Consider the heat equation in a radially symmetric sphere of radius unity:(adsbygoogle = window.adsbygoogle || []).push({});

[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]??

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# Consider the heat equation in a radially symmetric sphere of radius

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