Solve the mixed value problem for $1$-D wave equation \begin{align*} u_{tt} & =u_{xx}\text{, }x>0\text{, }t>0,\\ u(x,0) & =g(x), u_t(x,0) =0,\,x>0,\\ u_t(0,t) & =\alpha u_x(0,t) ,~t>0, \end{align*} where $\alpha\ne-1$ is a constant and $g\in C^2(\mathbb{R}^{+}) $ vanishes near $x=0$.
I can get the fundamental solution for homogeneous and non-homogeneous but I am unable to solve this mixed value problem. Please help.