Mathematical Methods in the Physical Sciences · Problem 7.7

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · sphere-temperature-legendre-piecewise

Problem

Find the steady-state temperature distribution inside a sphere of radius 1 when the surface temperature is:

\begin{cases} \cos\theta, & 0 < \theta < \pi/2, \quad \text{that is, upper hemisphere,} \\ 0, & \pi/2 < \theta < \pi, \quad \text{that is, lower hemisphere.} \end{cases}