Thread: bounded solution to Laplace's equation

1. bounded solution to Laplace's equation

Laplace’s equation in planar polar co-ordinates (r, µ) is
Frr + 1/r Fr + 1/r^2 Fµµ = 0.

Show, for any integer k, that
F (r, µ) = r^k cos (kµ) and F (r, µ) = r^k sin (kµ)
are solutions to Laplace’s equation.

(i) Find a bounded solution to Laplace’s equation in the region r <= 1 which satisfies the boundary condition F (1, µ) = cos^2 µ for 0 <= µ < 2pi.

(ii) Find a bounded solution to Laplace’s equation in the region r >= 2 which satisfies the boundary condition F (2, µ) = sin^2 µ for 0 <= µ < 2pi

I can do the first part, showing that they are solutions to Laplace's equation. I'm totally stuck as to how to do i) and ii).. is this to do with Dirichlet's Problem and seperable solutions? Thanks!

2. Originally Posted by billybobby26
Laplace’s equation in planar polar co-ordinates (r, µ) is
Frr + 1/r Fr + 1/r^2 Fµµ = 0.

Show, for any integer k, that
F (r, µ) = r^k cos (kµ) and F (r, µ) = r^k sin (kµ)
are solutions to Laplace’s equation.
Just show that $\displaystyle F_{rr} + \tfrac{1}{r} F_r + \tfrac{1}{r^2}F_{\mu \mu} = 0$ on whatever domain you are working on.

(i) Find a bounded solution to Laplace’s equation in the region r <= 1 which satisfies the boundary condition F (1, µ) = cos^2 µ for 0 <= µ < 2pi.
Note, $\displaystyle 2\cos^2 \mu = 1 + \cos 2\mu$
Let $\displaystyle F(r,\mu) = \tfrac{1}{2} + \tfrac{1}{2}r^2 \cos 2\mu$

(ii) Find a bounded solution to Laplace’s equation in the region r >= 2 which satisfies the boundary condition F (2, µ) = sin^2 µ for 0 <= µ < 2pi
Do you mean perhaps $\displaystyle r\leq 2$?

Note, $\displaystyle 2\sin ^2 \mu = 1 - \cos 2\mu$
Let $\displaystyle F(r,\mu) = \tfrac{1}{2} - \tfrac{1}{8} \cos 2\mu$