Radial diffusion in the sphere

In the diffusion equation (8.6.3) with radial flux and constant diffusion coefficient, let us introduce the new variable u r,t) = u=Cr 

Let us calculate the concentrations in a sphere with a uniform initial distribution C0 and zero surface concentration. Initial and boundary conditions are u(r,0)=C0r, u(a, t) = 0, u(0, f) = 0. The diffusion equation in one dimension (8.6.4) admits 

In order to make the solution consistent with initial and boundary conditions, we will use for u(r,0) the ramp function defined in Chapter 2. For 0 r a, u(r,0) = C0r, while u(r, a)=0. Using the results of Section 2.6, the Fourier expansion of this function 

The amount of diffusing substance M(t) still present in the sphere at t is obtained upon integration of the concentration times the volume (47tr2 dr) of the infinitesimal shell of thickness dr 

Let us call Jn the integral on the right-hand side of the last equation. Jn can be integrated by parts as 

---