I have the diffusion equation $u_t -D\nabla^2u + \lambda u = 0$ with $u = u(r,t)$ in spherical coordinates with the boundary conditions (BC) $u(r_1,t) = u_0~~u(r_2,t)=0$ and initial conditions (IC) $u(r,0)=0$ on the sphere described by $r_1 < r < r_2$.
My approach is to let $u(r,t) = \sum u_n(t)f_n(r)$ where $f_n$ is the orthogonal basis vectors for the Laplace operator with eigenvalues $\mu_n$ and $u_n(t)$ are the expansion coefficients.
Using the PDE yields $u_n(t) = Ae^{-(D\mu_n + \lambda)t}$, and we know that the radial part $f_n = j_n(\sqrt{\mu_n} r) + y_n(\sqrt{\mu_n} r)$ are the spherical Bessel functions.
The problem now is to fit the BC. We know that $f_n(r_2) = 0$, but how should I proceed?