Extra Example 1.  Find the temperature [Graphics:Images/TemperaturesMod_gr_223.gif] in the upper half-plane [Graphics:Images/TemperaturesMod_gr_224.gif], where the temperature at points on the boundary satisfies  

            [Graphics:Images/TemperaturesMod_gr_225.gif]  

[Graphics:Images/TemperaturesMod_gr_226.gif]

Explore Extra Solution 1.

Enter the function U[t] and use the Poisson integral to construct  [Graphics:../Images/TemperaturesMod_gr_227.gif].  

[Graphics:../Images/TemperaturesMod_gr_228.gif]



[Graphics:../Images/TemperaturesMod_gr_229.gif]

 

 

 

Use Mathematica to make a contour plot of the solution.

[Graphics:../Images/TemperaturesMod_gr_230.gif]





[Graphics:../Images/TemperaturesMod_gr_231.gif]

[Graphics:../Images/TemperaturesMod_gr_232.gif]

 

 

 

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

[Graphics:../Images/TemperaturesMod_gr_233.gif]





[Graphics:../Images/TemperaturesMod_gr_234.gif]

[Graphics:../Images/TemperaturesMod_gr_235.gif]

Therefore, the function [Graphics:../Images/TemperaturesMod_gr_236.gif] is harmonic in the upper half-plane  [Graphics:../Images/TemperaturesMod_gr_237.gif],  and takes on the desired boundary values.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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