Consider the heat equation in a radially symmetric sphere of radius

[mcfc]

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 $$\mu_n$$??

 

[tiny-tim]

Hi mcfc!

(have a mu: µ and an infinity: ∞ )

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) < ∞