Exercise 5.  Find the temperature function in the semi-infinite strip  ,  

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

                      

Solution 5.

See text and/or instructor's solution manual.

Answer.   .

Solution.   The transformation    maps the semi-infinite strip  

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 construct the solution.

Therefore,   

                    .    

Use Identity  (5-34)     and get  

                         

Therefore,   

                    .   

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function      

                     where      for   .  

                     A graph of the function   ,  

                      

We are 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