‘Like’ this comment if you want to see a derivation of the solution of Poisson’s equation. UPDATE: Thanks for your likes, the sequel is now up on: th-cam.com/video/o23hJ8JX-aY/w-d-xo.html Also in 19:10, it should be r/n-2 not r times n-2. Thanks Bonnardot Philippe for noticing!
Finally, I understand what’s going on here after passing the half of this semester. I love mathematics but it need a great time to understand her because there is a big gap between us.
Why didn't you solve the last part i.e. finding values of A and B and having fundamental solution. I was waiting for just that part from last 24 minutes 🙁🙁
In your summary, you put the radial part as N>=3, but it also works for N=1 😊. Thanks for the awesome video Dr. Peyam! I watched your video a few times and now it makes sense.
Omfg that was soo long. I mean not your video, that was the most entertaining thing in my day. So thank you! But I felt a bit unsatisfied, because although I could believe the rotation stuff in the beginning, I felt I need a proof for this, and so I prove it... Finally... At some point there were three sums in my equation and it seemed pretty awkward... But I finally did it and I feel so good now. The whole thing is complete now in my head. The pieces are all together! So thank you for this "homework" you (technically) suggested!
Fantastic. I guess there are other videos showing (examples, intuition) why Laplace's equation is important/interesting. In particular, I believe that there is a connection with complex analysis (Dirichlet' principle, Riemann mapping theorem, Cauchy-Riemann equations, harmonic functions), it would be nice to see some videos about these topics. Thank you so much.
It was difficult to see what you were writing over on the far right side of the blackboard, and also on the top left. More substantively, this video was interesting, and I enjoyed watching it. Thank you very much for your video work. However, I need to see an example application of this fundamental solution to something tangible like a physics problem. Without an application, for me it is just algebra and calculus, which obviously you enjoy, and for which you have great talent. For those of us who do not have much talent for algebra, and always try to use geometry and visualization instead, the presentation you made does not give immediate insight to the (stunningly important) Laplace's equation.
Great stuff. In engineering classes they usually tell, "well if you solve this thing you actually get this thing..." Nice to see why. Maybe unrelated topic, but how can you know when you can write the infinite series solution as a combination of simple functions? Have seen in these pdes sometimes pop weird integrals with no formula like integral of sin/x or gaussian monsters
Sadly it’s hard to tell because some methods work for some PDEs and not for others, it’s a very ad hoc field of math; it’s not like other math subjects where you can just apply a theorem and get a formula. It’s a very beautiful field nonetheless!
Ryan Roberson Yeah, it means that if u is a solution, then v is a solution as well. And it should be V’’, I think that extra ‘ is just a smudge on the blackboard :P
also try E.i. "et inventa" for 'to be found', just like I.e. 'id est' and Etc 'et cetera' you could be the first to coin this very useful acronym that more relevantly translates to "to be determined"
Very cool video! Please do the proof you mentioned in the end! (The only way I know would be to say Laplace(Phi)(x-y)=delta(x-y), but I think that's cheating)
how is this related to the wave equation (basically the laplace equation but with one of the coordinates having a negative sign upto a constant)? it seems like it would be very related so i was expecting the solution to this to have either trig functions or exponentials, was definitely not expecting it to be what it ended up being.
The wave equation has surprisingly very different properties from Laplace’s equation, and its derivation is completely different, even though the only difference is a minus sign! It’s related to what’s called d’Alembert’s formula
The U( x ) = V( |x| ) really threw me into a loop because I wasn't quite listening and | x | resembled 1 ⨯ 1 to my tired brain. It's amazing to me to think that functions satisfying the Laplacian condition also satisfy the conditions in any rotation. Simultaneously, it kind of makes sense because no matter how you rotate an object, its geometry/curvature is still the same. Rotations just change your POV, they never extend/contract. Also, the notion of radial functions f( |x| ), that's a power tool in itself. Can we extend this "radiality" to any norm in a vector space, and would that be productive? In any case: Thanks Dr. Peyam. You made me spend a couple of hours on research, which is all good. :)
How about Legendre's equation? It's "just" an ordinary differential equation of order 2, but the fact that it has variable coefficients introduces some complication ;J Here's how it looks like: (1-x²)·d²y/dx² - 2·x·dy/dx + A·y = 0 where `A` is some constant parameter. I've seen some solution methods that involved infinite power series, and they are not only ugly, but also quite useless, because once you go into power series, there's no going back (there's usually no obvious way to convert a resulting power series into a combination of known functions). So I'm yet to see a solution that would not involve infinite power series, just algebra.
Bon Bon i think there's not an elementary function that solves Legendre's equation. My ode's teacher told us that, because the coefficients weren't as cool as those of Cauchy-Euler's equation, and because it was a non linear second order ordinary differential equation with singular regular points at x=1,-1, you must use the Frobenius theorem to solve it. And that theorem assures that there exist at least one solution in series form. That solution leads to what is called the "Legendre's polynomials" which are commonly used in mathematical physics.
"there's not an elementary function that solves Legendre's equation" I don't think that's true. Legendre polynomials are ordinary polynomials, made of simple powers and arithmetic operations, so they _are_ "elementary functions". And as you pointed out, the equation has _regular_ singular points, which means it is not _that_ nasty. But the common technique used to solve id (Frobenius method) introduces power series which then are quite hard to "reverse". That's why I was looking for some alternative, more algebraic approaches. (Even the fact that something cannot be solved in terms of elementary functions, doesn't mean it can't be solved algebraically, right?)
Can anyone tell me how to find the Laplace transform of tan(x) and could this formula be used for all unidimensional functions to find the slope of the line tangent better known as the derivative/tangent vector
One way to solve the laplace equation in 2d is to realise that it is the same as the wave equation when the parameter is i =sqrt ( -1 ) as uxx + uyy=0 means uxx = -uyy which means uxx = i^2 uyy
i really liked this video, you saying i-th derivative if x, wont it be more accurate to say the derivative in respect to i-x?(you make it sounds like f^(i)(x) and not f_x_i(x)) and when you got to the ODE, multiplying by r^2 will get you to Euler‐Cauchy equation, which solve for V(r) without getting V'(r) first(no integrals) and i want a video about how the convolution U=fnd.eq*f solves -∆U=f last thing, the regular reminder of me waiting for the proof that pi and e are Transcendental numbers
f i is faster to say than f xi :) Also since we’re doing multivariable calculus, f^(i) wouldn’t really make much sense anyway! And yep, your way works too! And I’ll working on it, hahahaha xD
‘Like’ this comment if you want to see a derivation of the solution of Poisson’s equation.
UPDATE: Thanks for your likes, the sequel is now up on:
th-cam.com/video/o23hJ8JX-aY/w-d-xo.html
Also in 19:10, it should be r/n-2 not r times n-2. Thanks Bonnardot Philippe for noticing!
this is why you solve it with Euler‐Cauchy equation! mistakes like this just can't happened(i just really like Euler‐Cauchy equation)
but it is still a constant so it doesn't matter
Yes! Please do more PDEs!!!! And solve that last eq you wrote!
I'm taking a course on PDE's and I just can't stop watching this playlist about PDE's hahaha please do more Dr Peyam!
Didn't understand a thing I love it.
Dr. Peyam, thanks for explainning interesting topics and greeting from Ecuador :D
Me: A 2ND ORDER LINEAR EQ WITH NO CONSTANT COEFFICIENTS
Me, seconds later: No wait it's separable ok crisis avoided
It's also just a Cauchy-Euler equation which are trivial
Awesome video Dr. Peyam!
Finally, I understand what’s going on here after passing the half of this semester.
I love mathematics but it need a great time to understand her because there is a big gap between us.
Laplace's equation! My favorite equation! Thank you!
19:04 shouldn’t it be 1 over N-2 instead of just N-2, it doesn’t matter since you redefine the constant but still
Hi Dr Peyam, at 19:10 when you integrate V' the -(N-2) should be over r and not in front of it, have a nice day !
Bonnardot Philippe You’re right, thanks for noticing :)
Why didn't you solve the last part i.e. finding values of A and B and having fundamental solution.
I was waiting for just that part from last 24 minutes 🙁🙁
Check out Inhomogeneous Laplace equation
Loved it. This is some tedious stuff sometimes seeming hopeless but a lovely result
In your summary, you put the radial part as N>=3, but it also works for N=1 😊. Thanks for the awesome video Dr. Peyam! I watched your video a few times and now it makes sense.
Yeah but it fails for n = 2 where you get log
Omfg that was soo long. I mean not your video, that was the most entertaining thing in my day. So thank you! But I felt a bit unsatisfied, because although I could believe the rotation stuff in the beginning, I felt I need a proof for this, and so I prove it... Finally... At some point there were three sums in my equation and it seemed pretty awkward... But I finally did it and I feel so good now. The whole thing is complete now in my head. The pieces are all together! So thank you for this "homework" you (technically) suggested!
Fantastic. I guess there are other videos showing (examples, intuition) why Laplace's equation is important/interesting. In particular, I believe that there is a connection with complex analysis (Dirichlet' principle, Riemann mapping theorem, Cauchy-Riemann equations, harmonic functions), it would be nice to see some videos about these topics. Thank you so much.
Yes check out the playlist
It was difficult to see what you were writing over on the far right side of the blackboard, and also on the top left.
More substantively, this video was interesting, and I enjoyed watching it. Thank you very much for your video work.
However, I need to see an example application of this fundamental solution to something tangible like a physics problem. Without an application, for me it is just algebra and calculus, which obviously you enjoy, and for which you have great talent.
For those of us who do not have much talent for algebra, and always try to use geometry and visualization instead, the presentation you made does not give immediate insight to the (stunningly important) Laplace's equation.
Thank you for this video.i understood the gist of fundamental solution of Laplace's and poison equations.
Great stuff. In engineering classes they usually tell, "well if you solve this thing you actually get this thing..." Nice to see why. Maybe unrelated topic, but how can you know when you can write the infinite series solution as a combination of simple functions? Have seen in these pdes sometimes pop weird integrals with no formula like integral of sin/x or gaussian monsters
Sadly it’s hard to tell because some methods work for some PDEs and not for others, it’s a very ad hoc field of math; it’s not like other math subjects where you can just apply a theorem and get a formula. It’s a very beautiful field nonetheless!
Is it possible to find the general solution? We’ve just looked for solution of the form
Nice video peyam! You should do the integral from -inf to inf of cos(x)/(1+x^2) , using feynman’s technique I can’t quite figure it out
Dr peyam how are you plz plz make videos on fourier and fourier transform
God damn PDES are a whole new creature... I love it
3:00 wait... does invariant just mean unchanging?? when was that introduced?
also, is there any reason why you do V'"(|x|) instead of V''|x|?
Ryan Roberson Yeah, it means that if u is a solution, then v is a solution as well. And it should be V’’, I think that extra ‘ is just a smudge on the blackboard :P
not the extra ', my keyboard's settings are all weird and i can't tell how many ' i type... but it's about the (||) vs ||
thé séttíng ís rélévánt fór óbvíóús réásóns :( why has windows gotta do this to me?
Oh! It’s just to emphasize that it’s V’ of |x|, not V’ times |x|
You must be "different" in some way, and Windows obviously knows it. (Just kidding!)
Anyway the result looks continental and rather charming!
also try E.i. "et inventa" for 'to be found', just like I.e. 'id est' and Etc 'et cetera'
you could be the first to coin this very useful acronym that more relevantly translates to "to be determined"
Please show us the cool proof you mentioned by the end!
It’s up now :)
PDEs tend to have a very different solution in 2D than other numbers of dimensions, is that right?
Yes
Very cool video!
Please do the proof you mentioned in the end!
(The only way I know would be to say Laplace(Phi)(x-y)=delta(x-y), but I think that's cheating)
Dr. Peyam, could you make a video on math logic.
There are some set theory videos on my channel if you’re interested!
how is this related to the wave equation (basically the laplace equation but with one of the coordinates having a negative sign upto a constant)? it seems like it would be very related so i was expecting the solution to this to have either trig functions or exponentials, was definitely not expecting it to be what it ended up being.
The wave equation has surprisingly very different properties from Laplace’s equation, and its derivation is completely different, even though the only difference is a minus sign! It’s related to what’s called d’Alembert’s formula
The U( x ) = V( |x| ) really threw me into a loop because I wasn't quite listening and | x | resembled 1 ⨯ 1 to my tired brain.
It's amazing to me to think that functions satisfying the Laplacian condition also satisfy the conditions in any rotation. Simultaneously, it kind of makes sense because no matter how you rotate an object, its geometry/curvature is still the same. Rotations just change your POV, they never extend/contract.
Also, the notion of radial functions f( |x| ), that's a power tool in itself. Can we extend this "radiality" to any norm in a vector space, and would that be productive?
In any case: Thanks Dr. Peyam. You made me spend a couple of hours on research, which is all good. :)
Machst du noch Videos auf Deutsch?
Nobody does PDEs like Dr. Peyam...
How about Legendre's equation? It's "just" an ordinary differential equation of order 2, but the fact that it has variable coefficients introduces some complication ;J Here's how it looks like:
(1-x²)·d²y/dx² - 2·x·dy/dx + A·y = 0
where `A` is some constant parameter.
I've seen some solution methods that involved infinite power series, and they are not only ugly, but also quite useless, because once you go into power series, there's no going back (there's usually no obvious way to convert a resulting power series into a combination of known functions). So I'm yet to see a solution that would not involve infinite power series, just algebra.
Bon Bon i think there's not an elementary function that solves Legendre's equation. My ode's teacher told us that, because the coefficients weren't as cool as those of Cauchy-Euler's equation, and because it was a non linear second order ordinary differential equation with singular regular points at x=1,-1, you must use the Frobenius theorem to solve it. And that theorem assures that there exist at least one solution in series form. That solution leads to what is called the "Legendre's polynomials" which are commonly used in mathematical physics.
"there's not an elementary function that solves Legendre's equation"
I don't think that's true. Legendre polynomials are ordinary polynomials, made of simple powers and arithmetic operations, so they _are_ "elementary functions". And as you pointed out, the equation has _regular_ singular points, which means it is not _that_ nasty. But the common technique used to solve id (Frobenius method) introduces power series which then are quite hard to "reverse". That's why I was looking for some alternative, more algebraic approaches. (Even the fact that something cannot be solved in terms of elementary functions, doesn't mean it can't be solved algebraically, right?)
Can anyone tell me how to find the Laplace transform of tan(x) and could this formula be used for all unidimensional functions to find the slope of the line tangent better known as the derivative/tangent vector
One way to solve the laplace equation in 2d is to realise that it is the same as the wave equation when the parameter is i =sqrt ( -1 ) as uxx + uyy=0 means uxx = -uyy which means uxx = i^2 uyy
But then you get complex solutions, which is a problem if your initial functions are only defined on R!
Thank you so much for this!!!
Yaaay Dr.Peyam
i really liked this video,
you saying i-th derivative if x, wont it be more accurate to say the derivative in respect to i-x?(you make it sounds like f^(i)(x) and not f_x_i(x))
and when you got to the ODE, multiplying by r^2 will get you to Euler‐Cauchy equation, which solve for V(r) without getting V'(r) first(no integrals)
and i want a video about how the convolution U=fnd.eq*f solves -∆U=f
last thing, the regular reminder of me waiting for the proof that pi and e are Transcendental numbers
oh and one last thing: nice, you didnt broke the chalk
f i is faster to say than f xi :) Also since we’re doing multivariable calculus, f^(i) wouldn’t really make much sense anyway!
And yep, your way works too! And I’ll working on it, hahahaha xD
@7:50, "blah blah blah". My pde prof says the same thing. lol.
Let Ω = {x = (x1, x2) : x1 > 0}.
1. Solve the Dirichlet problem
∆u(x) = 0, x ∈ Ω,
u(x) = g(x), x ∈ ∂Ω.Is it possible to solve this problem?
Yes, with green’s functions
Do you have a video explaining that?
Not yet, maybe in a year or so
Thank you prof
I love when he says Chen Lu :DDD
Normal TH-cam videos do not need to congratulate viewers on surviving/sticking around to a certain point. :P
Hello senpai!
Actually Dr. Peyam is my senpai...
Chen-Lu FTW
24:12 very woke. I have a lot of maths to learn lol