Talk:Heat equation

New applications

I added a sentence mentioning the equations use in image analysis. I have also once seen it mentioned in a textbook on population genetics. Can anyone corroborate that this is a common usage? In that case that should be mentioned too on this page. 130.235.35.201

the matrix A governing heat transfer

Do the eigenvectors of the matrix have physical meaning? ie. do they yield the direction of highest heat flow?

--24.84.203.193 28 June 2005 05:21 (UTC)

Clarification

I think it might be a good idea for someone to explain what situations the heat equation works in. For example, it may just be me, but I didn't understand whether what was being talked about with "propagation" was whether the heat came from a point source, or a source of finite volume is. The equation doesn't make sense to me because it seems like you could have to rooms full of air that were "isotropic" and "homogeneous" and they still could be different temperatures and have different levels rates of change of temperatures. Right? [unsigned]

and also, wouldn't k need units of some sort?

This article starts out overly technical from the beginning. Anyone without a degree in physics or math will get virtually nothing out of this article as they are derailed from the get-go, is that what is desired? One could easily write a simple conceptual paragraph or two for the lay person, another section on the 1D heat equation for say undergrads, and then get into all the gory math detail you wanted later. Mentioning parabolic partial differential equations, causality, the Riemann conjecture, and Ricci flow in the "General-audience description" is pretty hilarious. [unsigned]

Another clarification suggestion: The first paragraph of Heat makes reference to Heat being energy transferred due to convection, conduction, radiation, friction... But which of these does the generic "heat equation" apply to? Should the very first paragraph make clear that we are talking only about conduction (or whatever the right scope statement would be)? Deep down in the article it finally mentions that additional terms can be added to cover radiative losses, etc, but the introduction of the article never clearly describes what physical situations the default discussion is and isn't applicable to (it just says "diffuses"). DKEdwards (talk) 19:35, 13 June 2022 (UTC)

Symmetry of Laplacian

I am a little puzzled by the following passage. Could someone explain what it is trying to say and perhaps we can clarify it.

Note also that the ability to use either ∆ or ∇2 to denote the Laplacian, without explicit reference to the spatial variables, is a reflection of the fact that the Laplacian is independent of the choice of coordinate system. In mathematical terms, one would say that the Laplacian is "translationally and rotationally invariant." In fact, it is (loosely speaking) the simplest differential operator which has these symmetries. This can be taken as a significant (and purely mathematical) justification of the use of the Laplacian and of the heat equation in modeling any physical phenomena which are homogeneous and isotropic, of which heat diffusion is a principal example.

On first read it sounds like this is saying something about two alternative notations for the Laplacian and that the freedom to use either *notation* is somehow related to the Laplacian's co-ordinate independence. Unless I am missing something, this is not what is meant.

Second, one is left with the impression that the Laplacian is somehow special (as a differential operator) in that it is co-ordinate independent, but this is surely try of div, curl etc (indeed anything based on d and * in the exterior calculus sense). Is that what is meant? I am not sure what it means to say it is translationally and rotationally invariant that goes beyond this fact.

Lastly, is it really true that all homogenous and isotropic phenomena follow the heat equation? That sounds like something that would need more highlighting.

Francis Davey (talk) 08:26, 27 December 2022 (UTC)

"Stochastic heat equation" listed at Redirects for discussion

The redirect Stochastic heat equation has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 July 5 § Stochastic heat equation until a consensus is reached. 1234qwer1234qwer4 20:59, 5 July 2023 (UTC)