Heat equation

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!

[@tylershepard4269]

Awesome video! I really enjoy your content. I would love to see a similar video for the wave equation!

[@drpeyam]

Coming within the next month or so :)

[@johnstotko6733]

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 :)

[@harikishan5690]

just was reading about this on feynman lectures :))

[@pythoner7138]

your channel has almost 10^4 subscribers & almost 1 year old
congratulates :)

[@alexanderjohn6908]

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

[@KidNamedVashin]

This video was straight up 🔥🔥🔥

[@guitar_jero]

Waltuh... put your PDE away Waltuh

[@KidNamedVashin]

@@guitar_jero I'm not solving you right now Waltuh

[@mauricepanero]

You lost me there. And I thought I knew the heat/diffusion equation(

[@leonardromano1491]

tbh Fourier transform seems like an easier approach to me lol :)

[@fatehalirana4194]

can you upload the video related to how would we convert any three dimensional PDEs into ODEs like heat equation, Poisson equation, Laplace equation

[@drpeyam]

I don’t think that trick works for any PDE

[@fatehalirana4194]

@@drpeyam is there any other method or way?

[@caldersheagren]

Can you do a video on Hodge theory? I think it's super cool stuff!

[@TheNachoesuncapo]

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

[@caldersheagren]

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!

[@TheNachoesuncapo]

yeah,i guess youre right

[@drpeyam]

I know very little about differential geometry, unfortunately

[@ffggddss]

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

[@drpeyam]

But then Delta^2 might be confused with the Hessian/Second derivative matrix 😉

[@ffggddss]

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

[@drpeyam]

I mean Del^2, the problem is that it might be interpreted as gradient squared (= applied twice), which is the Hessian!

[@ffggddss]

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

[@ssdd9911]

really is difficult

[@ryanlian7656]

Woah pdes look difficult! Amazing vid tho

[@MrBoubource]

30:15 was it meant to be a cut? Ahah love you pi m

[@cicciobombo7496]

I understood nothing

[@quickmath8290]

Yeah a bit more explanation would have been nice for those people who have never heard about it

[@gabrielmello3293]

No idea what you're talking about.