Chapter 13.7, Problem 15P

A sphere initially at $0^{°}$ has its surface kept at ${100}^{°}$ from $t=0$ on (for example, a frozen potato in boiling water!). Find the time-dependent temperature distribution. Hint: Subtract ${100}^{°}$ from all temperatures and solve the problem; then add the ${100}^{°}$ to the answer. Can you justify this procedure? Show that the Legendre function required for this problem is $P_0$ and the $r$ solution is $(1/\sqrt{r})J_{1/2}$ or $j_0$ [see (17.4) in Chapter 12]. Since spherical Bessel functions can be expressed in terms of elementary functions, the series in this problem can be thought of as either a Bessel series or a Fourier series. Show that the results are identical.