# Thread: What are the boundary conditions?

1. ## 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 $\displaystyle \mu_n$??

2. 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.

3. Originally Posted by Opalg
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?

4. Originally Posted by bigdoggy
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.

5. Originally Posted by Opalg
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...