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Consider the heat equation in a radially symmetric sphere of radius

  1. Aug 27, 2009 #1
    Consider the heat equation in a radially symmetric sphere of radius unity:
    [tex]u_t = u_{rr}+{2 \over r}u_r \ for \ (r,t) \in (0,1) x (0,\infty)[/tex]
    with boundary conditions [tex]\lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \ t >0[/tex]

    Now, using separation of variables [tex]u=R(r)T(t)[/tex] leads to the eigenvalue problem [tex]rR''+2R'-\mu rR=0 \ \ with \ lim_{r \rightarrow 0}R(r) < \infty \ and \ R(1)=0[/tex]

    Then using the change of variable [tex]X(r)=rR(r) [/tex]this becomes [tex]X''-\mu X = 0[/tex]

    Now to find all the eigenvalues I need to know what the boundary conditions are, and from the info above, how do I determine the sets of boundary conditions to use to find the eigenvalues [tex]\mu_n[/tex]??
     
  2. jcsd
  3. Aug 27, 2009 #2

    tiny-tim

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    Hi mcfc! :smile:

    (have a mu: µ and an infinity: ∞ :wink:)

    u(1,t)=0 for t >0, and u = R(r)T(t), so what does that tell you about R or T?

    Also, R(r) = X/r, so put that into your solution for X, and then use u(r,t) < ∞ :wink:
     
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