I'm having trouble with this PDE
$$u_{tt}-4u_{xx}=0$$
$$u(x,0)=u_t(x,0)=0$$
$$u(0,t)=t$$
with $x\geq 0$, $t\in R$.
Making $w=u-t$ I get
$$w_{tt}-4w_{xx}=0$$
$$w(x,0)=0$$
$$w_t(x,0)=-1$$ $$w(0,t)=0$$
Applying d'Alembert's formula
$$w(x,t)=\frac{1}{2c}\int_{x-t}^{x+t}g^*(s)ds=\frac{-2t}{4}$$
But this doesn't fit the contour conditions. What went wrong?