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The equation is $$u_{tt} - u_{xx} = 0, x>0, t>0$$ $$u(x,0)=f(x); u_t(x,0)=0$$ $$u_x(0,t)=ku_t(0,t)$$ and the third condition is hard to deal with, as it isn't three standard type of boundary condition (Dirichlet, Neumann and Robin).

It is trivial to solve it in the region $x\geqslant t$, because it is just the case of d'Lambert formula and doesn't involve the boundary condition. Is there some hint to solve it in $x\leqslant t$?

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  • $\begingroup$ Boundary conditions should not affect your approach. Find a general solution first then just apply boundary conditions. $\endgroup$
    – Vasili
    Sep 17, 2021 at 15:16
  • $\begingroup$ @Vasya Do you mean the general solution $F(x-at)+G(x+at)$? I try to use this by extending $f(x)$ to $x\leq 0$, but failed: the solution may be not second-order differentiable$. $\endgroup$
    – user779130
    Sep 17, 2021 at 17:32
  • $\begingroup$ You can use separation of variables, look for a solution in a form $u(x,t)=X(x)T(t)$. In this case $u_{tt}=u_{xx}$ $\endgroup$
    – Vasili
    Sep 17, 2021 at 17:40

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Your idea to extend $f$ to the whole line is correct I think. An extension like $$ f(-y) = cf(y), \qquad (y>0) $$ seems to work, with $c$ depending on $k$ (except for $k=-1$, that I'll have to think more). Problem Unknown time-dependent boundary data for a BVP involving the wave equation and the reference there is somewhat relevant. In particular, differentiability is not required for a weak soluion.

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