Question

How to solve 2-dimessional laplace equation?

Answer 1

Let’s start out by solving it on the rectangle given by , .  For this geometry Laplace’s equation along with the four boundary conditions will be,

                                                                       



(1)

 

One of the important things to note here is that unlike the heat equation we will not have any initial conditions here.  Both variables are spatial variables and each variable occurs in a 2nd order derivative and so we’ll need two boundary conditions for each variable.

 

Next, let’s notice that while the partial differential equation is both linear and homogeneous the boundary conditions are only linear and are not homogeneous.  This creates a problem because separation of variables requires homogeneous boundary conditions.

 

To completely solve Laplace’s equation we’re in fact going to have to solve it four times.  Each time we solve it only one of the four boundary conditions can be nonhomogeneous while the remaining three will be homogeneous.