- #1
mcfc
- 16
- 0
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??
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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