Radial heat equation in three dimensions ($u_t=D{r}^{-1}(ru{)}_{rr}$)

For a spherically symmetric field $u(r,t)$ in three dimensions,Thus the heat equation becomes $u_t=D{r}^{-1}(ru{)}_{rr}$. Introducing $w=ru$ reduces its spatial part to the one-dimensional second derivative $w_{rr}$, while regularity at the centre requires $w(0,t)=0$.

On a shell $0<r_{-}<r<r_{+}$ with homogeneous Dirichlet boundary conditions, put $L=r_{+}-r_{-}$ and $u=r\psi$. The radial heat equation in three dimensions becomes $u_t=\kappa u_{rr}$, with zero values at both endpoints. Expanding $u(0,r)=r\psi (0,r)$ in a Fourier sine series gives the displayed solution, with $b_n=(2/L){\int }_{r_{-}}^{r_{+}}r\psi (0,r)\sin [n\pi (r-r_{-})/L]dr$. Each radial mode decays at the square of its wave number. The positive inner radius makes division by $r$ harmless throughout the shell.