Exercise 7.  Find the temperature function in the domain  ,  

that satisfies the following boundary conditions (shown in Figure 11.27).

                    

Solution 7.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   The transformation      maps the domain   

onto the rectangle   .  

                      The mapping   .   

The new boundary value problem in the rectangle is

                      

The solution      has the form   ,   where  

                      

Solve and get      and   .

Then use      and obtain

                        

Therefore,

                    .   

We are 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