Conduction-diffusion

The solution for the zeroth order mass fraction and temperature equations, (8.14) and (8.15) with $i=0$, is obviously

$$m_0 = T_0 = x = r\sin\theta\cos\phi = rP_1^1(\cos\theta)\cos\phi.$$ (8.27)

The functions $P_n^m(\cos\theta)$ are defined and tabulated in appendix B.