The Laplace equation is given by
This site gives links to files that may be used in conjunction with Mathematica to solve various forms of the Laplace equation. You may click on the following links to download Mathematica NoteBooks to study its varoius forms.
Dirichlet conditions Inside a Rectangle.
Dirichlet conditions Inside a Circle.
Neuman conditions Inside a Rectangle.
Neuman conditions Inside a Circle.
Various Robin conditions for rectangles and circles.