Determine a formula for the position $\displaystyle u(x,t)$ each point $\displaystyle x$ of the string at time $\displaystyle t$ in the form of a trigonometric series. Also, determine a formula for the coefficients of the series for $\displaystyle f$ and $\displaystyle g$.

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

$\displaystyle u(0,t) = 0$; $\displaystyle u(L,t) = 0$; $\displaystyle t>0$

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

$\displaystyle f(x)$ is continuous split and regular period 2L which describes the initial position of the string