Consider a metallic spherical shell in three spatial dimensions of inner radius $r = 1$ and outer radius $r = 2$, and let $u(r, t)$ denote the temperature inside it.
The inner and outer surfaces of the shell are placed on ice so that the temperature at $r = 1$ and $r = 2$ is always equal to zero so that $u(1, t) = u(2, t) = 0$ for all $t \ge 0$. At $t = 0$ the temperature inside the shell is given by $u(r, 0) = − \sin(\pi r), 1 \le r \le 2$.
Using the boundary conditions supplied, I have deduced that the radial eigenfunctions of the separable spherically symmetric solutions are given by: $$X(r) = \frac{\sin(n \pi r)}{r}$$ where $n$ is any positive integer. However I am having trouble finding the coefficients that admit the initial condition $u(r, 0) = −\sin(\pi r)$.
I can't seem to find a sensible answer for the coefficients in the sum:
$$−\sin(\pi r) = \sum_{n=0}^{\infty}C_n\frac{\sin(n \pi r)}{r}\cdot e^{-n^{2}\pi^{2} t}$$
Can anyone help?
Thanks