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A very simple solution to the diffusion equation is $u(x,t)=x^2+2 t$

My question:
How can this be a solution to the diffusion equation when nothing really diffuses, but just stays the same - see plot: Here

I can see that it is a solution but can't believe my eyes... Is there some kind of intuition other than: "it satisfies the PDE"?

Are there other examples of solutions to well-known equations that behave completely un-intuitive in some situations?

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3 Answers 3

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Here are a few pointers, from the more obtuse to the more direct.

  1. "Meeting a friend in a corridor, Wittgenstein said: 'Tell me, why do people always say it was natural for men to assume that the sun went round the earth, rather than that the earth was rotating?' His friend said, 'Well, obviously, because it looks as if the sun is going round the earth.' To which the philosopher replied, 'Well, what would it have looked like if it had looked as if the earth was rotating?'" Apparently from Tom Stoppard's play Jumpers. More to the point: what does your intuition think the solution to the diffusion equation with initial conditions $u(x,0) = x^2$ should have looked like?

  2. $u_1(x,t) = \text{const}$, and $u_2(x,t) = x$ are also solutions of the diffusion equation. Do you find them more or less surprising than the $x^2 + t$ solution?

  3. The standard intuitive interpretation of the diffusion equation is that it "smooths out" variations in the solution. The way it finds these variations is by looking at the second derivative. But the second derivative of $x^2$ is constant everywhere, so what can the diffusion equation do?

  4. Locally, every point "wants to be" equal to the average of the value in its neighbourhood. If the second derivative at a point is positive, the function looks like a valley, so the point increases its value over time; if the second derivative is negative, the point is on a peak, and tries to decrease. With $x^2$, all of $\mathbb{R}$ is a valley.

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Well, concerning 1. I would have thought that some flattening out takes place ending in a flat line, but for 2.: This really dropped my jaw to the floor! Esp. with $u_2$ there is such an disequilibrium (and it could be multiplied by any const. to make it even worse) and the whole situation doesn't change over time - now, where is the intuition here?!? –  vonjd Oct 20 '10 at 9:54
    
@vonjd: Well, diffusion is certainly taking place, in that heat is flowing at a constant rate from the right to the left. But the temperature $u$ cannot change on any region, because exactly as much heat flows out from the left as flows in from the right. A more intuitive variant may be to think of diffusion on an interval $[-a,a]$ with the boundaries fixed at temperatures $-a$ and $a$. Then $u(x,t) = x$ on $x \in [-a,a]$ is a solution: heat comes in from the right, goes out the left. My example is what you get when you let $a \rightarrow \infty$. –  Rahul Oct 20 '10 at 18:58
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It is difficult to interpret a PDE without knowing its boundary conditions and/or initial values.

But, at first guess, I'd say you should look at individual time slices of the solution on the same graph and see if that helps your interpretation. The 3D image is nice for showing everything at once, but that can obscure an interpretation.

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But there is diffusion! Heat – or whatever quantity $u$ represents – flows from the far left and right ($x\to\pm\infty$, where there's lots of it!) towards the middle ($x=0$). For every fixed point $x$ there is always more heat flowing towards it from the outside than what it emits inwards, and the result is that the heat keeps increasing at each point.

Now this might seem unphysical, and that's because it is. For this particular solution, the total amount of heat, $\int_{-\infty}^{\infty} u(x,t) dx$, is infinite for each $t$.

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This is a very intuitive explanation - the fascinating thing is that it keeps it form which is lifted upwards over time. –  vonjd Oct 20 '10 at 9:56
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