Exercise 4.  Find the temperature function inside the unit disk  ,  

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

                      

Hint.  Use  .  

Solution 4.

See text and/or instructor's solution manual.

Answer.   .  

Alternative Answer.   .  

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  

                       

Thus,   

                    .   

Since      the temperature function inside the unit disk will be

                    .  

Use      and

and      and      and      and get

                      

Therefore,  

                    .  

An alternative solution is

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  For illustration purposes we can graph the solution   .   

                     A contour graph of the function     

                     where      for   .  

                    A graph of the function   ,  

                      

                    A graph of the function   ,  

                      

We are really really done.   

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

                     A contour graph of the alternative solution   

                     where      for   .  

                    A graph of the alternative solution   .  

                      

                    A graph of the alternative solution   .  

                      

We are really really really done.   

Aside.  We can also graph the intermediate function   .

                    A contour graph of the intermediate function   .

                    where      for   .  

                    A graph of the intermediate function   .

                       

This solution is complements of the authors.

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