why doing a video with such a strict initial conditions? say that we are working on a body with one dimensional for x in L(L=[0, C]), Ut=aUxx, U(x, 0)=f(x) for some function f and that U(0, t)=U(c, t) for t>0.(well if you do this the Fourier series part will probably be bigger part than the Separation of Variables so i guess it is okay)

(btw, still waiting!)

[bonbonpony]

03:18 I always wondered: what if there are some funky solutions that are NOT in that form? What justifies this assumption anyway? And when is the separation of variables method applicable? Can we know that upfront, before we even start solving the equation? Because if it has to turn out later that our assumption was wrong and it is impossible to solve it by this method, we will have lost several hours of pointless work and short on paper :q