Solve the following initial value and boundary problem.
$u_{tt} -c^{2}u_{xx} = 0$, $0<x<T$, $t>0$
$u(x,0) = \varphi(x)$, $u_{t}(x,0) = \psi(x)$, $0 \leq x \leq L$
$u(0,t) = a(t)$, $u(L,t) = b(t)$, $t \geq 0$
This problem models the motion of a vibrating string with the initial position given by $\varphi$ and initial velocity given by $\psi$, with free ends whose motion is controlled by functions $a$ at the left end and $b$ at the right end.
I solved this problem in the case in which the vibrating string is fixed at the ends, that is, when $u(0,t) = 0$ and $u(L,t) = 0$. In this case, I obtain the solution in terms of Fourier series and is given by
$$u(x,t) = \left[\sum_{n=1}^{\infty}{a_{n}\cos\left(\frac{n\pi ct}{L}\right) + b_{n}\sin\left(\frac{n\pi ct}{L}\right)}\right]\sin\left(\frac{n\pi x}{L}\right)$$ Where the Fourier coefficients are given by $$a_{n} = \frac{2}{L}\int_{0}^{L}{\varphi(\epsilon)\sin \left(\frac{n\pi \epsilon}{L} \right)}d\epsilon$$ and $$b_{n} = \frac{2}{n\pi c}\int_{0}^{L}{\psi(\epsilon)\sin \left(\frac{n\pi \epsilon}{L} \right)}d\epsilon$$
However, in the general case, when $u(0,t) = a(t)$ y $u(L,t) = b(t)$ I have not been able to find a solution yet. How can I do this? I need some help!