I have seen the heat kernel a million times, but have never seen it's derivation. Fun to see it finally. Thanks for the interesting content... Love your channel. Keep it up. Mahalo.
Next step : solving the Navier-Stokes equations ;) haha Will you talk about the solutions of differential equations thanks to green functions? I learnt it with the example of the 3D heat equation and I think it's a beautiful, intuitive and elegant way to solve and understand it. Thanks for everything!
I second that exploration! That's another equation where a length-square per time quantity - 'viscosity', typically denoted by a 𝜈 [and which, as @Ottmar555 will know, isn't the actual viscosity usually denoted with μ but which needs to be divided by a density, ρ, to get to length-square per time], is often used in a dimensionless scaling of distances and times as a 'viscosity factor' in determining how important the awkward non-linear convective terms are likely to affect approximate solutions.
Off topic, but do you have any idea of what has happened to "Mr Chen Lu"? (AKA the mathematician who calls himself *blackpenredpen* ). Has he taken a sabattical to focus on the business of becoming _Doctor_ BlackPen RedPen? Or did he just get sick of TH-cam, like Jon Lajoie once did, and broken the habit, cold turkey?
OMG! It's really interesting result because it's very similar as Green's function for the heat equation! :-) UPD: This is Green's formula)) I glad to see things that I know and find new methods and properties of these things with Dr Peyam, hah))
Maybe too late, but I'm very surprised and excited about this, the solution depends on the fundamental solution (Gaussian distribution) to do diffusion ，regardless of the initial function ！
The equation with the 2nd time derivative is very different from the equation with only a 1st derivative because the second time derivative specifies a property similar to inertia. When U(x) is moving up or down at time t, it will keep moving in that same direction and rate of travel, because its first time derivative at that point will remain constant until a non-zero 2nd space derivative somehow appears at that x location.
In physics, quantities like mass, length, time etc matter. So x has dimension of length and t has dimension of time and thermal conductivity k has dimension of square-length-per-time, so the simplest way to abstract the units is to manufacture a dimensionless variable (with no physical units) out of what's to hand and x²/kt does the trick as all the lengths and times cancel each other out. As the physics cannot 'care' how we measure physical quantities (e.g. metres or feet or furlongs, or seconds or hours or tortoise-lifetimes) it's often revealing to, um, 'dimensionless-ise', or de-physicalise, the independent variables.
@@Ottmar555 You're correct of course, but it would have required a diversion into contributions from density and specific heat capacity used to bring actual thermal conductivity into the length-squared per time κ used here in this video, and it wouldn't really have been relevant to the non-dimensionalisation, which is to explain why it makes sense to exponentiate (or sine, or cosine, or anything really) a thing like x²/4κt = ρCx²/4kt - a pure number without physical units. Thanks for the opportunity to better explain!
Glad I found this video. But can you do heat equation on 1d spherical coordinate please. And by the way, how do we choose boundary and initial conditions for PDEs based on real world phenomena? I can't quite grasp it. Thanks btw :)
@@drpeyam ow sorry, i was intending to say, can you do one video on how to solve heat equation but on spherical coordinate. I was doing the seperation of variable for that equation, and I was stuck on something called Sturm-liouville problem when I get an ODE for some function as a function of radius only, and i dont know anything about that Sturm-Liouville problem. About real life thing, just forget it. Pardon my english :)
@@chymoney1 well this was my comment a month ago, and i threw some stupid questions all over the internet, while i studied to get more deeper understanding in PDEs myself and also studied Sturm-Liouville. Thanks for the feedback tho :)
I just learned that the solution you derived is also the Dirac delta function (δ function). It's "used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one".... Oops! I had not watched the last 30 seconds of your video when you mention that!! :) Sorry!
Thank you for the motivational explanation! Would you make the video with the 3 dimentional case? Is there any other way to solve the wave eq. and heat eq. other than separation of variables?
I will understand this one day!
G luck on your interviews!!!
Thank you!!!
I have seen the heat kernel a million times, but have never seen it's derivation. Fun to see it finally. Thanks for the interesting content... Love your channel. Keep it up. Mahalo.
Next step : solving the Navier-Stokes equations ;) haha
Will you talk about the solutions of differential equations thanks to green functions? I learnt it with the example of the 3D heat equation and I think it's a beautiful, intuitive and elegant way to solve and understand it.
Thanks for everything!
I second that exploration! That's another equation where a length-square per time quantity - 'viscosity', typically denoted by a 𝜈 [and which, as @Ottmar555 will know, isn't the actual viscosity usually denoted with μ but which needs to be divided by a density, ρ, to get to length-square per time], is often used in a dimensionless scaling of distances and times as a 'viscosity factor' in determining how important the awkward non-linear convective terms are likely to affect approximate solutions.
love this video! every time the Dirac delta function appears I get shivers down my spine
"It's not math, it's different math." - Dr. Peyam, 2020
These videos are simply great. Thanks for the content!
Off topic, but do you have any idea of what has happened to "Mr Chen Lu"? (AKA the mathematician who calls himself *blackpenredpen* ). Has he taken a sabattical to focus on the business of becoming _Doctor_ BlackPen RedPen? Or did he just get sick of TH-cam, like Jon Lajoie once did, and broken the habit, cold turkey?
Amazing explanation sir …. 🇮🇳
OMG! It's really interesting result because it's very similar as Green's function for the heat equation! :-)
UPD: This is Green's formula))
I glad to see things that I know and find new methods and properties of these things with Dr Peyam, hah))
Maybe too late, but I'm very surprised and excited about this, the solution depends on the fundamental solution (Gaussian distribution) to do diffusion ，regardless of the initial function ！
Yep :)
The equation with the 2nd time derivative is very different from the equation with only a 1st derivative because the second time derivative specifies a property similar to inertia. When U(x) is moving up or down at time t, it will keep moving in that same direction and rate of travel, because its first time derivative at that point will remain constant until a non-zero 2nd space derivative somehow appears at that x location.
In this PDEs series, will you be covering the Schrodinger equation (for some potential) at some point?
Probably not, I don’t know anything about it
Dr Peyam I can teach you its easy
Thanks a lot for the math videos. Is there any general theory for factoring differential equations to solve them ?
Yeah Differential Equations the COOL way th-cam.com/video/NutSdXr1it0/w-d-xo.html
Will you be covering the general Fokker-Planck equation in relation to the heat equation pls?
Probably not
@@drpeyam oh ok :(
In physics, quantities like mass, length, time etc matter. So x has dimension of length and t has dimension of time and thermal conductivity k has dimension of square-length-per-time, so the simplest way to abstract the units is to manufacture a dimensionless variable (with no physical units) out of what's to hand and x²/kt does the trick as all the lengths and times cancel each other out. As the physics cannot 'care' how we measure physical quantities (e.g. metres or feet or furlongs, or seconds or hours or tortoise-lifetimes) it's often revealing to, um, 'dimensionless-ise', or de-physicalise, the independent variables.
You're confusing thermal conductivity with energy duffusivity.
@@Ottmar555 You're correct of course, but it would have required a diversion into contributions from density and specific heat capacity used to bring actual thermal conductivity into the length-squared per time κ used here in this video, and it wouldn't really have been relevant to the non-dimensionalisation, which is to explain why it makes sense to exponentiate (or sine, or cosine, or anything really) a thing like x²/4κt = ρCx²/4kt - a pure number without physical units. Thanks for the opportunity to better explain!
@@LemoUtan Thank you for you thoughtful response.
Glad I found this video. But can you do heat equation on 1d spherical coordinate please.
And by the way, how do we choose boundary and initial conditions for PDEs based on real world phenomena? I can't quite grasp it. Thanks btw :)
1d spherical coordinates? And I don’t know anything about real life 😂
@@drpeyam ow sorry, i was intending to say, can you do one video on how to solve heat equation but on spherical coordinate. I was doing the seperation of variable for that equation, and I was stuck on something called Sturm-liouville problem when I get an ODE for some function as a function of radius only, and i dont know anything about that Sturm-Liouville problem. About real life thing, just forget it. Pardon my english :)
@@halbmannhalbsib9881 I think he means spherical coordinates but only depending on the radius and not other angles.
Afri Wahyudi if you wanna learn PDES you should know strum liouville it’s crucial to diff eq and hence PDES
@@chymoney1 well this was my comment a month ago, and i threw some stupid questions all over the internet, while i studied to get more deeper understanding in PDEs myself and also studied Sturm-Liouville. Thanks for the feedback tho :)
This is so true and can relate to it so much so if you can show more video it will be better and really appreciate more explanation of all of this
Partial Differential Equations th-cam.com/play/PLJb1qAQIrmmDoNqmBDkgJ6-vN-j-aJFIm.html
Please make a video about the integral of (e^2x)/(2x+1) . Love your videos by the way .
No antiderivative
I just learned that the solution you derived is also the Dirac delta function (δ function). It's "used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one".... Oops! I had not watched the last 30 seconds of your video when you mention that!! :) Sorry!
Thank you for the motivational explanation! Would you make the video with the 3 dimentional case? Is there any other way to solve the wave eq. and heat eq. other than separation of variables?
I think PDE is a field demanding mathematical imagination.
Heat equation th-cam.com/video/kFX1K7vO8Xc/w-d-xo.html
Wave equation th-cam.com/video/KFS_Fs1ZGRw/w-d-xo.html
You are great
Good video
Merci :)
Nice
It's hard getting used to the "freedom" involved in solving PDEs. It doesn't feel right.
Takes a while getting used to
Dirac intensifies
20 years ago
WTF does not mean "want to find". That abbreviation is, unfortunately, taken. ;)
This one is more polite so you can use this one to de-brainwash yourself from the other one.
@@OndrejPopp Not a bad idea, thanks!
Verbazingwekkend!
I found a solution in 3 seconds... u=0
like an extra chromosome :)
Omg you're showing signs of baldness.
😭😭😭
Nice video