Solve the initial value problem $$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=1$$ subject to $u(x,0)=f(x)$ for the semi-infinite plane $-\infty<x<\infty$ and $y>0$ .
We use the Green's function technique . We know that the associated equation $$-\nabla^2G(x,y;\xi,\eta)=\delta(x-\xi,y-\eta)$$ and Green's identity would give $$u(x,y)=-\int_Sf(\xi)\frac{\partial G}{\partial n}d\xi-\int\int_RG(x,y,\xi,\eta)d\xi d\eta$$ and using $G=\dfrac{1}{4\pi}\log\dfrac{(x-\xi)^2+(y+\eta)^2}{(x-\xi)^2+(y-\eta)^2}$ we find the solution as $$\color{blue}{u(x,y)=\frac{y}{\pi}\int_{-\infty}^\infty\frac{f(\xi)}{(x-\xi)^2+y^2}d\xi \ - \ \dfrac{1}{4\pi}\int_{\xi=-\infty}^\infty\int_{\eta=0}^\infty\log\dfrac{(x-\xi)^2+(y+\eta)^2}{(x-\xi)^2+(y-\eta)^2}d\xi d\eta}$$ But I am not so sure about the correctness of the solution since I have just learned the method myself and I have doubt whether the second integral would converge for the limit for $\eta$ . If someone checks the solution , it will be very helpful . Regards .