Thanks for reply, but I am not sure I understand why is this so.

Mr. Blinder presumes that you use in 2D a point source for diffusion, so the problem has (for a homogenous material) radial symmetry and he mentions the radial form of the Laplacian, the angle does not matter because of the radial symmetry. You can use Mathematica to find out the form of the Laplacian in diverse co-ordinate systems. Symmetry is always an important point in reducing computational complexity of problems, you know.

Make the following changes in the diffusion differential equation: For 1D -> 2D, make substitutions x -> r and d^2/dx^2 -> d^2/dr^2 + (1/r)d/dr For 1D -> 3D, x -> r and d^2/dx^2 -> d^2/dr^2 + (2/r)d/dr