Exercise 8.  Find the temperature function in the domain  ,  

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

                      

Hint.  Use  .  

Solution 8.

See text and/or instructor's solution manual.

Answer.   .

Solution.   Use the result of Example 11.2 in Section 11.1.  

The temperature is    where   and  .  

Thus the solution is  .  

Therefore,   .

We are done.   

Aside.   We can explore a specific example of the general case.

For convenience we can choose  .  For this choice we will find the temperature function

in the domain  ,  

that satisfies the following boundary conditions:  

                      

                    A contour graph of the function   ,

                    where     for   .

                    A graph of the function   .  

                    In Cartesian coordinates   ,  
                    
                                          

This solution is complements of the authors.

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