Exercise 3.  Find the temperature function in the first quadrant  ,  

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

                    

Solution 3.

See text and/or instructor's solution manual.

Answer.   .  

Alternative Answer.   .

Remark.   Notice that the isothermals        are identical to the level curves of     that were constructed in Exercise 3 in Section 11.2

where we investigated the function   .  

Also notice that the isothermals        are identical to the streamlines      that will be constructed in Exercise 1 in Section 11.11

where we will investigate the function   .  

Solution.   The transformation    maps the unit disk onto the upper half plane     where the boundary values are

                       

Apply Theorem 11.2 in Section 11.2 to construct a Dirichlet solution in the upper half-plane

use formula (11-5)      with   ,  

                    .  

Now we substitute      and      to obtain  

                       

Therefore,   

                    .  

Now use   ,    ,  and      and construct

                        

Therefore,   

                    .   

We are done.   

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function   .

                     A graph of the function   .

                      

We are really done.   

Alternative Solution.   Start with    and write

                    .   

Next, make the substitution  .

Therefore,

                           

Aside.  For illustration purposes we can graph the alternative solution   .   

                     A contour graph of the alternative solution   .

                     A graph of the alternative solution   ,   

                      

We are really really done.   

Aside.  We can also graph the intermediate function   .

                     A contour graph of the intermediate function   .

                     A graph of the intermediate function   ,

                       

This solution is complements of the authors.

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