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Thread: 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:
    $\displaystyle u_t = u_{rr}+{2 \over r}u_r \ for \ (r,t) \in (0,1) x (0,\infty)$
    with boundary conditions $\displaystyle \lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \ t >0$

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

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

    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 $\displaystyle \mu_n$??
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  2. #2
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    If $\displaystyle R(1)=0$ and $\displaystyle X(r) = rR(r)$ then $\displaystyle X(1) = 0$. Also, if R stays finite as $\displaystyle r\to0$ then $\displaystyle rR(r)\to0$ as $\displaystyle r\to0$, and so $\displaystyle X(0) = 0$. Those are the boundary conditions for X.
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    Quote Originally Posted by Opalg View Post
    If $\displaystyle R(1)=0$ and $\displaystyle X(r) = rR(r)$ then $\displaystyle X(1) = 0$. Also, if R stays finite as $\displaystyle r\to0$ then $\displaystyle rR(r)\to0$ as $\displaystyle r\to0$, and so $\displaystyle X(0) = 0$. Those are the boundary conditions for X.
    So considering $\displaystyle \mu<0,\mu>0 \ and \ \mu=0 \$
    I find the eigenfunction to be $\displaystyle R_n = \sin{n\pi r}$ is that correct?
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  4. #4
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    Quote Originally Posted by bigdoggy View Post
    So considering $\displaystyle \mu<0,\mu>0 \ and \ \mu=0 \$
    I find the eigenfunction to be $\displaystyle R_n = \sin{n\pi r}$ is that correct?
    Yes.
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  5. #5
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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
    $\displaystyle \lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \ t >0$ confused me(hence my post)... $\displaystyle R_n = \sin{n\pi r}$ is the answer I wrote...
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