Talk:Heat equation/Archive 1

Laplace operator
Let $$u = u(x_1,\ldots,x_n,t)$$. Would be better to say:
 * "when we write $$u_t = k \Delta u$$, 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)

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)

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.

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)

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

If it is, there it is.

Shouldn't the method of using Fourier transformations (mentioned below this line!) and convolutions be included in the Methods of solution section?

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

$$ {\partial u \over \partial t} = D $$

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 $$u(x,0)=\delta(x)$$. 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 $$\int_{-\infty}^\infty e^{ - D t s^2 } ds = \sqrt$$ That gives the solution.

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

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)

Restructure the article
As Oleg has said (in a recent edit summary) this article is a mess. I suggest it be reorganized as follows: --CSTAR 16:03, 30 October 2005 (UTC)
 * 1) Heat equation in a finite 1-dimensional medium. Derivation from Fourier's law.
 * 2) Solution by Fourier series
 * 3) Extension to Heat equation in three dimensional regions
 * 4) Fundamental solution (as for example discussed in the German WP page)
 * 5) General Heat equation in inhomogeneous anisotropic media. Derivation as in article now.
 * 6) Friedrichs extension of Laplacian and solution by Borel functional calculus
 * 7) Heat eqn on manifolds.
 * 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)

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)

Animated gif
I think the animated gif is cool. However can you make it stop after the first cycle. It tends to be as distracting as some animated gif ads when you are trying to read the text.


 * I think you're probably right. I'll take it off for now, and see about adding a non-animated thumbnail that links to the animated version. --Wtt 22:38, 1 May 2007 (UTC)


 * Well you can always press Esc to stop animated gif --Novwik (talk) 19:03, 22 December 2007 (UTC)
 * The animation illustrates what heat flow is about. It can be stopped, as mentioned above, and besides, it is next to the table of contents. If you actually scroll down to the article text, you won't see it. Oleg Alexandrov (talk) 04:43, 23 December 2007 (UTC)

Notation
To me it is confusing that temperature is denoted u while internal energy is denoted x. Isn't the convention that temperature is called T and internal energy U?

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))


 * you are missing convection, and the equation is not in its genrelised form (you have written it in cartesian) you should probably write it in terms of the Material derivative and Laplacian. also i believe it is convention to write the derivitives on the right hand side and the source terms on the left giving:


 * $$\frac{D u}{D t}

- k \nabla^2 u = q $$


 * RichardMathie (talk) 20:45, 10 August 2008 (UTC)

Heat capacity
In the derivation of the 1-dimensional equation, the heat capacity at constant pressure $$c_p$$ is used. I think that the stated condition of "no work" usually refers to constant volume, so $$c_v$$ should be used. Is a different definition of zero work being used? Or should this be changed?

Of course, these quantities are very similar for liquids and solids, while heat flow in gases is generally dominated by convection. But if $$c_p$$ and $$c_v$$ are to be used interchangeably because the difference is irrelevant, why not just use $$c$$? Woodford (talk) 15:07, 29 August 2008 (UTC)

I agree, the text is misleading as formulated. $$c_v$$ should be used for internal heat, i.e. the $$dT/dt$$ term, and $$c_p$$ should be used for heat due to material flowing across the surface of the control volume (such as advection in CFD). Since we are not treating advection here, we should only be using $$c_v$$. In general, you should NEVER use $$c_p$$ unless you need to account for $$PV$$ work due to material crossing the control volume surface. But it could be generalized to just a $$c=c_p=c_v$$ for the special case of an incompressible substance.

information propagation at infinite speed
in the "General-audience description" "and they are stuck together end to end, then instantaneously the temperature at the point of connection is 50 and the graph of the temperature is smoothly running from 0 to 100. This is not physically possible, since there would then be information propagation at infinite speed,"

I believe this section could do with being reworded, as "at the point of connection" the distance between them is 0, so mathematically information is not propagating at infinite speed, true, the limit of the rate of change goes to infinity as the distance goes to 0 for a finite temperature difference, but isn't that simply a statement that 2 parts of the same object, 0 distance apart, can't be different temperatures?

the reason we don't observe infinite heat flows is because materials are not continuous. They do however come quite close to it - if you touch a hot object and a cold object the interface comes to equal temperature quite quickly (although the rest of the body of the materials may not for quite some time).

Andy t roo (talk) 04:45, 22 September 2008 (UTC)


 * I took out that silly diversion. Dicklyon (talk) 05:10, 22 September 2008 (UTC)

Animated GIF caption
I was checking the matlab code for the animation and it sets the box boundary conditions as T=0. Why? The caption says "hot body is placed in a box of cold water". I would expect that a hot body, let's say initial T=100 placed in a box with initial T=0 would equalize to a T between 0 and 100. But with boundary T=0 this will not happen. With no source and boundary T=0, the steady state solution is T=0 all over the box. Maybe the boundary condition should be something like dT/dx = 0? Italo Tasso (talk) 00:29, 25 November 2008 (UTC)

Proposal to split solution and some tables
The article is too long and technical in my opinion. I suggest to split an article on solution to the heat equation which would include methods using Fourier series, Fourier transform, fundamental solutions. (Igny (talk) 23:25, 28 December 2008 (UTC))


 * Sounds like a good idea. Go for it.  siℓℓy rabbit  (  talk  ) 23:31, 28 December 2008 (UTC)


 * How about the following names Solution to heat equation (Fourier methods) and Fundamental solutions to heat equation?(Igny (talk) 19:33, 29 December 2008 (UTC))


 * Both suggestions seem eminently reasonable.  siℓℓy rabbit  (  talk  ) 20:11, 29 December 2008 (UTC)

discrete finite-element analysis/simulation
Seems like this article should at least mention/linkto discrete finite-element analysis/simulation. -96.237.10.106 (talk) 14:36, 25 May 2009 (UTC)

Water Capacity Calculation
Hello All,

here i met one question about the water capability calculation. For exzample, one heat exchanger, the water flow rate is 100GPM. Water entering temperature is 50F and the leaving temperature is 40F. Water entering pressure is 50 PSIA. The differential pressure though the heat exchanger is 10PSID.

How to calculate the water capability?

Which method we should use? Q=CpM(DeltaT) Cp-Water specific heat M-Water mass flow DeltaT-Entering T -Leaving T Note: Do we need to calculate the water facor led to temperature rising?

Q=M(Hi-Ho) M-Water mass flow Hi-Entering water enthalpy Ho-Leaving water enthalpy

Do not know which method we should use. Please help. thanks, Yours Lkvyang (talk) 07:48, 15 July 2009 (UTC)

Possible error in: Problem on (0,∞) with homogeneous initial conditions and non-homogeneous Dirichlet boundary conditions
I'm not sure the solution given can be correct. If I substitute x=0, I get u = 0, not h(t). For the case h(t) = 1, I find u(x,t) = erfc(x/sqrt(4t)) but do not get this answer by using the general form on the page. Does someone have a reference for this? Thank you-- 134.34.19.174 (talk) 18:39, 22 November 2009 (UTC)


 * Now I'm lazy to go and look for a reference, but I made the computation by myself. Let $$K$$ be the fundamental solution of $$\partial_t-k\partial_{xx}$$. I found as a solution with homogeneous IC and $$h(t)$$ as BC, a convolution of $$h(t)$$ with the distribution $$-2k\,\partial_x K$$ :
 * $$u:=h*(-2k\,\partial_x K).$$
 * Now, the point is that for x>0 the exponential makes it a smooth convolution of functions; so we can write,  for all x>0
 * $$u(x,t)=\int_{0}^{t}\frac{x}{\sqrt{4\pi k(t-s)^\mathbf{3}}}\exp\left(-\frac{x^2}{4k(t-s)}\right)h(s)\,ds.

$$
 * So by the way it seems that the formula in the article is indeed incorrect, because the exponent 3 is lacking. And this gives an explication to your question: we are not allowed to substitute x=0 in the integral to represent the convolution of h with the distribution $$-2k\,\partial_x K(t,0)$$ (notice that as a function, now it has a non-integrable singularity). Is true that the integral for u(t,x) converges to h(t) as x → 0, however. I'm pretty sure about my computation, but maybe one could make a double check in some classic text on the subject. --pma (talk) 22:57, 23 November 2009 (UTC)


 * Looking at the edit history, it seemed that this formula had quite a lively past. I have fixed the error in the article and added a few lines of explanation. --pma (talk) 20:25, 5 December 2009 (UTC)

Section "Physical problem and equation"
The section Physical problem and equation seems to have been edited by too many hands. This needs to be reorganized somehow. In particular, I think the basic discussion of the equation in the Three-dimensional problem section should be moved upwards in the article (minus irrelevant commentary on Black-Scholes and special relativity). The "General description" section could easily house this content. At present, it doesn't really discuss the equation itself, rather some "interesting properties" (like the maximum principle and smoothness properties). As an aside, shouldn't content specifically about the maximum principle be moved to the appropriate subsection? This doesn't help anyone understand the heat equation itself. Sławomir Biały (talk) 12:20, 1 October 2010 (UTC)

Animated GIF is confusing
I find this most confusing. For a start we have no idea what each axis is. We have no idea what the shape of the hot body is. We have no idea what the colours mean. Finally if as I suspect the z-axis is the temperature, why does this not flatten out as thermal equilibrium is reached? Drkirkby (talk) 09:03, 10 February 2011 (UTC)
 * I agree; without labels, it is confusing. __ Just plain Bill (talk) 17:44, 10 February 2011 (UTC)

Section title "Solving the heat equation using Fourier series" seems misnamed
The section titled "Solving the heat equation using Fourier series" nowhere mentions Fourier series. It does, however, mention that Fourier solved the problem in the manner described in the section, using separation of variables. Perhaps the section should be titled "Solving the heat equation using separation of variables"? 192.114.23.210 (talk) 10:16, 3 March 2011 (UTC)

λ → -λ
The section about solving the equation says: "Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value λ." Shouldn't it be -λ, as the rows below read "T'(t) = -λαT(t)" and " X''(x) = -λX(x) "? ✎ HannesP · talk 16:33, 24 May 2011 (UTC)

Schrodinger Equation Connection: Wick Rotation
whoever wrote the section did not take enough advanced physics to be told about the Wick Rotation. That is the connection between imaginary time and inverse temperature, which is what gives us a "sort of" relationship between classical diffusion and time evolution in quantum mechanics. See the Wick Rotation article.68.6.76.31 (talk) 03:06, 21 June 2011 (UTC)

Notation with Integral
In the section of "Heat conduction in non-homogeneous anisotropic media" appears this equation: $$ q_t(V) = \int_V Q(t,x)\,d x \quad $$

But this notation is a little bit misleading, it seems like a one-variable integral. But it is a volume integral, isn't it?

I think the followin notation would be better:

$$ q_t(V) = \int_V Q(t,x,y,z)\,dV \quad $$

or $$ q_t(V) = \int_V Q(t,x,y,z)\,dxdydz \quad $$.


 * Not at all misleading: the use of "dx" for integrals wrto n-dimensional Lebesgue measure is customary between all mathematicians. That the variable "x" belongs to the domain of the integrand is in any case clear. The first solution you wrote is a bit strange, and should be avoided, for it mixes the notation for the domain of integration V and the notation for the measure and the variables. The last you wrote, with dxdydz, should be better used together with a triple iterated integral, which is correct but would make the notation unnecessarily complicated and unnecessarily coordinate-dependent (the use of the 3 dimensional Lebesgue measure also allows the simpler notation you wrote, which is also the more correct. --pm a 17:16, 21 April 2010 (UTC)


 * That may be true with respect to mathematicians, but because this article is neither pure math, nor pure physics, but applied math/theoretical physics in its relevance, it would be useful to include a more direct terminology. there are many alternatives to "dx" with the context being the guide...  I think the most obvious one is:  "dnx" which is used all the time in physics and mathematical physics texts.  so that you would have "d3x" for R3.  There are other alternatives such as dσ, dτ, dζ, and dξ which have various traditional meanings (not all of them relavent to this specific discussion of dimensionality)68.6.76.31 (talk) 03:15, 21 June 2011 (UTC)68.6.76.31 (talk) 03:17, 21 June 2011 (UTC)

fundamental solution:need for 4*Pi*k*t vs. just t?
When the following script:

[snip]

Indicating that t works just as well as 4*Pi*k*t as the argument to the pow function in divisor of u. Is there any reason for using the more complicated expression?

Answering myself, the reason is that the differential operator is linear and derivative_of(c*f,x) = c*derivative_of(f,x) hence for any solution, f, of the linear homogeneous differential equation, c*f is also a solution of same equation.

Sorry for noise :( If I knew how to delete this post, I would.

Cppljevans (talk) 20:59, 21 September 2011 (UTC)

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
 * $$\rho$$ is the material density
 * $$c_p$$ is the material heat capacity

&mdash;The preceding unsigned comment was added by Dapanara (talk &bull; contribs). Daniele


 * There needs to be some definition of k. Currently the article states "k is a material-specific quantity depending on the thermal conductivity, the density and the heat capacity." Why not give the actual definition rather than this vague and unenlightening statement?  —The preceding unsigned comment was added by Special:Contributions/ (talk)

Comment added:

There is a mix of thermal conductivity (in Europe often defined by λ (unit = W/(m.ºK), and the thermal diffusivity, most often noted by α (unit m2/sec. In a lot of the equations the symbol k is used initially defined as the thermal conductivity, where α was menat. It would be a good idea to list the symbols used, and give their eaxct definition.

As an example. At the end of the one dimensional paragraph this statement appears:

The coefficient k/(cpρ) is called thermal diffusivity and is often denoted α.

Then in the fundamental one dimensional solution this equation appears:

In one variable, the Green's function is a solution of the initial value problem



\begin{cases} u_t(x,t) - k u_{xx}(x,t) = 0& -\infty<x<\infty,\quad 0<t<\infty\\ u(x,t=0)=\delta(x)& \end{cases} $$

where δ is the Dirac delta function. The solution to this problem is the fundamental solution


 * $$\Phi(x,t)=\frac{1}{\sqrt{4\pi kt}}\exp\left(-\frac{x^2}{4kt}\right).$$

This should be:



\begin{cases} u_t(x,t) - \alpha u_{xx}(x,t) = 0& -\infty<x<\infty,\quad 0<t<\infty\\ u(x,t=0)=\delta(x)& \end{cases} $$

where δ is the Dirac delta function. The solution to this problem is the fundamental solution


 * $$\Phi(x,t)=\frac{1}{\sqrt{4\pi \alpha t}}\exp\left(-\frac{x^2}{4 \alpha t}\right).$$

See also the German article on this subject.

Gerben49 (talk) 11:21, 20 December 2011 (UTC)


 * Yes, note however this issue is about the physical model, and should not enter the more mathematical sections where the PDE is treated. While the coefficient in front of $$\Delta$$ has an important meaning in each physical or probabilistic model, from the mathematical point of view, of course, it is just a positive real number, that can be denoted by whatever letter, and that may be assumed to be 1 with no loss of generality, as explained in the introduction. (Also, the notation $$u(x,t=0)$$ is never used in mathematics, and should be avoided. The meaning of the Dirac $$\delta$$ is a distribution, and needs no special notation). --pm a 14:41, 20 December 2011 (UTC)

Name
In my undergraduate (Engineering) studies, this equation was referred to as 'the Fourier equation'. Should we not mention this name? --Moemin05 (talk) 18:51, 28 May 2012 (UTC)
 * You are the first person I have ever heard that refers to it in that way. I think the Heat Equation is by far the most conventional name. Hamsterlopithecus (talk) 05:34, 29 December 2012 (UTC)

Historical Background
I was looking for the historical background of the heat equation but this article doesn't seem to cover it. Is there a reason for that or should we make a History section? Hamsterlopithecus (talk) 05:41, 29 December 2012 (UTC)

What is u?
Nowhere does it say what u physically means... 76.254.50.99 (talk) 15:38, 2 September 2009 (UTC)

The first line covers it really... "The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature)" It is the temperature at a point in space and time —Preceding unsigned comment added by 132.181.247.158 (talk) 16:06, 7 September 2009 (UTC)

The first line is nonsense. Anyone would understand "equation which describes the distribution of heat" as saying that u is heat. Which is of course false. 76.254.50.99 (talk) 20:29, 7 September 2009 (UTC)

Why is u first the temperature, then it becomes 'excess temperature' when introducing the radiative term? This treatment is misleading. — Preceding unsigned comment added by 79.119.49.89 (talk) 05:37, 3 August 2013 (UTC)

k wrongly used instead of α?
I think that starting from the paragraph 6 (Fundamental solutions) what is called k should be called α, according to the definitions given in the article until that point. — Preceding unsigned comment added by 141.89.115.240 (talk) 16:46, 24 February 2015 (UTC)

Special case k=const
I think it's worth mentioning that the derivation assumes constant thermal conductivity, which is a special case. Hennui (talk) 14:24, 29 July 2015 (UTC)

Vector/scalar inconsistency
In the paragraph where Fourier's law is introduced the 1D special case is presented.

On the left hand side q is written in bold (vector) font. While it is true that a one-dimensional vector is a scalar, I think it will be clearer if either: or or
 * A unit vector in the x-direction is added to the right hand side,
 * q is written non-bold to clarify that it is scalar,
 * The situation is otherwise clarified.

130.225.98.190 (talk) 13:59, 26 August 2015 (UTC)

Agent-Based Model Interpretation
Hello all,

According to some agent-based modeling textbooks and classes (I'll see if I can find some references, until then I'll provide some math), one can roughly interpret the heat equation (for uniform materials) as follows. Let each atom, cell, or agent replace its own temperature with the average temperature around it each time interval. This is approximately correct. It follows from finite differences and euler integration.

In 2 dimensions using cartesian coordinates, $$\nabla^2 u = \left(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}\right)$$. Using finite differences, the second partial derivatives are $$\frac{\partial^2 u}{\partial x^2} \approx u(x - h, y, t) + u(x + h, y, t) - 2 u(x, y, t) $$ $$\frac{\partial^2 u}{\partial y^2} \approx u(x, y - h, t) + u(x, y + h, t) - 2 u(x, y, t) $$

Substituting in we have, $$\nabla^2 u = \left[u(x - h, y, t) + u(x + h, y, t) - 2 u(x, y, t)\right] + \left[u(x, y - h, t) + u(x, y + h, t) - 2 u(x, y, t)\right]$$ $$\nabla^2 u = u(x - h, y, t) + u(x + h, y, t) + u(x, y - h, t) + u(x, y + h, t) - 4 u(x, y, t) $$

Integrating in time, using Euler's method we have: $$ u(x, y, t + dt) = u(x, y, t) + dt \nabla^2 u $$ $$ u(x, y, t + dt) = u(x, y, t) + dt \left(u(x - h, y, t) + u(x + h, y, t) + u(x, y - h, t) + u(x, y + h, t) - 4 u(x, y, t)\right) $$.

If we choose dt appropriately the $$ u(x, y, t) $$ term vanishes: $$ u(x, y, t + 1/4) = \frac{\left(u(x - h, y, t) + u(x + h, y, t) + u(x, y - h, t) + u(x, y + h, t)\right)}{4} $$

In 3 dimensions, the appropriate $$ dt $$ is 1/6.

This provides a very intuitive meaning for the heat equation. Given the appropriate time interval, the heat equation says each atom/cell/agent is just updating its temperature with the average surrounding temperature. I think this agent-based interpretation would be a useful addition to this article. This may even be suitable for the simple-English wiki.

If I or the community finds some references, so this isn't counted as original research, should we include this? Does this "intuitive" understanding as the averaging of nearby temperatures provide anything meaningful? If other users think it would be worthwhile including, where in the article should it go?Mouse7mouse9 06:36, 6 January 2016 (UTC)

Assessment comment
Substituted at 17:21, 29 April 2016 (UTC)

Article is too long and polluted
The article is too long and has extraneous material that does not belong here. The article should be about the mathematical equation $$\dot u = \alpha \nabla^2 u$$. It should describe its mathematical properties and methods of solution, and discuss lightly its extensions and applications. It is very unfortuante that it is named "heat equation", because it seems that editors have mistaken the topic of this article to be "physics of heat flow". That is only one application, whose theory goes well beyond the "heat equation". That topic should be trated in full in a separate article (or several). All the best, --Jorge Stolfi (talk) 08:20, 12 April 2019 (UTC)

Wikipedia is not a textbook or lecture notes
Just a reminder: Wikipedia is meant to be a reference to answer specific questions that general readers may have. It is not meant to contain self-study textbooks, or of lecture notes for courses.Therefore, relevant mathematical facts should be stated without derivations or proofs (unless the topic of the article is a notable proof, like that of Poncaré's conjecture). If disputed or disputable, such statements should be backed by references instead. --Jorge Stolfi (talk) 08:32, 12 April 2019 (UTC)

Infinite speed of propagation
I am not an expert in the area, but should there not be an section on the compartibility with relativity and why that is/is not a problem in practical applications. — Preceding unsigned comment added by 2601:280:4B80:A490:2109:FFAA:6848:FDD (talk) 21:28, 22 May 2020 (UTC)

Heat Flow.
Damorbel (talk) 13:22, 1 January 2021 (UTC)Surely the use of the term flow is to be deprecated in an article about heat?

Physical Interpretation
This seems wrong: *Informally, the Laplacian operator ∆ gives the difference between the average value of a function in the neighborhood of a point, and its value at that point. Thus, if u is the temperature, ∆ tells whether (and by how much) the material surrounding each point is hotter or colder, on the average, than the material at that point.*

Reason being, ∆ is a 2nd order derivative in space. *First derivate* of temperature over space accounts for heat flow around an observed spot; *Second derivate* or the derivative of heat flow over space indicates how much heat is retained in the observed spot. And this determines temperature change at that spot.


 * It is explained in the first sentence of the following "Mathematical interpretation of the equation" section. Gumshoe2 (talk) 21:26, 9 April 2021 (UTC)

I think what the Physical Interpretation says is downright wrong; and it is confusing to speak of the "difference of the average value". It is the "difference of the differential value"; the current description matches NABLA, not DELTA.

I propose this more intuitive text as an alternative starting point:
The variables $$u_{xx}$$ are second derivates along (in this case) the dimension $$x$$:


 * $$u$$ represents the temperature in a point;
 * $$u_x$$ represents the gradient of temperature, and heat will flow in a linear ratio to that;
 * $$u_{xx}$$ represents the difference in heat flow (for example left in and right out) and determines how much heat is retained in each point;
 * the retained heat is linear to how much the temperature rises, in other words the value of $$u_t$$.

During this computation a few material-dependent linear coefficients are used; these factors combined in the thermal diffusivity $$\alpha$$:


 * $$k$$ determines how much heat will flow through a material as the result of a temperature gradient;
 * there is no factor as a result of determining the difference between heat flows;
 * $$1/{\rho\cdot c_p}$$ determines how much the temperature of a material rises as a result of heat retention.

Need of standard notation
This article presents annoying incongruence in notations indeed. The Laplacian should be denoted Δu everywhere, avoiding double notations like $$\nabla^2$$. Moreover, the awful and unclear notation "Δx" to denote an increment of the variable x should be banned once for all - it also fights with the notation for the Laplacian.


 * (Reply to above unsigned comment:) In physics, it seems that the standard notation for the Laplacian is $$\nabla^2$$ while mathematicians tend to use $$\Delta$$. As heat transfer is fundamentally a physical concept I'd propose that $$\nabla^2$$ be used consistently throughout. (As a bonus this avoids the confusion with the $$\Delta x$$ notation, which is also standard and commonly used in physics.) I'll go ahead and make these changes in the next few days unless there are objections here. DrPippy (talk) 21:36, 20 April 2022 (UTC)


 * This article mostly about mathematics (and the heat equation with its infinite speed of propagation is somewhat questionable physics). I object to your suggestions. The confusing notation $$\nabla^2$$ is also used for second (sometimes covariant) derivatives (as $$\nabla\circ\nabla$$ instead of $$\nabla\cdot\nabla$$. Confusion with the "$$\Delta x$$" notation is avoided by not using it. —Kusma (talk) 22:57, 20 April 2022 (UTC)
 * I'm inferring from your unusual objection to using $$\Delta$$ to indicate increments (a notation that was apparently developed in the early eighteenth century and has been standard in math, physics and chemistry over multiple centuries) that you're also the author of the unsigned first statement? If so, I think it would be helpful to wait for a third perspective here.
 * I agree with you that the article is mostly about mathematics; since the page for heat diffusion redirects here, I see this as a bad thing! Either a separate article treating the physical phenomenon of heat diffusion could be created (which I might enjoy taking a crack at), or this article should be edited to include a more physically motivated treatment of the heat equation. I'd probably lean towards the first approach. DrPippy (talk) 16:27, 28 April 2022 (UTC)
 * No, I'm not the author of the original statement, that was @PMajer, more than ten years ago. But more perspectives are always good. Wikipedia's applied/applicable mathematics content is often a bit messy because mathematicians, physicists and engineers all use the same words but have completely different interests, and many articles tend to reflect the point of view of the group that first started them, with other views tacked on as an appendix. It is not an easy problem to solve (and so I usually write about other things than mathematics here...) —Kusma (talk) 16:57, 28 April 2022 (UTC)
 * Sorry, a little slow getting back here! I appreciate the historical perspective, and agree that it's difficult to disentangle the different approaches taken by mathematicians, physicists, and engineers in a way that keeps everyone happy! It sounds like it may be wisest to let this one be for the moment. DrPippy (talk) 21:51, 10 May 2022 (UTC)

Fourier series solutions?
The Fourier series solution to the heat equation would a good thing to include in the article.

I hope someone knowledgeable about this subject will do this. 2601:200:C000:1A0:7097:9C72:4BFB:717D (talk) 20:00, 17 May 2022 (UTC)


 * As Heat_equation is already in the article, it might help if you could elaborate what you wish to include. —Kusma (talk) 20:09, 17 May 2022 (UTC)