Example 11.16.  Find the temperature [Graphics:Images/TemperaturesMod_gr_96.gif] at each point in the upper half-disk  [Graphics:Images/TemperaturesMod_gr_97.gif],  if the temperature at points on the boundary satisfies

            [Graphics:Images/TemperaturesMod_gr_98.gif]    

Figure 11.19  The temperature [Graphics:Images/TemperaturesMod_gr_109.gif] in a half-disk.

Explore Solution 11.16.

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

[Graphics:../Images/TemperaturesMod_gr_110.gif]



[Graphics:../Images/TemperaturesMod_gr_111.gif]

 

 

 

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

[Graphics:../Images/TemperaturesMod_gr_112.gif]


[Graphics:../Images/TemperaturesMod_gr_113.gif]

 

 

Use Mathematica to make a contour plot of the temperature.

[Graphics:../Images/TemperaturesMod_gr_114.gif]





[Graphics:../Images/TemperaturesMod_gr_115.gif]

[Graphics:../Images/TemperaturesMod_gr_116.gif]


Hence the temperature function  [Graphics:../Images/TemperaturesMod_gr_117.gif]  is harmonic in the upper half-disk  [Graphics:../Images/TemperaturesMod_gr_118.gif],  and has the desired boundary values.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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