# Derivation of heat equation

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.

• It uses the law of conservation of energy and the Fourier's law connecting the heat flux with a temperature gradient. – Alex Trounev Oct 5 '19 at 12:10
• @Alex Yes I know that but formally. Thank you! – Migalobe Oct 5 '19 at 15:54

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.

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