Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2007 / Paper 1 / 2

Separation of variables gives $X^{\prime \prime }=-k^{2}X$ and $T^{\prime }=-\kappa k^{2}T$. The periodic matching of both the function and its first derivative selects $k=n\pi /L$, with the constant mode at $n=0$. Thus the periodic diffusion on a circle solution iswhere the Fourier coefficients of the initial function areThe formula for $a_n$ also gives $a_0$ at $n=0$. The constant mode preserves the initial spatial mean; every nonconstant Fourier mode decays. For initial data in L2 space, the series approaches those data in L2 space as $t\downarrow 0$ and is smooth at positive times.

The initial function is odd, so all $a_n$, including $a_0$, vanish. Set $k_n=n\pi /L$. For $k_n\ne 1$, product-to-sum and integration by parts giveConsequently the coefficients of the periodic sine with an endpoint-correcting linear term areInsert them into the sine terms of the preceding solution. If $L=m\pi$ for a positive integer $m$, the original initial profile is simply $\sin x$, with $k_m=1$. The apparent $0/0$ in the displayed coefficient has removable limit $b_m=1$, while $b_n=0$ for $n\ne m$. In that case