Find a formula for the temperature $\displaystyle u(x,t)$ in the form of a series with general formula for the coefficients of the series.

$\displaystyle \alpha^2 \frac{ \partial^2 u}{ \partial x^2} = \frac{\partial u}{\partial t}$ and $\displaystyle 0<x<L$ ; $\displaystyle t>0$

$\displaystyle u(0,t) = 0$ and $\displaystyle \frac{\partial u}{\partial x}(L,t) = 0$ and $\displaystyle t>0$

$\displaystyle u(x,0) = f(x)$

$\displaystyle f(x)$ is assumed to be piecewise continuous and periodic of period 2L