I got lost in PDEs after 1D separation of variables.

August 8, 2007 11:23 PM

Yeah, this could be the lamest applied math/engineering question ever - is there a good resource that can show me how to solve the heat equation in two (or more) spatial coordinates, with a source?

Infinite, semi-infinite, and finite domains are all cool. The examples I've seen solve this equation in one dimension, or in the steady-state case, or without a source, or some combination thereof. I know that the beyotchitude of this problem is why numerical methods were invented, but I'm curious as to what the analytical solution looks like.

And no, you're not doing my homework! I'm just curious. Really.

posted by universal_qlc to Science & Nature (12 answers total)

If I understand what you're asking about, my understanding is that it can't be solved in two dimensions with an equation. You have to use brute force.

"Take a square steel plate at ambient temperature and put a lit candle under one corner. After five minutes, what is the temperature gradient over the whole plate?" -- is that the kind of thing you're talking about?

This was one of the first problems they took on with ILLIAC IV, back during early research into massively parallel computing.

Going from one dimension to two dimensions in this problem is like going from the "two body problem" to the "three body problem" in orbital dynamics. The two body problem has an elegant solution. The three body problem (or any larger number) can only be solved by brute force simulation, which is ugly and inelegant and subject to all kinds of horizon limitations.
posted by Steven C. Den Beste at 12:45 AM on August 9, 2007

Response by poster: Seriously? I honestly thought that given boundary conditions for a finite region, it could at least be solved through some sort of separation of variables solution, ugly eigenfunctions and inhomogeneities included. But then again, when all you have is a hammer...
posted by universal_qlc at 1:00 AM on August 9, 2007

Actually, it's easier to solve for an infinite plate than for a finite one. The boundaries are what make it hellish.
posted by Steven C. Den Beste at 1:21 AM on August 9, 2007

It may be useful to consult Crank's "Mathematics of Diffusion", since the diffusion equation is analogous to heat. There is some discussion of approximate two-dimensional solutions. Otherwise it is probably easier to solve numerically.
posted by Comrade_robot at 5:42 AM on August 9, 2007

I'm pretty sure my heat transfer textbook has a few solutions for 2D heat transfer. Numerical methods would be my first instinct if I had to find a usable/robust solution.

I have an exam this morning, but if you nobody else gives you a satisfactory answer I'll dig out my book and let you know.
posted by KevCed at 6:04 AM on August 9, 2007

Best answer: Some thoughts:

The reason the finite case is harder is that you need to solve the time-dependent or time-independent heat equation, plus satisfy the condition that the heat flux must be zero at your boundary.

One solution is to use the method of images: assume a mirror-image pseudo-source on the other side of the boundary so that symmetry requires the flux to be zero at the edge. However, this approach screws things up at the opposite edge; the flux there is now non-zero. So you need two pseudo-sources on that side to fix it. But now you need more on the first side... Fortunately, the additional pseudo-sources are farther away, and just provide correction terms. The upshot is, there is an analytical solution, but it has an infinite number of terms. You include as many terms as you need for the desired accuracy. And of course, the solution is different for different geometry. That's why you see strategies instead of solutions.

I also recommend Crank.
posted by Mapes at 6:39 AM on August 9, 2007

I just checked my old Heat and Mass transfer text (Incropera & DeWitt), and they don't really seem to go over the solution methods -- they say "The solution can be expressed as ..." My Advanced Engineering Mathematics book (Kreyszig) isn't very helpful either.

Crank solves a special case: (Again, these are the diffusion equations, which are analogous)

2.5 Solution in two and three dimensions
2.5.1

Consider the equation of diffusion

d2c/dx1^2 + d2c/dx2^2 + d2c/dx3^2 = 1/D dc/dt (where the 'd's are partials)

in the rectangular parallelepiped

a1 < x1 b1, a2 x2 b2, a3 x3 b3br>
For certain types of initial and boundary conditions, the solution of [the first equation] is the product of the solutions of the three one-variable problems, and thus can be written down immediately if these are known. The following proof is given by Carslaw and Jaeger (1959, p. 33).

Suppose cr(xr,t),r=1,2,3, is the solution of

d2cr/dxr^2 = 1/D dcr/dt, ar < xr br (d's are partials again)br>
with boundary conditions

alphar*dcr/dxr - betarcr = 0, xr = ar, t > 0,
alpha'r*dcr/dxr - beta'rcr = 0, xr = br, t > 0, (d's are partials again)

where the alphar and betar are constants, either of which may be zero (so that the cases of zero surface concentration and no flow of heat at the surface are included) and with initial conditions

cr(xr,t) = Cr(xr), t = 0, ar < xr br.br>
Then the solution of [first equation] in the regsion defined by [second equation] with

c = C1(x1)C2(x2)C3(x3), t = 0,

and with boundary conditions

alphar dc/dxr - betarc = 0, xr = ar, t > 0 , r = 1,2,3,
alpha'r dc/dxr - beta'rc = 0, xr=br, t > 0, r = 1,2,3, (d's are partials again)

is c = c1(x1,t)c2(x2,t),c3(x3,t).

For subsituting (above equation) in [first equation] gives

c2c3d2c1/dx1^2 + c3c1d^2c2/dx2^2 + c1c2d2c3/dx3^2 - 1/D(c2c3dc1/dt + c3c1dc2/dt + c1c2dc3/dt) = 0,

using [third equation]. Clearly the initial and boundary condtions [seventh equation, eighth equation, ninth equation] are satisfied.

An essential condition is [seventh equation], namely that the initial condition must be expressible as a product of the initial conditions for the one-variable problems taken separately. Carslaw and Jaeger (1959) give solutions for a rectangular corner, rectangles, parallelpipeds, cylinders and some examples of isotherms are shown graphically.

Carslaw and Jaeger is:

CARSLAW, H.S. and JAEGER, J. C. (1959) Conduction of Heat in solids Clarendon Press, Oxford.
posted by Comrade_robot at 6:42 AM on August 9, 2007

My partial diff eq class (long ago) did heat, though we had not text for me to reference for you. It's true that you can solve it under certain restrictive conditions. If you google "heat equation two dimensions" you get dozens of articles on both analytic and numerical methods. For example. If you pick up ANY textbook on partial differential equations it will contain the heat equation. Please google first.
posted by a robot made out of meat at 8:21 AM on August 9, 2007

you can solve them, I just spent last semester doing so, but they'll end up as an infinite sum or some crap like that depending on the IC and BC. Your best bet is to sit down with a mathematical methods for physical sciences book (Boas is the one we used) and figure out seperation of variables, steady state solutions, etc. Once you've got 1D down, 2D is cake because it's just 1 extra step, but it takes forever to do a problem for the first time.
posted by Large Marge at 9:04 AM on August 9, 2007

Response by poster: Thanks everyone. Yes, I'm definitely aware that it's much easier to solve numerically (see original question), and indeed that what's I've been doing since school, but was simply trying to see how awful the analytical solution was.

To robot made out of meat: try not to get all cranky pants on me; realized now that that what I typed in my original question was ambiguous. I was looking for a solution to the heat equation (or general diffusion equation) whose conditions included a HEAT or MASS or whatever source. My bad.

Comrade_robot - I wholly appreciate the sentiment, and I'm very sorry you had to type all the out. See above. I will definitely check out those books you mentioned though.

Steven and Large Marge: yep, I indeed am curious to see the godawful solutions.
posted by universal_qlc at 9:24 AM on August 9, 2007

Oh, my bad. I read your question as for a reference with sources (as opposed to just an answer), not a heat source.
posted by a robot made out of meat at 10:29 AM on August 9, 2007

Just in passing, it occurs to me that the approach that Mapes describes (which is really cool) isn't practical if the plate has any curved edges. It only works if the edges are straight.

And it may be unreasonably complicated to apply on anything other than a rectangle. What about a plate which has 6 edges and is L-shaped?

Not to mention cases where there are more than one heat sources, and maybe one or more heat sinks.

I think the reason that in practice this is usually solved by brute force is that brute force is a general solution which works equally well (or badly) for any shape, and any number of heat sources and sinks. The programming is almost certainly bitch, but once you've done it, you don't have to do it again.
posted by Steven C. Den Beste at 8:26 PM on August 9, 2007