Heat Equation

Fundamental Solution of the Heat Equation

In this video, I derive the fundamental solution of the heat equation u_t = k u_xx by assuming the solution has a special form, and therefore turning the PDE into an ODE, which has an explicit solution. Enjoy!

[blackpenredpen]

I will understand this one day!
G luck on your interviews!!!

[Pradowpradow]

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!

[Chariotuber]

love this video! every time the Dirac delta function appears I get shivers down my spine

[paulkohl]

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.

[guitar_jero]

"k" here is typically "alpha": thermal diffusivity, the ratio of conductivity to heat capacity (times density).
It explains why, for example, a metal and a plastic feel like having different temperatures despite being in thermal equilibrium with each other :)

[BlokenArrow]

"It's not math, it's different math." - Dr. Peyam, 2020

[sicko_the_ew]

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 YouTube, like Jon Lajoie once did, and broken the habit, cold turkey?

[pohljaviermorenoroncancio]

These videos are simply great. Thanks for the content!

[RalphDratman]

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.

[KalininEvgen]

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))

[euva]

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!

[nicholasbohlsen]

In this PDEs series, will you be covering the Schrodinger equation (for some potential) at some point?

[kamalmarket]

Please make a video about the integral of (e^2x)/(2x+1) . Love your videos by the way .

[afriwahyudi]

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 :)

[marouaniAymen]

Thanks a lot for the math videos. Is there any general theory for factoring differential equations to solve them ?

[LemoUtan]

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.

[atrumluminarium]

Will you be covering the general Fokker-Planck equation in relation to the heat equation pls?

[ericchen]

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 ！

[georgettebeulah]

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

[정대영-le]

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?