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Wave equation on $[0,1]$ with mixed boundary conditions

Consider the wave equation $u_{xx}-u_{tt}=0$ on the unit interval $x\in[0,1]$. Take mixed boundary conditions ($\alpha_{1,2}^2+\beta_{1,2}^2 \neq 0$) \begin{align*} \alpha_1 u(0,t) + \beta_1u_x(0,...
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