Exercise 15.  Find the temperature function in the portion of the upper half-plane     

that lies inside the ellipse    and satisfies the following boundary conditions (shown in Figure 11.35).

                           

Hint.  Use   .  

Solution 15.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   Map the given region onto the rectangle    with the function   .  

                      The mapping   .  

        The new boundary value problem in the w-plane is

                      

This is similar to Example 11.1  in Section 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  v  alone; that is,

                    ,    for    and for all  u.

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.    

Thus,   

                    .  

Now use the substitutions   ,   ,  and      and get

                    .  

In equations (10-26) and (10-27) in Section 10.4 we found the real and imaginary parts of  ,  respectively. Thus

                    .  
Therefore, the temperature function is

                    .

We are really done.   

Aside.   We can graph the function   .   

                    A contour graph of the function   ,

                    where     for   .

                    A graph of the function   ,    

                           

                    A graph of the function   ,    

                           

                    A graph of the function   ,    

                    In Cartesian coordinates   ,    

                           

This solution is complements of the authors.

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