Boundary Values & Fourier Series · Challenge problem

A rod $0 \le x \le \pi$ obeys $u_t = u_{xx}$ and starts at $u(x, 0) = 0$. Its left end is held at $0$ while its right end is heated steadily: $u(0, t) = 0$, $u(\pi, t) = t$. For large $t$ the midpoint lags behind the linear profile: find $\displaystyle\lim_{t\to\infty}\left[u\!\left(\dfrac\pi2, t\right) - \dfrac{t}{2}\right]$.