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

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The discussion focuses on applying the D'Lembert method to solve the wave equation with specified boundary and initial conditions. The equation is given as u_tt = u_xx for x > 0 and t > 0, with boundary condition u(0,t) = 0 and initial conditions u(x,0) = x e^(-x^2) and u_t(x,0) = 0. A participant expresses uncertainty about incorporating the boundary condition u(0,t), typically applying the method with only initial conditions. It is suggested to ignore the boundary condition initially and verify that the final solution satisfies it. The conversation concludes with a commitment to apply the method and observe the results.
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$\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?
 
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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.
 
Okay, I'll apply it then and see how it works, thanks!
 

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