finding a solution of a PDE Unfortunately I can't find a solution $u:[0,\infty)\times[0,\infty)\rightarrow\mathbb R$ of this PDE:
\begin{array}{rl}
u_{tt}&=&u_{xx} &\text{in }(0,\infty)\times(0,\infty)\\
u(0,t)&=&0 &\forall t>0\\
u(x,0)&=&\varphi(x) &\forall x\ge0\\
u_t(x,0)&=&\psi(x) &\forall x\ge0\\
0=\varphi(0)&=&\psi(0)
\end{array}
Can someone help me? Thank you very much!

If you didn't have the condition $u(0,t)=0$ and the domain for the $x$ variable was $\mathbb{R}$ instead of $(0,\infty)$, then you just would use D'Alembert's formula. The trick in your situation is to think of $u$, $\varphi$ and $\psi$ to be extended from $(0,\infty)$ to $\mathbb{R}$ by odd reflection, that is, $u(-x,t)=-u(x,t)$, and likewise for $\varphi$ and $\psi$. The extended $u$ satisfies the wave equation with extended initial conditions $\varphi$ and $\psi$ (check this!), and thus you can apply D'Alembert's formula in this situation.
