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Math Help - What are the boundary conditions?

  1. #1
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    What are the boundary conditions?

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

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

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

    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 \mu_n??
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  2. #2
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    If R(1)=0 and X(r) = rR(r) then X(1) = 0. Also, if R stays finite as r\to0 then rR(r)\to0 as r\to0, and so X(0) = 0. Those are the boundary conditions for X.
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    Quote Originally Posted by Opalg View Post
    If R(1)=0 and X(r) = rR(r) then X(1) = 0. Also, if R stays finite as r\to0 then rR(r)\to0 as r\to0, and so X(0) = 0. Those are the boundary conditions for X.
    So considering \mu<0,\mu>0 \ and \ \mu=0 \
    I find the eigenfunction to be R_n = \sin{n\pi r} is that correct?
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    Quote Originally Posted by bigdoggy View Post
    So considering \mu<0,\mu>0 \ and \ \mu=0 \
    I find the eigenfunction to be R_n = \sin{n\pi r} is that correct?
    Yes.
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    Quote Originally Posted by Opalg View Post
    Yes.
    It was a question in an exam and I had no idea if my answer was correct, the way the boundary conditions were given
    \lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \  t >0 confused me(hence my post)... R_n = \sin{n\pi r} is the answer I wrote...
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