# why I cannot find explicit finite difference for elliptic equation

Let us think on the Poisson equation $$nabla^2 u(bf{x})=rho(x)$$ with Neumann boundary conditions, with $$bf{x}=it (x,y)$$ in 2D.

Here is a stencil with central differences in both $$x$$ and $$y$$ (forgive my label $$t$$ on the vertical, it is supposed to be $$y$$) directions.

with $$beta=Delta x/Delta y$$. More specifically we have the
equation:

$$begin{eqnarray*} w_{i+1 j} -2 w_{ij} left ( 1 + beta^2 right) + w_{i-1 j} + beta^2 (w_{i j+1} + w_{i j-1}) = Delta x^2 rho(x,y) end{eqnarray*}$$

We can get a similar stencil in the (hyerbolic) wave equation in 1D where
time is the $$y$$ (vertical axis). We can always find, recursively
$$w_{i j+1}$$ in terms of all other $$w_{lm}$$ where $$l=i,i+1,i-1$$ and
$$m=j,j-1$$. The points 1 cell away from the boundary can be computed with
a ghost boundary condition (assuming Neumann BC with derivative equal to 0).

I have been searching for hours and all literature shows me an implicit method where I need to invert a huge $$(n-1 times m-1)$$ matrix. Is there a specific reason why the elliptic equation is not solved as we do the 1D wave explicit solution centered in space and time?

Or…if it could be solved, is there a place to look for that solution? Thanks.