Heat equation
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- Опубликовано: 25 авг 2024
- In this video, I derive the fundamental solution for the heat equation, just like I did for Laplace’s equation, by simply making a clever guess and plugging that guess into the PDE. It’s a delight for differential equations-lovers!
Awesome video! I really enjoy your content. I would love to see a similar video for the wave equation!
Coming within the next month or so :)
Ha! I was just reviewing how to tackle some applications of the heat/diffusion equation. It's nice to see a different approach to it :)
I'll be sure to go through and find the alpha and beta as you mentioned and derive the expressions at 17:15... and see what happens if you keep the constant at 24:15. Should be fun.
Keep up the good work! Love the vids :)
just was reading about this on feynman lectures :))
your channel has almost 10^4 subscribers & almost 1 year old
congratulates :)
Who the hell can come up with these kinds of guesses?? I mean seriously?? is there any analytical way to solve this? my intro pde class didnt cover higher dimensional wave eqn. All I learned for the eqns mentioned in this vid is hit 1D heat eqn with a seperation of varibles and multiple D laplace with a greens function
This video was straight up 🔥🔥🔥
Waltuh... put your PDE away Waltuh
@@guitar_jero I'm not solving you right now Waltuh
You lost me there. And I thought I knew the heat/diffusion equation(
tbh Fourier transform seems like an easier approach to me lol :)
can you upload the video related to how would we convert any three dimensional PDEs into ODEs like heat equation, Poisson equation, Laplace equation
I don’t think that trick works for any PDE
@@drpeyam is there any other method or way?
Can you do a video on Hodge theory? I think it's super cool stuff!
do you really know all the maths needed to understand hodge theory?I honestly not.And I guess the average viewer neither do,but obviously I could be wrong
Nacho I know enough to understand the Hodge decomposition theorem on Kahler manifolds, but not everyone does. That's why I want to see Peyam's take on it, as he's great at motivating it for general audiences!
yeah,i guess youre right
I know very little about differential geometry, unfortunately
On a personal notation note, I find the all-too-common practice of using "∆" for the Laplacian, 2nd-derivative operator,
∇•∇ = ∇ ²
unsatisfactory & a bit misleading, because it obscures the 2nd-derivative nature of that operator. And because it is also in common use as a finite-difference operator.
But hey! - - That's just me!
Nice presentation!
Fred
But then Delta^2 might be confused with the Hessian/Second derivative matrix 😉
Delta-squared? That would be ∆ ² (capital delta, squared).
I'm advocating for del, ∇ ² - it's used all over the place for the Laplacian, ∑ᵢ₌₁ⁿ(∂²/∂xᵢ²), without confusion.
Fred
I mean Del^2, the problem is that it might be interpreted as gradient squared (= applied twice), which is the Hessian!
Well, no, actually, that's the tensor product, ∇⨂∇.
∇ ² follows the usual notation for vectors, in which the symbol ( F = |F⃗| ), without the horizontal line or arrow above it, means the magnitude (which is a scalar) of that vector (F⃗); and the magnitude squared, F² = |F⃗|² = F⃗•F⃗, is the dot product of F⃗ with itself.
I believe it's important to keep to conventions of this sort, as much as possible & practical, anyway, in order to avoid confusion.
Sadly, however, I don't have a symbol to type here, for ∇ with a vector-arrow over it.
Anyway, I have no quarrel with the substance of your talk; it flowed quite well to its conclusion, and was engaging along the way.
And I didn't find that the "∆" symbol was confusing in this case, because you made quite clear several times, what it was supposed to mean.
Fred
really is difficult
Woah pdes look difficult! Amazing vid tho
30:15 was it meant to be a cut? Ahah love you pi m
I understood nothing
Yeah a bit more explanation would have been nice for those people who have never heard about it
No idea what you're talking about.