Talk:Heat equation

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1 New applications
2 Notation
3 solving heat equations with fourier
4 the matrix A governing heat transfer
5 Laplace operator
6 diffusion
7 Heat and Schrodinger equations
8 Fundamental solution for the heat operator
9 heat equation and special relativity
10 Restructure the article
11 Mistake?
12 k should be the thermal diffusivity
13 Clarification

[edit] 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

[edit] Notation

Do you think:

${\partial u\over \partial t} = k \left({\partial^2 u\over \partial x^2 } + {\partial^2 u\over \partial y^2 } + {\partial^2 u\over \partial z^2 }\right)$

would be better? Dmn 20:40, 28 Mar 2004 (UTC)

yes I think it would be better if the article mentioned the equation at least once in this form (and I wouldn't mind if it was in this form throughout the whole article)ThorinMuglindir 17:32, 30 October 2005 (UTC)

Done, on 02:21, 23 February 2006 (UTC), by anon IP 150.140.211.76. (comment by Oleg Alexandrov (talk) 02:21, 23 February 2006 (UTC))

[edit] solving heat equations with fourier

is there a solved example where we solve/prove the heat equation using the fourier series

Is the question the solution of ${\partial u \over \partial t} = D {{\partial^2 u \over \partial x^2}}$ which is ${1 \over \sqrt{4 \pi D t}} e^{-{x^2 \over {4 D t}}}$ ?

If it is, there it is.

We start with the partial differential equation (pde), one dimensional diffusion equation

${\partial u \over \partial t} = D {{\partial^2 u \over \partial x^2}}$

Take Fourier transform the pde with respect to x using $u(k,t)=\int_{-\infty}^\infty e^{-i k x} u(x,t) dx$ . And so

$\int_{-\infty}^\infty e^{-i k x} {\partial^2 \over \partial x^2} u(x,t)= \int_{-\infty}^\infty (-k^2)e^{-i k x} u(x,t) dx=-k^2 u(k,t)$ by separation by parts.

The pde becomes ${\partial \over \partial t} u(k,t)=-k^2 D u(k,t)$ . This is an 1st order differential equation. The solution is simply $u(k,t)=C(k) e^{-D k^2 t}$

First we need to find what is $C (k)$ . For $t = 0$ , we see that $C (k) = u (k,0)$ and

$u(k,0)=\int_{-\infty}^\infty e^{-i k x} u(x,0) dx$

The initial condition is $u (x,0) = δ(x)$ . Since it is a delta function the answer will be the Green function.

So from the above equation simply $u (k,0) = 1 = C (k)$ . And so

$u(k,t)=e^{-D k^2 t}$ . This is the solution of the pde but we need to convert it back to x space.

The inverse Fourier will be $u(x,t)={1 \over 2 \pi}\int_{-\infty}^\infty e^{i k x} u(k,t) dk$

So, $u(x,t)={1 \over 2 \pi}\int_{-\infty}^\infty e^{i k x} e^{-D k^2 t} dk = {1 \over 2 \pi}\int_{-\infty}^\infty e^{ - D k^2 t + i k x} dk$

Let's play a little with the exponential part.

$- D k^2 t + i k x = -D t ( k^2 - {i k x \over D t} ) = -D t \left[ (k - {i x \over 2 D t})^2 + {x^2 \over 4 D^2 t^2} \right] = -D t \left[ (k - {i x \over 2 D t})^2 \right] - {x^2 \over 4 D t}$

So, $u(x,t) = {1 \over 2 \pi} e^{- {x^2 \over 4 D t} } \int_{-\infty}^\infty e^{ - D t (k - {i x \over 2 D t})^2 } dk$

Set $k - {i x \over 2 D t} = s$ so $d k = d s$ . The integral is $\int_{-\infty}^\infty e^{ - D t s^2 } ds = \sqrt{{\pi \over D t}}$ That gives the solution.

$u(x,t) = {1 \over 2 \pi} e^{- {x^2 \over 4 D t} } \sqrt{{\pi \over D t}} = {1 \over \sqrt{4 \pi D t}} e^{- {x^2 \over 4 D t} }$ --71.96.115.55 15:51, 30 October 2005 (UTC)

[edit] 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)

[edit] Laplace operator

Let $u = u(x_1,\ldots,x_n,t)$ . Would be better to say:

"when we write

u t = k Δ u

, we consider

$\Delta u = u_{x_1 x_1} + \cdots + u_{x_n x_n}$

instead of

$\Delta u = u_{x_1 x_1} + \cdots + u_{x_n x_n} + u_{tt}$ "? --nosig

Would my recent addition make it more clear? Oleg Alexandrov 22:24, 20 July 2005 (UTC)

[edit] diffusion

I'll change the following sentence:

"The heat equation also describes other physical processes, such as diffusion."

Because the heat equation governs the diffusion of heat, which is already a diffusive process. --anon

I don't think that's a good idea. People usually don't think of heat as diffusion. Saying that the heat equation governs both heat and diffusion would be best of understanding I think, even if a bit reduntant. Oleg Alexandrov (talk) 09:34, 25 October 2005 (UTC)

[edit] Heat and Schrodinger equations

This article has now become very confusing with its discussion of the Schrodinger eqn. Mentioning imaginary time later on in the article might have been OK.--CSTAR 12:24, 25 October 2005 (UTC)

Agree with C*. Also, there is some good new material in this article, but it needs better integration and more work. Oleg Alexandrov (talk) 12:56, 25 October 2005 (UTC)

I agree that the reference to the Schrödinger equation should be removed. As I understand it the theory for the Schrödinger equation is very different from the theory for the geat equation.

[edit] Fundamental solution for the heat operator

I think the formula for the solution to the heat operator on Rⁿ should come after the Fourier series solution. And moreover, its relation to the Fourier method should be clarified. --CSTAR 02:12, 26 October 2005 (UTC)

Agree with that. This because the Fourier series thing is simpler, so it should come before. Oleg Alexandrov (talk) 03:21, 26 October 2005 (UTC)

[edit] heat equation and special relativity

Is there an invariant form of the heat equation? --MarSch 10:10, 26 October 2005 (UTC)

Why is the title Heat Equation not Diffusion Equation? Heat is just one example of diffusion.

I couldn't agree more on this. If we're speaking physiscs, heat transfers are just one among many diffusive phenomena, which all involve this equation of a generalization of this equation.ThorinMuglindir 17:40, 30 October 2005 (UTC)

[edit] Restructure the article

As Oleg has said (in a recent edit summary) this article is a mess. I suggest it be reorganized as follows:

Heat equation in a finite 1-dimensional medium. Derivation from Fourier's law.
Solution by Fourier series
Extension to Heat equation in three dimensional regions
Fundamental solution (as for example discussed in the German WP page)[1]
General Heat equation in inhomogeneous anisotropic media. Derivation as in article now.
Friedrichs extension of Laplacian and solution by Borel functional calculus
Heat eqn on manifolds.

--CSTAR 16:03, 30 October 2005 (UTC)

As a physicist I (of course) disagree with this overly mathematical treatment, that leaves aside other diffusion phenomena. This equation is relevant to all diffusion calculations, of which the heat equation is nothing but a particular case.ThorinMuglindir 17:27, 30 October 2005 (UTC)

Well the solutions are obviously the same. What is it that you object to? Certainly heat flow in anistropic media is physically important. If you want to say that diffusion is a similar mathematical problem, well that's OK, but maybe you should have a different article with a discussion of the physics of diffusion. This is also related to the Wiener process. Also I think the treatment of Schrodinger eqn here is misplaced. There already is a separate article on the Feynman-Kac formula.--CSTAR 18:17, 30 October 2005 (UTC)

The Feynman-Kac formula is apparently about any partial differential equation and stochastic processes, not Schrodinger equations and stochastic processes. Have you seen how vague it is? As it reads all we can say is that this might be related to Green functions of the Scrodinger equation, or not. It doesn't mention the analogy between Schrodinger and diffusion equations either. If you feel you can demonstrate to me that this is related to the Green functions, I might well learn something, but then I'll just note that this formalism needs not be deployed to introduce green functions of the diffusion and Scrodinger equations.ThorinMuglindir 19:30, 30 October 2005 (UTC)

Now my objections to the proposed organisation of the rewrite: If this article is to be completely rewritten, first I object to leaving heat equation as its title. The title should be diffusion equation, because in physics heat transfers are just a subset of diffusion. Second, although the term fundamental solutions might be mentioned, the sections about Green functions should mention Green functions instead, because that's the term physicists and engineers use. Third, still about Green functions, I object to introducing them in the framework of heat transfer: they are better introduced in the framework of particle diffusion, because the intial condition corresponding a Dirac delta function is more easily understood in this framework, as it corresponds to the initial condition of a brownian particle in a known position. This initial condition is at the base of the decomposition of the solution on Green funtions. Can you explain simply what a temperature field corresponding to a Dirac delta function means physically? Can you realize that initial condition in an experiment?ThorinMuglindir 19:30, 30 October 2005 (UTC)

About your proposal of moving contents to a new article: this is too early to speak of this. We all agree that the diffusion equation can be applied to particle diffusion, heat conduction, and many other things. Let's try to find a common ground. My position remains that the article is not that bad as is, though some minor things could be reorganised, and renaming it to diffusion equation would be benefitial, though it would require some careful work.ThorinMuglindir 19:30, 30 October 2005 (UTC)

A word about my proposed title change: imagine a guy who doesn't know too much about physics. He types heat equation in the search engine, and is directly redirected to an article called diffusion equation. He's just learned something, because heat exchange is diffusion, while not all diffusion is heat exchange. In the way it stands now, there's nothing to learn, and someone who types diffusion equation on the search engine may be led to thinking that he's obtained the particular when he was searching for the general.ThorinMuglindir 19:30, 30 October 2005 (UTC)

The Feynman Kac formula is a path integral representation of the fundamental solution of the Schrodinger eqn extended to imaginary time; N.B. this isn't what currently is in the Feynman-Kac formula article, but that article should in my view also be rewritten. Green's functions and fundamental solutions are basically the same thing. Your section on the Schrodinger eqn is is the extension of the fundamental solution for the heat operator to imaginary time.--CSTAR 20:58, 30 October 2005 (UTC)

OK. As I wrote to Oleg on my talk page, I also intended to mention in this page that Green functions are the a simplification of path integrals for the case where there is no source term or other extra term in the equation, such as the equation that the heat equation article is about. You'll note path integrals are a much more complex mathematical object. And, someone who needs info about Green functions won't always be interested in path integrals. There are numeric methods which are based on the Green functions alone.ThorinMuglindir 21:19, 30 October 2005 (UTC)

[edit] Mistake?

It seems that there is a mistake in section 1 and in what follows this section. Why there is k under the square root? There is no k in Eq.6.

You're absolutely right. It should be fixed now. Thanks very much for mentioning this. Cheers, Jitse Niesen (talk) 19:10, 10 February 2006 (UTC)

[edit] k should be the thermal diffusivity

In my opinion k should be the thermal diffusivity not the thermal conductivity

$k = \frac{K}{\rho c_p}$

where

$K$ is the thermal conductivity
$ρ$ is the material density
$c p$ is the material heat capacity

—The preceding unsigned comment was added by Dapanara (talk • contribs) . Daniele

[edit] 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?

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

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