Second-order elliptic equations of the Laplace and Poisson type can be solved by centraldifference-based finite-difference schemes. Thus, if we employ a square mesh, show that the temperature at a mesh point \((i, j)\) is given by the averge of the adjacent mesh values:

\[T_{i, j}=\frac{1}{4}\left(T_{i+1, j}+T_{i-1, j}+T_{i, j+1}+T_{i, j-1}\right)\]

This provides an iterative method of solving for the temperature field using the GaussSiedel iterative scheme.

Solve the temperature field for a 2D square geometry and compare your result with the analytic solution. Note that the scheme should not be used on the boundary nodes. The temperature is simply set as the boundary value at these points.

How would you modify this if there is also a heat generation in the slab?