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Bessel's Differential equation

  1. Apr 17, 2010 #1
    1. The problem statement, all variables and given/known data

    A solid ball at 30 degrees C with radius a=1 is placed in a refrigerator that maintains a constant temperature of 0 degrees C. Take c (speed)=1 and determine the temperature u(r,theta,phi,t) inside the ball

    2. Relevant equations

    partial differential heat equation in spherical coordinates

    3. The attempt at a solution

    In this case, I started out with setting all the partials of the function u (assuming u is the solution) with respect to theta and phi as 0 since the temperature is uniform and doesn't depend on either. I then set u(r,t)=R(r)T(t), and then plugged back into the partial equation in order to separate variables, and I got T(t)=exp(mu*t). However, R turns into a bessel equation that I can't figure out how to solve, with the coefficient of R double prime equal to r^2, the coefficient of R prime being 2r, and the coefficient of R being mu. Mu is the separation constant. The book is generally unhelpful. Can anyone tell me how to solve that equation?
     
  2. jcsd
  3. Apr 17, 2010 #2

    LCKurtz

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    The Bessel's equation is solved with series methods. You can read about how to do it here:

    http://www.ucl.ac.uk/~ucahhwi/MATH7402/handout9.pdf [Broken]
     
    Last edited by a moderator: May 4, 2017
  4. Apr 18, 2010 #3
    I guess I should've been more descriptive. I have all that information already (even though this is for spherical coordinates and not polar). The equation I got for the last term multiplied by R is mu*r^2. This is why I'm having trouble, since it doesn't really agree completely with the normal bessel equation, and I don't understand how the helmholtz equation is related to this equation, and why I should use it.
     
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