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The heat equation

The heat equation describes how the temperature $u(x,t)$ in a rod evolves over time, and is often solved by separation of variables. The solution becomes a sum of modes that each decay at their own rate - higher modes fastest.

u_t = c^2u_{xx}

the heat equation (1D)

u_n(x,t) = \sin\frac{n\pi x}{L}\,e^{-c^2(n\pi/L)^2t}

mode n, with

u=0

at both ends

Symbols

$c^2$	diffusion coefficient
$L$	length of the rod	m

Example

$c^2=4$ , $L=\pi$ , $n=3$ : the decay rate is $\lambda=c^2(n\pi/L)^2=4\cdot9=36$ .

Because higher modes decay fastest, the temperature profile smooths out faster as time passes.

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Part of Laplace, Fourier and PDEs: Partial differential equations.