Revisited Example 11.6.  Find the temperature  [Graphics:Images/TemperaturesMod_gr_185.gif]  in the upper half-plane  [Graphics:Images/TemperaturesMod_gr_186.gif]  where the temperature at points on the boundary satisfies Figure 11.5.  That is

            [Graphics:Images/TemperaturesMod_gr_187.gif]  

Figure 11.5  The  Dirichlet solution for the steady state temperatures.

Explore Revisited Solution 11.6.

Enter the boundary values and construct the  Dirichlet sum.

[Graphics:../Images/TemperaturesMod_gr_195.gif]




[Graphics:../Images/TemperaturesMod_gr_196.gif]


First use Mathematica to make a contour plot of the solution.

[Graphics:../Images/TemperaturesMod_gr_197.gif]





[Graphics:../Images/TemperaturesMod_gr_198.gif]

[Graphics:../Images/TemperaturesMod_gr_199.gif]

 

 

 


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

[Graphics:../Images/TemperaturesMod_gr_200.gif]





[Graphics:../Images/TemperaturesMod_gr_201.gif]

[Graphics:../Images/TemperaturesMod_gr_202.gif]


The function  [Graphics:../Images/TemperaturesMod_gr_203.gif]  is harmonic in the upper half-plane  [Graphics:../Images/TemperaturesMod_gr_204.gif],  and takes on the desired boundary values.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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