Exercises for Section 11.5.  Steady State Temperatures

Exercise 1 (a).  Show that  [Graphics:Images/TemperaturesModHome_gr_1.gif]  satisfies Laplace's equation  [Graphics:Images/TemperaturesModHome_gr_2.gif]  in three-dimensional Cartesian space,

and that   [Graphics:Images/TemperaturesModHome_gr_3.gif]   does not satisfy Laplace's equation   [Graphics:Images/TemperaturesModHome_gr_4.gif]   in two-dimensional Cartesian space.

Solution 1 (a).

Exercise 1 (b).  Show that  [Graphics:Images/TemperaturesModHome_gr_47.gif]  does not satisfy Laplace's equation  [Graphics:Images/TemperaturesModHome_gr_48.gif]  in three-dimensional Cartesian space,

and that   [Graphics:Images/TemperaturesModHome_gr_49.gif]   satisfies Laplace's equation   [Graphics:Images/TemperaturesModHome_gr_50.gif]   in two-dimensional Cartesian space.

Solution 1 (b).

Exercise 2.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_93.gif] in the infinite strip bounded by the lines   [Graphics:Images/TemperaturesModHome_gr_94.gif],  

that satisfies the following boundary values (shown in Figure 11.22).

                    [Graphics:Images/TemperaturesModHome_gr_95.gif]  

Solution 2.

Exercise 3.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_133.gif] in the first quadrant  [Graphics:Images/TemperaturesModHome_gr_134.gif],  

that satisfies the following boundary values (shown in Figure 11.23).

                    [Graphics:Images/TemperaturesModHome_gr_135.gif]  

Solution 3.

Exercise 4.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_181.gif] inside the unit disk  [Graphics:Images/TemperaturesModHome_gr_182.gif],  

that satisfies the following boundary values (shown in Figure 11.24).  

                    [Graphics:Images/TemperaturesModHome_gr_183.gif]  

Hint.  Use  [Graphics:Images/TemperaturesModHome_gr_184.gif].  

Solution 4.

Exercise 5.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_242.gif] in the semi-infinite strip  [Graphics:Images/TemperaturesModHome_gr_243.gif],  

that satisfies the following boundary values (shown in Figure 11.25).  

                    [Graphics:Images/TemperaturesModHome_gr_244.gif]  

Solution 5.

Exercise 6.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_280.gif] in the domain  [Graphics:Images/TemperaturesModHome_gr_281.gif],  

that satisfies the following boundary values (shown in Figure 11.26).  

                    [Graphics:Images/TemperaturesModHome_gr_282.gif]  

Hint.  Use   [Graphics:Images/TemperaturesModHome_gr_283.gif].

Solution 6.

Exercise 7.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_410.gif] in the domain  [Graphics:Images/TemperaturesModHome_gr_411.gif],  

that satisfies the following boundary conditions (shown in Figure 11.27).

                    [Graphics:Images/TemperaturesModHome_gr_412.gif]

Solution 7.

Exercise 8.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_444.gif] in the domain  [Graphics:Images/TemperaturesModHome_gr_445.gif],  

that satisfies the following boundary conditions (shown in Figure 11.28).  

                    [Graphics:Images/TemperaturesModHome_gr_446.gif]  

Hint.  Use  [Graphics:Images/TemperaturesModHome_gr_447.gif].  

Solution 8.

Exercise 9.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_468.gif] in the first quadrant  [Graphics:Images/TemperaturesModHome_gr_469.gif],   

that satisfies the following boundary conditions (shown in Figure 11.29).  

                    [Graphics:Images/TemperaturesModHome_gr_470.gif]  

Solution 9.

Exercise 10.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_506.gif] in the infinite strip  [Graphics:Images/TemperaturesModHome_gr_507.gif],  

that satisfies the following boundary conditions (shown in Figure 11.30).   

                    [Graphics:Images/TemperaturesModHome_gr_508.gif]  
        
Hint.  Use  [Graphics:Images/TemperaturesModHome_gr_509.gif].

Solution 10.

Exercise 11.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_550.gif] in the upper half-plane  [Graphics:Images/TemperaturesModHome_gr_551.gif],  

that satisfies the following boundary conditions (shown in Figure 11.31).

                    [Graphics:Images/TemperaturesModHome_gr_552.gif]  

Solution 11.

Exercise 12.  Find the temperature function [Graphics:Images/TemperaturesModHome_gr_579.gif] in the first quadrant  [Graphics:Images/TemperaturesModHome_gr_580.gif],  

that satisfies the following boundary conditions (shown in Figure 11.32).  

                    [Graphics:Images/TemperaturesModHome_gr_581.gif]  

Solution 12.

Exercise 13.  For the temperature  [Graphics:Images/TemperaturesModHome_gr_625.gif]  in the upper half-disk  [Graphics:Images/TemperaturesModHome_gr_626.gif],  

show that the isothermals   [Graphics:Images/TemperaturesModHome_gr_627.gif],   for  [Graphics:Images/TemperaturesModHome_gr_628.gif],   are portions of circles that pass through the points  [Graphics:Images/TemperaturesModHome_gr_629.gif]  and  [Graphics:Images/TemperaturesModHome_gr_630.gif],  as illustrated in Figure 11.33.  

Solution 13.

Exercise 14.  For the temperature function  [Graphics:Images/TemperaturesModHome_gr_665.gif]  in the upper half-plane  [Graphics:Images/TemperaturesModHome_gr_666.gif],  

show that the isothermals  [Graphics:Images/TemperaturesModHome_gr_667.gif]  are portions of hyperbolas that have foci at the points  [Graphics:Images/TemperaturesModHome_gr_668.gif]  and  [Graphics:Images/TemperaturesModHome_gr_669.gif],  as illustrated in Figure 11.34.

Solution 14.

Exercise 15.  Find the temperature function in the portion of the upper half-plane  [Graphics:Images/TemperaturesModHome_gr_694.gif]   

that lies inside the ellipse  [Graphics:Images/TemperaturesModHome_gr_695.gif]  and satisfies the following boundary conditions (shown in Figure 11.35).

                    [Graphics:Images/TemperaturesModHome_gr_696.gif]       

Hint.  Use   [Graphics:Images/TemperaturesModHome_gr_697.gif].  

Solution 15.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2008 John H. Mathews, Russell W. Howell