1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Bessel equation

  1. May 15, 2005 #1
    Im trying to solve Helmholtz equation

    [tex]
    \nabla ^2u(r,\phi,z) + k^2u(r,\phi,z) = 0
    [/tex]

    in a hollow cylinder with length L and a < r < b
    and the boundary conditions:

    [tex]
    u(a,\phi,z) = F(\phi,z)
    [/tex]
    [tex]
    u(b,\phi,z) = G(\phi,z)
    [/tex]
    [tex]
    u(r,\phi,0) = P(\phi,z)
    [/tex]
    [tex]
    u(r,\phi,L) = Q(\phi,z)
    [/tex]
    [tex]
    u(r,\phi,z) = u(r,\phi + 2\pi,z)
    [/tex]

    Solution:

    With
    [tex]
    u(r,\phi,z) = v_1(r,\phi,z) + v_2(r,\phi,z) + v_3(r,\phi,z)
    [/tex]

    i get three problems which i can solve separately.
    Separation of variables gives 9 d.e. Three of them are bessel equations.

    [tex]
    r\frac {d} {dr}(r\frac {dR_i(r)} {dr}) + (\mu_i^2r^2-m_i^2)R_i(r) = 0
    [/tex]

    i = 1,2,3. and [tex] \mu, m [/tex] are separation constants.

    The boundary conditions are

    [tex]
    R_1(a,\phi,z) = F(\phi,z),
    R_1(b,\phi,z) = G(\phi,z)
    [/tex]

    [tex]
    R_2(a,\phi,z) = 0,
    R_2(b,\phi,z) = 0
    [/tex]

    [tex]
    R_3(a,\phi,z) = 0,
    R_3(b,\phi,z) = 0
    [/tex]


    The general solutions of Bessels equation are

    [tex]
    R = C_1 J_m(nr) + C_2 N_m(nr)
    [/tex]

    where J_m is the mth bessel function of the first kind and N_m is the mth neumann function (or bessel function of the second kind)

    I dont know how to continue with the boundary conditions.
    Any ideas?
     
  2. jcsd
  3. May 15, 2005 #2

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    There are some theorems which under certain conditions allow you expand any function (namely the functions which appear as boundary conditions) in terms of Bessel functions.Note that these functions are not orthonormal polynomials (so no Hilbert space here),but that still doesn't prevent this from happening.

    So i suggest you read more on the Bessel functions.Gray & Matthews wrote a monography.And there are tons of other useful books.

    Daniel.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Bessel equation
  1. Bessel Functions (Replies: 2)

Loading...