PDE 101: Separation of Variables! ...or how I learned to stop worrying and solve Laplace's equation

This video introduces a powerful technique to solve Partial Differential Equations (PDEs) called Separation of Variables. I demonstrate this technique to solve Laplace's equation in two-dimensions for the steady state heat distribution on a rectangle. It can be used for a huge variety of other problems in physics and engineering.

@eigensteve on Twitter

This video was produced at the University of Washington

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0:00 Overview and Problem Setup: Laplace's Equation in 2D
5:12 Linear Superposition: Solving a Simpler Problem
8:38 Separation of Variables
15:39 Reducing the PDE to a system of ODEs
33:26 The Solution of the PDE
36:21 Recap/Summary of Separation of Variables
42:06 Last Boundary Condition & The Fourier Transform

[YassFuentes]

Masterclass. How much I do love your videos. They are so much enjoyable! Thank you for your effort and time ❤️

[Chandra-hwhv]

I couldn't understand the whole Fourier thing for the longest time in my college, huge thanks to you for this gem of a video!

[mostafaatalla]

You are fantastic, Steve! Thanks so much for all of your videos. I love it!

[xue]

Best title, my PDE class starts in 2 weeks, this video is such good timing woo-hoo

[sodaangel]

An amazing video coming to save me two days before my final. This lecture is Superb!!

[noahr]

This is incredibly smart...
Well explained 👍

[teodorbabic]

Great video and extremely useful for anyone in younger scientific community! Greetings from Serbia :D

[muthukamalan.m]

thanks for phenomenal works and make public to know more

[fernando]

Brilliant, mate. Lovely. Preparing for the finals. Regards from Chile

[lgl_noname]

One can only hopre that eventually there will a Brunton book integrating Brunton's view and treatment of PDE's

Thank You.

[saadmansakib]

Thank you so much for the video, Sir !

[rogelvtd]

Thank you, very good lecture.
Love your videos

[Gullinnova]

It just so happens im reading the end of certainty right now and all that you teach is helping to illuminate the pages. I feel like i have a really good handle on his explanations thanks to you. I still dont understand what he means by "deterministic chaos" it just seems like a contradiction on its face.

[sdpenning]

Very usefull, thank you. I wonder though how it would work if you considered the surface as another boundary. Like a thin square of copper foil where the edges and the top and bottom surfaces are in contact with air.

[lorhancosta]

Honest to goodness this content is guuuhhhd!!!

[ShinjiCarlos]

At about 15:00, I would say that if you take the derivative in relation to, let's say x, to both sides, the derivative of the right hand side should be 0 by definition, just like the partial derivative of a function of y regarding to x should be. In the other hand, the derivative of a function of x regarding to x should be zero only in the case that this function is a constant.

[WesleyDevlin]

Youtube is pushing the limits on advertisements. It's sad.

[indrasismitra]

This is just for a little information. Recently, while working on a problem involving heat transfer in cylindrical coordinates, I have found (from other published literature) that for some Boundary Value Problems (i.e., steady-state problems) the multiplicative, separation of variables i.e., u=F(x)*G(y) is insufficient. Another alternative that can be used is u = F(x)*G(y) + H(x) + I(y), although both the multiplicative (F(x)*G(y)) and the additive (H(x) + I(y)) parts have to be separately substituted into the original pde to find two different sets of odes consisting of different separation constants.

[LucasVieira-obfx]

I told myself I would only watch Steve Brunton's favorite lectures... Turns out I'm watching all of them

[ShinjiCarlos]

I have not checked, but I think that we could also extract the solutions components at the boundary as done, but alongside the functions sinh. It only remains to check that those functions form a complete set of eigentunctions. But since each of them belong to a unique $\lambda$, they do form a complete set of eigentunctions as well and could properly be used on its own right, in an equivalent Fourier trick.