Solving the Heat Equation with the Fourier Transform

This video describes how the Fourier Transform can be used to solve the heat equation. In fact, the Fourier transform is a change of coordinates into the eigenvector coordinates for the heat equation.

These lectures follow Chapter 2 from:
"Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz

This video was produced at the University of Washington

[Vss.alex]

Big respect to Pr. Brunton, i think such kind of projects should be supported.

[BoZhaoengineering]

the convolution, here it comes pretty intuitive. clearly explained the initial condition acts at the Gaussian solution. After a interpretation to convolution integral, it would be benefit more from this video. Thank you for your excellent teaching Professor.

[thellvll]

I was mind blown by how nicely you described this! Thank you for the amazing video :)

[roxanabusuioc]

The best description I've seen on the topic.

[nocode]

This is really cool. Connecting the diffusion phenomenon to the convolution kernel was a really nice touch. I never thought about it that way before.

[amribrahim]

Really amazing and simple explanation. I never understood Fourier transform as I did from your videos. Thank you!

[jinyunghong]

Thank you so much for your video! Your explanation using the convolution is much easier to understand than what I have ever studied so far!!

[yosephtekle]

Thank you very much! You are the type of teachers who we really need in my country. I hope I will be one in the future and your helps are enormous. Thank you again and again.

[ngocnguyn]

I can't tell how grateful I am. Thank you so much

[ahmedhassaine]

So essentially, when we analyze the Fourier series of a function and derive its coefficients, we gain insight into how each frequency contributes to the energy of the signal. In a sense, each frequency is linked to a quantum of energy. Consequently, the total energy of a signal can be represented as the sum of all its coefficients multiplied by their corresponding harmonics. In simpler terms, it's the sum of all frequencies in the Fourier space, leading to the emergence of this identity.

[Moonlight-emmq]

Thank you so much for your lessons, I passed 2 of my exams using them !! It actually saved my semester and I will recommend you everywhere ! :) (I'm a master student in applied mathematic master in France)

[vjnbarot]

Thank you! This was a great explanation of the heat equation as a smoothing operator.

[altuber_athlete]

What a professional video. The details were in depth, the speed was moderately low (perfect), no background music, equations + words + diagrams. I loved it. Could you do a video with the wave equation, please? (I’m an electrical engineer and we use it in electromagnetics and in lossless transmission lines.)

[alexeyl]

Thank you! Would be nice to see this linked to a video explaining convolution identities in detail.

[Andrew-rcvh]

Interesting. It reminds me of the problem of heat distribution through the earth. If we take the surface temperature as a sinusoidal then we only have a single omega and hence this would be a special case of your method, and measuring at a depth x, we should see another sinusoidal but with diminished amplitude via the Gaussian function.

[shobhanpaul]

Excellent. Explained it well., Should have also identified the error function.

[sergiomanzetti]

Absolutely superb lecture, but how did you find the Gaussian in x, t-coordinates ?

[andinosa]

How does this method work when you have two or more space variables?

[kevinshao]

Hi Steve, one question at 4:49, how did you get d U_hat/ dt on the left hand side. do you for F(tranform) on both sides? or do you mean d / dt is independent linear operator on t, so you can take it out? Thank you so much!

[harshvardhan]

Very precise and clear explanantion thank you sir