Results 1 to 2 of 2

Math Help - bounded solution to Laplace's equation

  1. #1
    Newbie
    Joined
    Jan 2009
    Posts
    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!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by billybobby26 View Post
    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 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, 2\cos^2 \mu = 1 + \cos 2\mu
    Let 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 r\leq 2?

    Note, 2\sin ^2 \mu = 1 - \cos 2\mu
    Let F(r,\mu) = \tfrac{1}{2} - \tfrac{1}{8} \cos 2\mu
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Nontrivial Solution/Laplace Transform
    Posted in the Differential Equations Forum
    Replies: 2
    Last Post: December 13th 2011, 01:15 PM
  2. Why bounded ode solution tends to a constant
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: October 16th 2011, 07:32 PM
  3. Solution of Differential equations by Laplace Transforms
    Posted in the Differential Equations Forum
    Replies: 12
    Last Post: August 14th 2010, 03:44 PM
  4. Most General solution of Laplace's equation
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: January 16th 2009, 12:14 PM
  5. Replies: 7
    Last Post: September 2nd 2008, 05:01 PM

Search Tags


/mathhelpforum @mathhelpforum