4. 1) Find the solution of the Laplace Equation

in the domain subject to boundary conditions:

and

2) Sketch the graph of the solution for , and

3) What is the solution for ? Sketch the graph.

Solution:

Laplace’s Equation. Basic case: 3 b.c. are homogeneous.

1) Separation of variables:

Sturm-Liouville Problem (equation with two homogeneous b.c.):

Second equation:

Apply

Therefore

Write the solution in the form of the infinite series:

Apply the last b.c. to determine coefficients :

Treat it as the cosine Fourier series expansion of function , then coefficients are:

Then solution is:

3) Case

then

for (orthogonality)

An solution becomes: