= Periodic diffusion on a circle
{title2=$u(x,t)=\bar f+\sum_{n\geq1}(a_n\cos nx+b_n\sin nx)e^{-Dn^2t}$}
For a circle of circumference $2\pi$, the <heat equation> $u_t=Du_{xx}$ has periodic matching of both $u$ and $u_x$ at $x=\pm\pi$. For $D>0$, the displayed <Fourier series> solves the initial-value problem, with $\bar f=(2\pi)^{-1}\int_{-\pi}^\pi f(x)dx$ and the usual real <Fourier coefficients>. Every nonconstant <normal mode> decays, leaving the conserved mean. In particular $f(x)=(\pi-|x|)^2$ gives $\bar f=\pi^2/3$, $a_n=4/n^2$ and $b_n=0$. The initial cusp is smoothed for every positive time.
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