Exercise 2.  Find the temperature function in the infinite strip bounded by the lines   ,  

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

                      

Solution 2.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   The mapping    will map the given domain

onto the infinite strip .   The new boundary value problem for   is

                    

The method in Example 11.14 (compare with Exercise in Section 11.2) can be used to construct the solution in the w-plane

                        where    ,  ,  ,   .  

Making these substitutions and get   ,   which can be simplified as   .  

Now use the substitution     and     to obtain the solution in the z-plane  

                       

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 contour graph of the function   

                     where      for   .  

                    A graph of the function   .  

                      

                    A graph of the function   .  

                      

This solution is complements of the authors.

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