What are the boundary conditions?

Printable View

August 27th 2009, 09:47 AM
bigdoggy

What are the boundary conditions?

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 $\mu_n$ ??
August 27th 2009, 12:41 PM
Opalg

If $R(1)=0$ and $X(r) = rR(r)$ then $X(1) = 0$ . Also, if R stays finite as $r\to0$ then $rR(r)\to0$ as $r\to0$ , and so $X(0) = 0$ . Those are the boundary conditions for X.
August 27th 2009, 01:03 PM
bigdoggy

Quote:

Originally Posted by Opalg

If $R(1)=0$ and $X(r) = rR(r)$ then $X(1) = 0$ . Also, if R stays finite as $r\to0$ then $rR(r)\to0$ as $r\to0$ , and so $X(0) = 0$ . Those are the boundary conditions for X.

So considering $\mu<0,\mu>0 \ and \ \mu=0 \$
I find the eigenfunction to be $R_n = \sin{n\pi r}$ is that correct?
August 27th 2009, 01:10 PM
Opalg

Quote:

Originally Posted by bigdoggy

So considering $\mu<0,\mu>0 \ and \ \mu=0 \$
I find the eigenfunction to be $R_n = \sin{n\pi r}$ is that correct?

Yes. (Clapping)
August 27th 2009, 01:18 PM
bigdoggy

Quote:

Originally Posted by Opalg

Yes. (Clapping)

It was a question in an exam and I had no idea if my answer was correct, the way the boundary conditions were given
$\lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \ t >0$ confused me(hence my post)... $R_n = \sin{n\pi r}$ is the answer I wrote...