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Helmholtz Equation with non-homogeneous b.c.

  1. Jan 31, 2010 #1
    1. The problem statement, all variables and given/known data
    In 2D: (del2 + k2) u(r,theta) = 0 with b.c u(R,theta) = f(theta)

    Starting from the general solution (by separation of variables) show that the solution can be rewritten as:

    [tex]\int[/tex] K(r,theta,theta')*f(theta') dtheta' from 0 to 2*Pi

    2. Relevant equations

    The general solution is Jm(k*r)*Pm(cos(theta)) where Pm are the Legendre polynomials and Jm are the Bessel functions

    3. The attempt at a solution

    Ok so I've attempted this on three different occasions, following the professor's instructions to the problem I end up with a relationship Jm(kR) Pm(cos(theta)) = f(theta) where I need to find the values for k, but this would require some type of inversion for the Bessel functions. So I am thinking this isn't the case.

    Alternatively, if I try to define the solution u(r,theta) = v(theta) + w(r,theta) I end up getting confused on how to properly deal with the v(theta) term. I know it needs to obey the boundary conditions so I can rewrite the equation as an inhomogeneous helmholtz equation in terms of w(r,theta) and then find the solution as an integral involving the appropriate Green's function.

    Any help?
  2. jcsd
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