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What are the main physical laws to derive the following heat equation: $$u_t -\Delta u=f(t,x)?$$ I'm wondering about the interpretation of the Laplacian $\Delta$ and its role in heat process.

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    $\begingroup$ It uses the law of conservation of energy and the Fourier's law connecting the heat flux with a temperature gradient. $\endgroup$ – Alex Trounev Oct 5 '19 at 12:10
  • $\begingroup$ @Alex Yes I know that but formally. Thank you! $\endgroup$ – Migalobe Oct 5 '19 at 15:54
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I think a very good intuition about what is the Laplacian "doing" is to look at the way of implementing it in a computational simulation, e.g., its finite-difference method's implementation. Since it is the sum of second, non-crossed derivatives, if the system is approximated with a square grid of stepsize $h$, the Laplacian is approximated as (you can think about this yourself as it is fairly easy or just google it):

$$\Delta u(x,y) \approx \frac{u(x-h, y) + u(x+h, y) + u(x, y-h)+ u(x, y+h) - 4u(x,y)}{h^2}$$

Looking carefully at this expression, it means computing the sum of the difference between all neighbouring sides. Imagine now that there is a gradient in heat $u$ from left to right. This will mean that the difference $u(x-h, y)-u(x, y)$ is positive while $u(x+h, y)-u(x, y)$ is negative. Thus the new value after a time $dt$ on the site $(x,y)$ will be a balance between heat that flows in from the left and heat that flows out to the right. So, basically, what the Laplacian is doing is "homogenizing" the value of $u$ in space. This is why you find the Laplacian operator in any equation that involves diffusion.

In the case where there are no sources or sinks in the system ($f(x,y) = 0$, $\forall(x,y)$), the equation basically comes to be $\partial_t u = \Delta u$ (diffusion equation), and the system distributes heat as homogenously as possible. With sources or sinks, the system tries to but the sources/sinks $f(x,y)$ keep them from doing it completely.

The subject is usually treated in books on Partial Differential Equations, usually it's one of the first (interesting) cases presented. It allows for a good introduction to Fourier series (historically originating in the problem) and Green's functions. The one by J. David Logan (Springer) has a treatment of the matter, and you can find the finite-difference approximations at the last chapter.

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  • $\begingroup$ Thank you for this good clarification! $\endgroup$ – Migalobe Oct 5 '19 at 15:54
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    $\begingroup$ There is a typo in the equation (should be $-4u(x,y)$ instead of $-uf(x,y)$ ). See youtube.com/watch?v=PE7oiOq_xig $\endgroup$ – Alex Trounev Oct 5 '19 at 16:24
  • $\begingroup$ thanks I made the correction $\endgroup$ – Marc Plana Caballero Oct 6 '19 at 17:07

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