MHB D'Lembert Method for Solving the Wave Equation with Boundary Conditions

[Markov2]

Solve

$\begin{aligned} & {{u}_{tt}}={{u}_{xx}},\text{ }x>0,\text{ }t>0 \\
& u(0,t)=0,\text{ }t>0 \\
& u(x,0)=x{{e}^{-{{x}^{2}}}},\text{ }0<x<\infty \\
& {{u}_{t}}(x,0)=0.
\end{aligned}
$

The condition $u(0,t)$ is new to me, since I usually apply the method when only having $u(x,0)$ and $u_t(x,0),$ what to do in this case?

 

[HallsofIvy]

The boundary and intial value conditions match at (0, 0) so I would just ignore it, then check to make sure my result satisfied that.

 

[Markov2]

Okay, I'll apply it then and see how it works, thanks!