Poisson equation

In this sequel to the Laplace-video, I solve Poisson’s equation by showing that Phi convolved with f solves the PDE (where Phi is the fundamental solution of Laplace's equation). Along the way we discover the coarea/onion formula, as well as a n-dimensional version of integration by parts. This is analysis and PDE theory at its finest, enjoy!

[mht]

Awesome! It could be interesting seeing this with the Dirichlet and/or Neumann boundary conditions.

[AndDiracisHisProphet]

Oh god, I am getting serious PDE-flashbacks....

Well, how about Navier Stokes equation next?


Also, how does a mathematician catch a lion?
.
.
.
.
He builds a small fence around him and defines himself as "being on the outside".

[marouaniAymen]

Thanks for your videos, they make me discover lot of new things in math. I remember, in Quantum Mechanics, we have the Hydrogen atom equation with f(r)=k/r, and a direct method has been used, it'd be more mathematically rigorous to re-examine again or cross-check the results of the two methods.

[muluegebreslasie]

Hi Dr Preyam, I thank you for making such an excellent video. I have a question at 40:43 while finding the derivative of Dphi(y), isn't it phi(y) a piecewise function for n=2 and n>=3, in all your work I think you have used for n>=3. Secondly, why do you use phi(x-y)*f what's behind it while constructing it (i.e you have said changing of variable if we let z=x-y, y=x-z..)?

[rapidracim]

love your channel !
is it possible that you make a video on the mean value property of harmonic functions ?
since you used the property here, it would be a nice follow up.

[choclegg]

Could you please do some videos on Greens Functions for the laplace equatio. 🙏

[powersurvive]

Hi Dr.Peyam
It´s nice to see how to solve PDEs if you know the approach for that specific PDE. But i would be more interested on a systematic way of solving PDEs.
Would it be possible to do a video focusing on symmetrics of diff. equations and Lie-theory?

[cricket]

wow having had a bit of functional analysis last semester finally pays off :D

[erfanmohagheghian]

Could you please give me some guide or a reference on how to apply BCs to the fundamental solution of Laplace/Poisson equation?
Thanks.

[rybaplcaki]

Please can you make video about calculing dzeta function values? For example for 1/2?

[TheRedfire]

Hi Dr. Peyam's I'm having a lot of trouble with set theory excercises and proofs. (The most basic ones like for example: (Ax(B\C))=(AxB)\(AxC)).
I've tried searching for literature on the matter but it's not working for me. Do you have any book suggestions? Im studying set theory for my topology course next semester.

[ashu]

Thanq u so much sir, u really have solved my all doubts, thanq thanq 1000 times thanq

[markjackson]

Hi Dr. Peyam, Could I make a request? Could you teach on how to solve the cubic equation (ax^3+bx^2+cx+d=0) just like solving a quadratic equation? By the way, you're one of the amazing math teachers! : )

[DeepakKumar-ztce]

Dear Sir,
Please tell me a book on elliptic partial differential equations so that I understand better to this topic( I am a research scholar of EPDE).

[Zeboss]

I thought you would refer to Green's functions properties, but maybe there is a dimensional problem? I am not sure of the limit of this theory.

[MiroslavMakaveli]

This is more advanced than just integrals

[uRiOmG]

Sht this looks sexy, let's see what black magic you came up with today

[matekichba]

v = nu (the greek letter, if someone is still asking it)

[deeptochatterjee]

If you rotate the phi it looks like a tie fighter

[cycklist]

That's not mu, Peyam. It's either upsilon or nu.