slab of material $\displaystyle -1 \leq x \leq 1$ is initally at a temperature given by

$\displaystyle T = T_{0}(1 + x) \ for \ -1 \leq x \leq 0$

$\displaystyle = T_{0}(1 - x) \ for \ 0 < x \leq 1$,

and the surfaces $\displaystyle x = \pm 1$ are maintained at zero temperature.

Show that the temperature is

$\displaystyle T=\frac{8T_{0}}{\pi^{2}} \sum^{\infty}_{n=0} \frac{1}{(2n+1)^2} cos \frac{(2n+1) \pi x}{2} exp[-k(n+ \frac{1}{2})^2 \pi^2 t]$