Are there implicit and explicit solution methods for elliptic equation?

Question

I know implicit schemes means that the unknown values are given in terms of known-and-unknown values, which leads to a system of equations. Explicit schemes means that the unknown values are given in terms of the known values which can be solved for directly. Implicit and explicit solution schemes are always explained using parabolic and hyperbolic equations but its never explained in terms of elliptic equations. Are these methods used only for parabolic and hyperbolic equation and if so why isn't it explained so(ie explicitly stated so). If these methods are used for solving elliptic equations as well, can you please explain the same.

Answer 1

There is a good reason for that... In elliptic equations you do not have the time variable, just space. You have for instance the values at the boundary but you must solve for the values in the interior, as they are not given. This is not the case in parabolic problems, where you know the solution in the whole space domain at the initial time.