Exercise 1.  Find the function    that is harmonic in the horizontal strip    and has the boundary values  

Solution 1.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   This is similar to Example 11.1.

Intuition suggests that we should seek a solution that takes on constant values along the horizontal lines of the form     

and that    be a function of  y  alone; that is,

            ,    for    and for all  x.

Laplace's equation,   ,   implies that   ,  

which implies   ,   where  c  and  m  are constants.   

The stated boundary conditions      and      produce the system of equations  

                    

The values     solve this system.    

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   .  

We are really really done.   

                     A contour graph of the function     

                     where      for   .  

                     A contour graph of the function     

                     where      for   .  

                    A graph of the function     

                       

This solution is complements of the authors.

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