Solve the initial boundary value problem: \begin{equation} \begin{aligned} u_{tt} &= 4u_{xx} \quad x > 0, \, t > 0 \\ u(x, 0) &= x^2/8, \\ u_t(x, 0) &= x \quad x ≥ 0 \\ u(0,t) &= t^2, \quad t ≥ 0.\end{aligned} \end{equation}
I don't think I can use d'Almbert's formula, so I'm stuck. Some help would be nice!