Consider the solution to the steady state concentration profile ∇²c = 0 using centered finite differences. The domain is discretized into nine nodes with a spacing Δx and Δy. The concentration at the top and right sides of the slab is c = 2, and the concentration on the bottom and left sides of the slab is c = 1. Answer the following questions:

(a) If Δx = Δy, what is the temperature at the center node?

(b) If Δx ≠ Δy, how does the temperature at the center node depend on the spacing? This is equivalent to discretizing a rectangular domain with an equal number of nodes on each side.

(c) If Δx ≠ Δy and the top side boundary condition is changed to c = 3, how does the temperature at the center node depend on the spacing?

(d) If Δx = Δy and the top side boundary condition is replaced with a no-flux condition ∂c/∂y = 0, what is the temperature at each node in the domain?