100 thoughts on “Solving the heat equation | DE3”

  1. Next up we finally get to Fourier series, which will be the beginning of a turn in the series towards understanding the surprising depth and importance of exponential functions for differential equations. Stay tuned!

  2. I don't know if here is the right place to ask, but I'd like a video on Simplex Algorithm in Linear Optimisation.

  3. A question that does not exist with the video, if I have an exponential function f (x) = a ^(x) e h(x)=a^(x/n), I say that the function h(x)=a^(x/n) undergoes horizontal (1/n) times or (n) times dilation with respect to h(x)? Note: neIN and sorry if i said anything wrong, i´m Brazilian.

  4. Please consider doing a lecture series on complex analysis. Heck, while you're at it, please do all of mathematics 🙂

  5. I spent 2 hours watching this video without ever not understanding something- a testament to the mulling potential portrayed by this video.

  6. This explanation is a must for chemical engineers! This is how any system equilibrates and describes mass stranfer (Fick's law) and heat transfer. This is at the heart of any chemical engineering problem. For those interested: The Fourier number relates to the general solution of this equation. So, this is not only mathematically interesting. Our lives, quite literally, depend on it!

  7. What's – 3² ?




  8. I love the way you visualize the topic where you gain a good intuition for why doing all this math. Thanks Grant!

  9. Very nice work again!! Loved it. However, I think you glanced over the boundary conditions too quick. There are conditions where for instance a side if the rod is kept fixed at a certain temperature or that the heat flow is determined by temperature. That is probably to detailed for this series but I would have appreciated a quick mentioning of that.

  10. You should do a series on stochastic processes – things like Brownian motion will probably be really interesting to the viewers 🙂

  11. Grant, watch out for Flammable Maths! He has insulted other members of the math community with well known victims include blackpenredpen, Dr. Peyam and presh talwalker! Beware!

  12. I have a B.S. In physics…in 1972 ! I have finally learned enough math to justify my degree ! Seriously,math and physics instruction is so much better on YouTube than it was in the 60's that even the few bucks it cost back then,that I am thinking of asking for my money back. With compounded interest of course.
    Then again that B. S. has me worth several mil at retirement so maybe I'll call it even.

  13. I watched through your videos on linear algebra a while ago and thought to myself "man, if only I had seen these while taking my linear algebra course, it would have given me a lot more intuition for the subject a lot easier" and now this, in the middle of my course on integral transforms and differential equations 😀
    Thanks a lot, you're doing a great job.

  14. wish I had a better math teacher in highschool, whenever I see formulas I just memorize them and skip the math. This was very educational, thank you

  15. Hi Grant,

    I really love your videos. They helped me a lot understanding complex math. Thanks to your explaination and animations I don't just see math as pure formulas anymore – I see vectors and functions in a transforming spatial space and this is really helpful. When this series is done I'd love to see an "essence of tensor calculus" or something like that. You have so much fun in creating animations – I guess tensors would be a good choice even for you 😉

    Thank you very much for your work 🙂

  16. would you say this is a high school topic? i need to do a research paper, and i am thinking of using this as my topic (the paper can be a thorough explanation of a proof / proving an equation), and I 'm not sure in terms of complexity if this is a viable option for me. I'm in Grade 11 (17y/o), any advice / opinions?

  17. @3blue1brown Hey Grant! Just a little feedback here. I think I found a jewel when I found your channel, for real. The reason why is because I have a tendency to understand things better when they're explained to me through graphs and animation, especially when it comes to math. I'm more of an artsy person myself, but I really love science.
    You see, I'm studying chemistry in Argentina, and despite the language barrier, I do think that it's worth the time and effort to try and understand the material you're producing, mainly because it's given me a deeper insight on the topic which I'm struggling to comprehend the most, that is, calculus. I'm currently doing the course for the second time… And last year my failure on every exam that I had was a bad hit to my self-esteem. But now, I'm starting to grasp more the concepts of divergence, curl, and all those things that seemed something that I had to memorize. Because of your videos, I do realise that I do have some kind of ingrained passion for math and physics, even though I do suck at them for the most part in terms of explaining my thoughts and reflecting it on paper.
    So, to wrap it up, thank you a lot for your passion and effort! I'm aware that the probability that you'll read this is rather scarce, but hey, a tiny dot in a tiny fraction of probability is not negligible, since it forms part of a bigger thing, right?
    Anyway, I'll go and try to watch the videos again. Listening to your lessons gets easier to understand the more time I let my brain traduce what you're saying, hahaha.
    Have a good one!

  18. When you have exam from this in 2 weeks,
    there are no more 3Blue1Brown videos on the topic,
    and this is the 666th comment :
    This must be the work of the devil!

  19. A solution of T = Cx is a real world solution to the problem just corresponding to different boundary conditions than the adiabatic (lnsulated object) boundary conditions. But it's not important for the purpose of the video. This is a really interesting video because I never understood the connection of how Fourier transform could be used to solve heat equation. Thank you for making such a great videos!

  20. A zero derivative at the boundary is only because you selected Neumann boundary conditions. We could have used sine waves if you wanted Diricelet Boundary Conditions

  21. Can the temperature distribution of a klein bottle over time be modeled with PDE's without using boundary conditions? 😉

  22. Walking into the bar I became aware of an exponentially decaying pencil of complex sinusoids flexing and writhing in the centre of the room, shedding colours across the watching conics.
    It was The Heat Equation.

    Disregarded, a bored Lissajous pattern spun idly in the corner.

  23. ugh I HATE messy nature why can't rods be infinitely long and then they'd DEFINITELY be like a sine wave forever :V

  24. I like the fact that music from the Essence of Linear Algebra series plays during this particular video. Just to give a slight hint of how important the subject is for Differential Equations.

  25. It sounds like a bell. Those wave functions describe the string on a guitar, or a bell, or the Moon itself, which also resonates like a bell in an specific frequency.
    No wonder Tesla said that in order to understand we need to think in terms of frequencies, the Spectrum.
    Math is proof of it.
    You are a great professor. You manage to explain complex into simple. Such a gift from Logos.

  26. Ohh la idea del seno modelando el eje x y el exponencial modelando el eje t me ayudó a entender porque se usa el método de separación de variables para resolver las edp

  27. Every math class into the future will be infinitely more useful with these videos, man. Like… before, math was the domain of those few who could easily translate it all into animations in their heads, and everyone else just had to struggle to try to do so, never really knowing if they had it right up top. Showing these animations to someone once is enough to make it all click so much more reliably. I sucked monumentally in my math courses in HS & college. I spent 10x the time for 10% of the understanding of the kids who just "got it" out of the box. Watching all of these videos almost a decade later, so much of it finally starts to make sense. I don't have to just cram-memorize formulas the night before an exam, only to dump them from my head immediately afterward, never actually able to get any real-world use out of them, lol.

  28. I get it that for t>0 the function have to satify the boundary condition, but nowhere in this video say it also have to at t=0. My question is for that condition to be satified, do the function at t=0 also have to satify it, or is it possible for us to generate a function such that the curves at boundaries is not 0 at t=0 and is 0 at all time t>0, and if not then why? Does anyone know?

  29. just finished a degree in ECSE and did not have an intuition for boundary conditions until just now, thank you for your content

  30. Thanks for the video!

    I'd like to ask how when there is a curvature i.e. change in space you say that there is a change in time as well?

  31. Hmm, so for an arbitrary function of temperature as a function of x, the arbitrary function of temperature in terms of x is approximated by the fourier series as an infinite summation of trigonometric curves. I wonder, would it have been easier if the temperature as a function of x be approximated by a taylor or maclaurin series? Would an infinite summation of monomials satisfy the boundary conditions?

  32. Hmm, so it seems that cosine functions (with zero derivative at boundary) makes sense for unbounded rods, where the rods are not subject to any temperature constraint. What would happen if we set the temperature of the rod at some constant temperature?

  33. This is the point of my life, where I realize how idle have I been 🙁 I have a PhD. in Maths and don't understand what he said about the flat thing in the boundaries (I could solve the equations as a student, yeah, but didn't understand the meaning); I will watch the video again… be dilligent and start again if necesary 🙁 Thanks a lot for the very illustrative videos

  34. I was trying to numerically solve the heat equation by substituting derivatives by its definitions and it looks like it is working, but how do I include the boundary condition stated in 9:33? How does the system even evolve at 9:04 if, by definition, the second derivative of the linear function is zero? Meaning that linear heat distribution will stay always the same as time passes by. Seems like the only way to solve the problem is to assume that boundary condition is also valid for t = 0 and then, after discretization of domain, we can assume that two points in the boundaries act as a single point (and have the same temperatures, so that the first derivative is zero) when heat is distributed, maintaining like that a slope of zero at boundaries as system evolves.

  35. SUGGESTIONS – 03:58 – In my humble opinion, You should have represented the "- alpha * sin(x)" representing the First Derivative of the Temperature Curve, then the Second Derivative, et cetera, and not// just the Arrows pointing down. You stick on the screen with the original sin(x) (or "c * sin(x)" curve while it is no more the subject and it is phase with sin(x) while the curve should be opposite phase which creates a bit of confusion.

    Also, I think it would have been Better that You represent, at least few seconds, the "cos(x)" curve, and in general, EACH AND EVERY OBJECT YOU WERE TALKING ABOUT, like also at 04:55, I think it would have been Better to trace the "-0.2*C*e^(-0.2*t)" curve, would be in dot line and very quickly.

    The problem, as I have seen is that even if a person think to Understand, one can lose some of the points and not really grasping all in its full depth and even leave with Little Misconceptions, of course not been Aware of but still having the Feeling having Greatly Understood all :/

    Nevertheless, it is still a A AWESOME WORK in term of Pedagogic Quality, thanks A Whole Lot For That 🙂

  36. What if you use an open subset of IR for our rod? Then every point has neighbours and thus no boundary-conditions are necessary. What am I missing?

  37. can any one suugest me on line lectures about real life applications of one dimensional,two dimensional and 3 dimensional heat and wave equations???i have to give presentation about this,,,,

  38. This video is amazing! But it leaves some things unmentioned. Adiabatic boundaries (dT/dx=0) are not the only boundary conditions in heat equations. Linear temperature profile T=c*x is perfctly good solution in some situations. It just states that the system is stationary and the boundaries are in constant temperatures Uniform temperature distribution is not the stationary solution of any other system than isolated systems. For the purpose of the video it doesn't matter tough.

  39. Anyone else wonder what is "c" in 4:00 and then what is c^2? I stopped the video and did a bit of algebra. I hope this helps someone else. We know the temperature at whatever point x starting from a time step 0 to be sin(x). If we use this recursively plus the fact that any future temperature value is the previous value plus (in this case minus because of the sign from second derivative) the rate of change times the time elapsed: T(x,0+dt) = T(x,0)-a*T(x,0+dt)*dt = sin(x)-a*sin(x)*dt=sin(x)*(1-a*dt). let's define c:=(1-a*dt). Thus for T(x,0+dt+dt)=T(x,2dt)=sin(x)-a*sin(x)*dt-a*(sin(x)-a*sin(x)*dt)*dt. Collect terms and you get sin(x)*(1-a*dt-a*dt-a^2*dt^2). The stuff in the last parenthesis is just c^2.

  40. Could the boundary condition be replaced by something related to energy conservation, ie. the total heat contained in the rod must remain constant?

  41. A mind-blowing clarity!
    I think it might be difficult to thank you enough for your contribution in spreading the love of (mathematical) beauty. I really agree beauty is the language of the universe.

  42. I like how the equation describing ME becomes more complicated, but less intricate, with time. Ultimately, I will just be a featureless infinite series of sines, indistinguishable from all the others.

  43. The stopping point at first did not bother me since I did not quite understand what happened after 6min into the video, but after rewatching it the cliff hanger tore my heart as I finally grasped the reason for the frequency change xD. Atleast the final video is already out !

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