Revisited Example 11.11.  Find the temperature [Graphics:Images/TemperaturesMod_gr_205.gif] in the upper half-plane [Graphics:Images/TemperaturesMod_gr_206.gif], where the temperature at points on the boundary satisfies  

            [Graphics:Images/TemperaturesMod_gr_207.gif]  

[Graphics:Images/TemperaturesMod_gr_208.gif]

Explore Revisited Solution 11.11.

The solution is similar to Example 11.7, but the method of solution is different.

Enter the function U[t] and use the Poisson integral to construct  [Graphics:../Images/TemperaturesMod_gr_212.gif].  

[Graphics:../Images/TemperaturesMod_gr_213.gif]



[Graphics:../Images/TemperaturesMod_gr_214.gif]

 

 

 

Use Mathematica to make a contour plot of the solution.

[Graphics:../Images/TemperaturesMod_gr_215.gif]





[Graphics:../Images/TemperaturesMod_gr_216.gif]

[Graphics:../Images/TemperaturesMod_gr_217.gif]

 

 

 

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

[Graphics:../Images/TemperaturesMod_gr_218.gif]





[Graphics:../Images/TemperaturesMod_gr_219.gif]

[Graphics:../Images/TemperaturesMod_gr_220.gif]

Therefore, the function  [Graphics:../Images/TemperaturesMod_gr_221.gif]  is harmonic in the upper half-plane  [Graphics:../Images/TemperaturesMod_gr_222.gif],  and takes on the desired boundary values.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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