Example 11.16.  Find the temperature at each point in the upper half-disk  ,  if the temperature at points on the boundary satisfies

                

            Figure 11.19  The temperature in a half-disk.

Explore Solution 11.16.

Construct the solution via a known conformal mapping. This is similar to Example 11.9.

We can check out some of the boundary values by taking limits.

Use Mathematica to make a contour plot of the temperature.

Hence the temperature function    is harmonic in the upper half-disk  ,  and has the desired boundary values.  

(c) 2006 John H. Mathews, Russell W. Howell