Elliptic boundary value problem
Solve the following problem:
Uxx means second derivative with respect to x
Uyy means second derivative with respect to y
Uxx + Uyy = 0, 0 < x < pi, 0 < y < pi
U(x,0) = 0, U(x,pi) = 1, 0 < x < pi
U(0,y) = 0, U(pi,y) = 1 0 < y < pi
I know the problem has to be broken into 2 separate problems using U = V + W with zero conditions on 3 sides for each problem W and V. So please show the problem in this manner.
Show all work including the derivation of any eigenvalues or eigenvectors.