Find a solution $u(x,t)$ of $$ u_{tt} - u_{xx} = 0, \ t>x>0$$ $$u_x(0,t) = f(t), \ t>0$$ $$u(x,x) = h(x), \ x>0$$ in terms of the two functions $f, h$. Is the solution unique?
This is the standard wave equation, so the general solution should be given by D'Alembert as $u(x,t) = F(x+t) + G(x-t)$ for any twice differentiable $F, G$. The boundary conditions are unusual, however.
Using the condition $u_x(0, t) = f(t)$ we obtain $f(t) = F'(t) + G'(-t)$. The condition $u(x,x) = h(x)$ leads to $h(x) = F(2x) + G(0)$. I know I need to "combine" these two equations to obtain $F$ and $G$ in terms of $f$ and $h$. I'm guessing that the first step is to integrate to write the first equation as $\int_0^t f(s)ds +C = F(t)-G(-t)$, but am stuck after that--the $G(0)$ is throwing me off in particular.
If $G(0)$ is troubling you, assume it's $0$. You have the freedom to subtract a constant from $G$ and add it to $F$; this doesn't change $u$. So we may assume $G(0)=0$.
Then $F(x) = h(x/2)$. And from the derivative condition $G'(-t) = f(t)-\frac12 h'(t/2)$ you find $G$.