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.
Alexey, see my second reply to Naresh Kumar below. Professor Brunton goes into greater detail explaining the specific convolution identity above and discusses convolution in general, although I don't recall that he actually derived any other identities.
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)
what concerns me is that I'm taking this as a sophomore applied physics student, and it's wayyy beyond my capabilities. I wish you all the best in the rest of your career!
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.
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.)
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.
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.
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!
Good question. Yes, we Fourier transform both sides with respect to the "x" variable, and so the "d/dt" can come out (so F(d/dt(u)) = d/dt(F(u)) since these operations can switch orders)
@@Eigensteve Thank you so much for clarifying Steve! And actually one more question, in DFT, you showed us FT matrix is a square matrix, does it have to be square? in another words, do we have to pick same number of points, n , in frequency space as in X space? And this is one of the greatest channel for lectures, we really appreciate it!
@@kevinshao9148 You are very welcome! In principle, there is no reason why we need to compute and use all of the Fourier coefficients, so we could consider a smaller "rectangular" matrix consisting of a subset of the rows of the DFT matrix. This is often done in random/sparse sampling. But for the computational benefit of the fast FT (FFT), the square structure is important.
@@Eigensteve Ah, I see!!! Now I kind of recall you might have mentioned this in your following FFT lecture. I am watching your series 2nd time, need multiple times to grip solid understanding. Again, really appreciate your great lectures and it's super pleasure to learn from you and your lectures!!! 👍👍👍
@@Eigensteve Hi Steve, but how do you guarantee the limited bases you selected from frequency space (the transform matrix) represent the major contribution in original f(x) ? I also see other python fft tutorial, they actually just call fftfreq(number of data points) and get the sample frequencies, but there is no theory backup that those are the major frequencies in your original signal. Am I missing something?
Often solving ODE's is about recognizing familiar patterns with known solutions. Here he doesn't derive the solution so much as he recognizes that the solution he gives is the "well-known" solution for an ODE of the given form.
First of all, thank you for nice explanation. I want to tell you that your lecture helps student in Korea a lot. I began to take an interest in Machine Learning since watching your lecture. Actually, I have a question at 05:25. I still don't understand how you solved ode. If you are still reading comments, it will be nice if you explain it for me. Thank you again.
Mr. Kumar, the book is in a reference in the description under the video. "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz I haven't bought the book yet, but it's in my wish list on Amazon. The preview of the book on Amazon shows a bibliography with source references also.
Professor Brunton also has a playlist on his RU-vid site listed as "Engineering Mathematics", which are videotaped lectures from that class at University of Washington. In that series of courses, he derives in greater detail some of the formulas you will see in the current series. I have watched those lectures and recommend them to you.
I still don't quite understand how the Fourier Transform of a derivative acts independently of the variable over which the derivative is taken. What if we transform u sub tt, not u sub xx, is it the same?
I belive it wouldn't be possible to transform u sub t because the fourier transform implies that the function goes from -infinity to +infinity and in that case the time starts at 0. Maybe we could assume that the time goes form -infinity to +infinity, idk.
See Deriving the Heat Equation: A Parabolic Partial Differential Equation for Heat Energy Conservation (ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-9d8PwnKVA-U.html) to understand Uxx.
