# Laplace Equation Polar Form

1. Apr 20, 2013

### middleramen

1. The problem statement, all variables and given/known data

Solve the BVP:

r$^{2}$u$_{rr}$ + ru$_{r}$ + u$_{ψψ}$ = 0

0 ≤ r ≤ 1, 0 < ψ < 2π

u(1,ψ) = 0.5(π - ψ)

2. Relevant equations

3. The attempt at a solution

I've derived the general solution of u(r,ψ) = C + r$^{n}$Ʃ$_{n}$a$_{n}$cos nψ + b$_{n}$sin nψ, where a,b, C are constants.

Attempts to determine fourier constants don't give meaningful results.

Also, it was my understanding that the boundary condition must be periodic, i.e. u(1,0) = u(1,2π), which is untrue for this problem.