Sovle this PDE with Neumanm boundary condition. \begin{cases} \begin{array}{l} u_{tt}-u_{xx}=0,f(t)<x<t. \\ u(t,t)=\phi(t)\\ u(f(t),t)=\psi(t) \end{array} \end{cases} Here $f(t)$ is a curve with the following features: $f(t)\in C^{\infty}(\mathbb{R})$; Its graphic is between $x=t$ and $x=-t$; $|f'(t)|\neq 1$ for all $t\in\mathbb{R}$
I have tried to solve this problem by letting $u(x,t)=F(x+t)+G(x-t)$. Therefore, I get \begin{cases} \begin{array}{l} F(2t)+G(0)=\phi(t)\\ F(f(t)+t)+G(f(t)-t)=\psi(t) \end{array} \end{cases} It's easy to obtain that $F(x)=\phi(\frac{x}{2})-G(0)$ from the first equation. However, it seems difficult to obtain $G(x)$ from the equations above.
