Example 11.15.  Find the temperature [Graphics:Images/TemperaturesMod_gr_63.gif] at each point in the upper half plane  [Graphics:Images/TemperaturesMod_gr_64.gif]  ,  if the temperature at points on the x-axis on the boundary satisfy

            [Graphics:Images/TemperaturesMod_gr_65.gif]  

Figure 11.18  The temperature [Graphics:Images/TemperaturesMod_gr_74.gif] in the upper half-plane where [Graphics:Images/TemperaturesMod_gr_75.gif].

Explore Solution 11.15.

Verify that  [Graphics:../Images/TemperaturesMod_gr_76.gif]  is the harmonic conjugate of  [Graphics:../Images/TemperaturesMod_gr_77.gif]  .

[Graphics:../Images/TemperaturesMod_gr_78.gif]

 

 

 

[Graphics:../Images/TemperaturesMod_gr_79.gif]

 

 

Enter the formula for the temperature function.  This is similar to Example 11.2.

[Graphics:../Images/TemperaturesMod_gr_80.gif]


[Graphics:../Images/TemperaturesMod_gr_81.gif]

 

 

For illustration we choose  [Graphics:../Images/TemperaturesMod_gr_82.gif].  
Then we can check out the temperature at points whose argument is  [Graphics:../Images/TemperaturesMod_gr_83.gif].  

[Graphics:../Images/TemperaturesMod_gr_84.gif]



[Graphics:../Images/TemperaturesMod_gr_85.gif]

 

 

Use Mathematica to make a contour plot of the temperature.

[Graphics:../Images/TemperaturesMod_gr_86.gif]





[Graphics:../Images/TemperaturesMod_gr_87.gif]

[Graphics:../Images/TemperaturesMod_gr_88.gif]

 

 

 

Hence the temperature function  [Graphics:../Images/TemperaturesMod_gr_89.gif]  is harmonic in the upper half plane  [Graphics:../Images/TemperaturesMod_gr_90.gif]  and has the desired boundary values.  

Then use Mathematica to make a 3D plot of the solution.

[Graphics:../Images/TemperaturesMod_gr_91.gif]





[Graphics:../Images/TemperaturesMod_gr_92.gif]

[Graphics:../Images/TemperaturesMod_gr_93.gif]

Hence the temperature function  [Graphics:../Images/TemperaturesMod_gr_94.gif]  is harmonic in the upper half plane  [Graphics:../Images/TemperaturesMod_gr_95.gif]  and has the desired boundary values.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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