Deriving the Heat Equation: A Parabolic Partial Differential Equation for Heat Energy Conservation

In this video we will derive the heat equation, which is a canonical partial differential equation (PDE) in mathematical physics.

@eigensteve on Twitter

This video was produced at the University of Washington

%%% CHAPTERS %%%
0:00 Overview
3:02 Statement in Words
6:52 Statement in Math
10:30 Heat Flux
16:00 Fourier's Law of Heat Conduction
18:54 The Heat Equation

[Mutual_Information]

Building advanced concepts from simple principles.. excellent. This channel is such a gem

[AG-pmtc]

I am lucky to live in this time, having access to such a great teacher for free and just before the exams. Thank you sir!

[dufflized]

That's a professor! Explains clearly because he knows deeply the topic. Thanks!

[chrisguiney]

always great to see a lecture push for common sense and knowhow 😄

it's actually reassuring, because it means that it's not always obvious how to map real world interactions that we have an intuitive understanding of into a formal system. It definitely intimidates me in a "not sure how to get started" sort of way.

my takeaways:
- identify the property you're modelling (heat over time and space)
- use a visual aid, draw out the physical setting
- set constraints on what you are & are not including in the model (e.g., not accounting for radiation)
- write down a laymen explanations of how the property changes in the dimensions (time and space)
- keep breaking those explanations down into smaller statements until you've got something isolated enough to model on its own (heat flux, sources/sinks)
- combine models
- know vector calculus
- ???
- win

Looking forward to future videos!

[brunoenricobignotti]

Steve... you are so great I have no words. I can only hope to have a fraction of your understanding and didactics. Thank God you exist, you've saved me so many times with your classes, you have no idea!

[ed.puckett]

No-no-no, thank YOU! Thank you for making this stuff so compelling and accessible.

[kevinshao]

You and your channel is the BEST online science education so far! What a wonderful lecture! Thank you so much for your knowledge, the way you teach and your greatness and kindness of sharing !!!

[hughca]

That was beautiful how he explained the heat equation.

[adrianl]

If you have a negative heat gradient from left to right, more heat is entering on the left than is leaving on the right. Therefore the heat will accumulate in the body. You could break it down as little cars bring heat people in and leaving with less heat people. That means the heat ppl are accumulating. 😀 Hope that helps with understanding the sign convention.

[fernandojimenezmotte]

Really enjoy your lessons in this You tube channel . Very nice explanation, pretty neat, very comprehensable Professor Steve. Among all scientific books that I have there is a really good one from Walter A. Strauss from MIT in Partial Differential Equations PDE´s. In another [Elementary Differential Equations] from Lyman M Kells, Mc Graw Hill, fith edition [1960], the derivation of Second Order Partial Differential Equation applied to NUCLEAR FISSION, which defines the NEUTRON FLUX and MATERIAL BUCKLING.

[reggieb]

Excellent lecture. Thank you from Nebraska!

[erickleuro]

@3:56, "...lets amplify that little segment here, shu, shu, shu, shu..." that was awesome!

[MrFazeFaze]

q(x + ∆x) is q(x) but with the additional change +(dq/dx)∆x. So written algebraically: q(x+∆x)=q(x) + (dq/dx)∆x you can quickly see that substituting this into the equation of transport for q(x+∆x) gives you the negative derivative

[gyorgyo]

A little bit confusing to me because both -(partial q / partial x) and q(x, t) are called 'heat flux'.
Your videos are very good. I like them. I have seen your entire series on vector calculus and partial differential equations.

[GeoffryGifari]

neat to see the equation being derived from the physics. i consider this several degrees more difficult than modelling a system when an equation is known

[umagrover]

Hi! This is very helpful. My friend and I were wondering how you write on that screen and it shows up the right way from our perspective. That’s so cool.

[Kong]

Thank you for these very clear explanations. Do you plan to make videos on the finite elements/volumes methods ?

[ethanmullen]

In case the origin of the heat energy term around 8:30 is confusing, the terms can be obtained by unit analysis:
c(x) = [Heat energy] / [Mass x Temp] (*this is interpreted as the amount of heat energy per unit mass the system gains as a result of a change in the system's temperature, hence "specific" heat capacity)
p(x) = [Mass] / [Length] (*in 1D)
u(x, t) = [Temp]
So heat energy distribution function is in units of [Heat energy] / [Length].

[erickleuro]

@17:05 "...there will be not heat flux, if the temperature is constant..." but only if the source term Q is equal to zero

[denm]

A good way to remember the signs is by making the normal unit vectors on the surfaces facing outwards and if the flow is opposite to the unit vector then - otherwise +
Ofc not reinventing the wheel, just a way on how I like to do analyze it but Nevertheless excellent video as always !