Example 11.17.  Find the temperature [Graphics:Images/TemperaturesMod_gr_152.gif] for the domain D consisting of the the upper half-plane  [Graphics:Images/TemperaturesMod_gr_153.gif]   where the temperature at points on the boundary satisfies  

            [Graphics:Images/TemperaturesMod_gr_154.gif]  

Figure 11.21  The temperature [Graphics:Images/TemperaturesMod_gr_165.gif] with [Graphics:Images/TemperaturesMod_gr_166.gif],

            and boundary values [Graphics:Images/TemperaturesMod_gr_167.gif], and [Graphics:Images/TemperaturesMod_gr_168.gif].

Explore Solution 11.17.

This is similar to Example 11.13, but the method of solution is different.

Construct the solution via a known conformal mapping.

[Graphics:../Images/TemperaturesMod_gr_169.gif]



[Graphics:../Images/TemperaturesMod_gr_170.gif]

 

 

 

For computational purposes we use the formula  [Graphics:../Images/TemperaturesMod_gr_171.gif]  
Then we can check out the temperature and the normal derivative at several boundary points.

[Graphics:../Images/TemperaturesMod_gr_172.gif]



[Graphics:../Images/TemperaturesMod_gr_173.gif]

[Graphics:../Images/TemperaturesMod_gr_174.gif]

 

 

Use Mathematica to make a contour plot of the temperature.

[Graphics:../Images/TemperaturesMod_gr_175.gif]





[Graphics:../Images/TemperaturesMod_gr_176.gif]

[Graphics:../Images/TemperaturesMod_gr_177.gif]

 

 

 

Hence the temperature function  [Graphics:../Images/TemperaturesMod_gr_178.gif]  is harmonic in the upper half plane  [Graphics:../Images/TemperaturesMod_gr_179.gif],  and has the desired boundary values.  

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

[Graphics:../Images/TemperaturesMod_gr_180.gif]





[Graphics:../Images/TemperaturesMod_gr_181.gif]

[Graphics:../Images/TemperaturesMod_gr_182.gif]

Hence the temperature function  [Graphics:../Images/TemperaturesMod_gr_183.gif]  is harmonic in the upper half plane  [Graphics:../Images/TemperaturesMod_gr_184.gif],  and has the desired boundary values.  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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