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

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

$\displaystyle u(0,t) = 0$

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

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

How can I resolve ?