Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 2 / ii / Solution

Separate variables as $u=X(x)T(t)$. The zero values at both plates give the Fourier sine basisExpand the initial profile using orthogonality. With $s=x/L$, its coefficients areConsequentlyOnly odd modes occur because the initial profile is symmetric about the midpoint. The coefficients are absolutely summable, so dominated convergence gives $\boxed{u(x,t)\to 0}$ uniformly as $t\to \infty$. The slowest decay rate is $\nu {\pi }^2/L^2$, associated with $j=1$.