I have a 1D Laplacian equation $\nabla^2 u(x) = 0$ with some boundary condition $u|_{\partial \Omega} = g$, where the domain $\Omega = [0,L]$ and the boundary condition is Dirichlet, that for some $x$, we set $u(x) = g(x)$. Now I would like to explore using Green's function to get a closed-form solution. We have two Dirichlet boundaries at the domain boundary $u(0) = u(L) = 0$ and the Green's function in 1D is: $$ G(x,y) = \begin{cases} y(1-x/L) & x > y\\ x(1-y/L) & x \leq y \end{cases} $$
For Poisson equations, we have: $$u(x) = \int_0^L G(x,y)f(y)dy$$ but I'm not sure how can we do the same thing for the Laplacian equation while still keeping the boundary condition $g$. Does anyone have any ideas?
Thank you very much!