Derivation of the Heat Equation

In this video, we derive the heat equation. This partial differential equation (PDE) applies to scenarios such as the transfer of heat in a uniform, homogeneous body. This video focuses on the derivation of the governing PDE only, other videos in this series discuss how to solve this PDE.

Additional videos in this series:

Associated videos on software tools relevant to PDEs include:

[KevinCastaneda]

AE501: Its amazing how two different problem setups like the 1d wave equation and 1d heat equation have very similar PDE that represent them. Great derivation!

[aidanschwartzentruber]

Thank you Christopher Lum, you saved me so much time looking for a value to use for arpha. Now I know! Cheers!

[jasonfranklin]

AE501: good breakdown of the derivation. The math on these has so many practical applications

[temssidesign]

I think you have a mismatch in the units...what you define as the velocity vector (J/sec)
is basically the heat flux vector at a point with units (J/sec*m^2)
and
q=-k*grad(u)
J/sec*K*m^2 = -(J/sec*K*m)*(K/m)

Other than that, a perfect derivation, thank you!

[mattpierucci]

AE 501 - Another great video. I find the derivation videos for the PDE very helpful in understanding the solutions.

[elijahleonen]

AE501: Very clear and concise explanation to derive the heat equation.

[matthew-bennett]

AE501 Seeing the fact that the integrand needs to be 0 from analysis leading to the simplified problem at 27:30 was very satisfying to see put together. This is what elegant math means to me and it was really cool to see.

[minaanderberg]

Great video the diagram of the body really helped me visualize what where/ what we were integrating over and how we got to the final equation.

[connorbaldwin]

AE 501 This derivation was very interesting to watch be derived. I am amazed at how the solutions come out after manipulation

[adamyousri]

AE 501: I was really confused at first, this really cleared things up

[anthonyz]

I now have a better grasp on 1d heat equations! Thank you for this content!

[jasondorn]

Thanks for the video. It was great seeing a bunch of previous lecture material coming together in the derivation

[Xyzzzz]

I am taking PDE over Zoom because of COVID restrictions. Your videos are so essential to me, thank you.

[ryoonoue]

AE501. Always great to see the derivation of equations before being used. Also loved the mario/luigi reference.

[lienchang]

It was my first time seeing the derivation and the video made it easier to understand it! :))

[daniellerogers]

So helpful to understand how the heat equation was derived, thank you!

[matthewbajamundi]

The derivation of the heat equation really helps me understand the overall concept. Thank you for posting this video.

[karlbalitaan]

Thanks for another great and helpful derivation

[robertdelossantos]

The video's helpful in understanding how the heat equation is formed.

[alisoncaprioli]

Good video! It's helpful to following along with the lecture notes. thanks!