Boundary Values & Fourier Series · Hard problem

A square plate $0 \le x \le \pi$, $0 \le y \le \pi$ is in steady state, so its temperature solves Laplace's equation $u_{xx} + u_{yy} = 0$. The bottom edge is held at $u(x, 0) = \sin x$, the left edge at $u(0, y) = \sin y$, and the other two edges at $0$. Find the temperature at the centre, $u\!\left(\dfrac\pi2, \dfrac\pi2\right)$.