This problem is the same as transient heat conduction with heat generation inside an insulated sphere. If we multiply both side of the differential equation by r2 and integrate between r = 0 and r = R, we obtain:
\frac{d(\int_0^R{r^2F(r,t)dr})}{dt}=\int_0^R{r^2g(r,t)dr}
This equation can be integrated between t = 0 and arbitrary t to yield:
\int_0^R{r^2F(r,t)dr}=\int_0^t{\int_0^R{r^2g(r,ξ)dr}}dξ
This suggests that it might be advantageous to express F(r,t) as:
F(r,t)=H(r,t)+\frac{[3\int_0^t{\int_0^R{r^2g(r,ξ)dr}}dξ]}{R^3}
The second term on the right here represents the temperature F averaged over the volume of the sphere at any time (assuming F = 0 at time zero). If we substitute this relationship into the original differential equation, we obtain:
\frac{\partial H}{\partial t}=\alpha \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial H}{\partial r}\right)+\left[g(r,t)-\frac{3\int_0^R{r^2g(r,t)dr}}{R^3}\right]
The term in brackets is the local rate of heat generation within the sphere minus the average rate of heat generation (averaged over the volume of the sphere). In addition, the function H satisfies the equation:
\int_0^R{r^2H(r,t)dr}=0
This equation means that the average value of H (averaged over the volume of the sphere) does not change with time.
If the heat generation rate g were constant, the term in brackets would vanish, which would make this problem very easy to solve. If the heat generation g were a function of r but not t, the problem would also be very easy to solve (since there would be a long time steady state solution). The time-dependence of the term in brackets makes the problem more difficult to solve (even though the average value of the term in brackets is equal to zero over the volume of the sphere).
Anyone else out there working on this problem. Thoughts?
I have another idea on how to approach this problem, but I'm not going to present it unless someone has more interest.
Chet
