My question:
I want to solve $u_{tt}-u_{xx}+\lambda u=0$ in $0 \leq |t|\leq x \leq 1$ with the conditions that $u(x,x)=u(x,-x)=1$.
My attempt:
I let $\xi=x+t$ and $\nu=x-t$. This transformed the equation into $$-4u_{\xi\mu}+\lambda u=0.$$
This can be written as an integral equation: $$\frac{u}{\lambda}4=\int_{0}^{\nu}\int_{0}^{\xi}u \ dy\ dz$$
What I want to know:
Is this correct so far? I feel that I've ignored the $u(x,x)=x(x,-x)=1$ condition.
Next, I want to use Picard iteration to get a series solution to this double integral but I do not know how to proceed. For the one dimensional ODE case, the procedure is to show that $u$ is Lipschitz and and then iterate with the initial guess to calculate $\int u.$