Exercise 11.  Find the function    that is harmonic in the quarter-disk    and has the boundary values  

Solution 11.

See text and/or instructor's solution manual.

Answer.   .  

Alternative Answer.   .  

Remark.   Notice that the level curves of     are identical to the isothermals      that will be constructed in Exercise 3 in Section 11.5,  

where we will investigate the function   .  

Also notice that the level curves of     are identical to the streamlines      that will be constructed in Exercise 1 in Section 11.11,  

where we will investigate the function   .  

Solution.   Use the result of Example 11.9 and observe that      has the boundary values

                      

Hence the function    is  

                        

Thus,   

                    .  

Now use the intermediate mapping      and get  

                       

Therefore,   

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

A More Detailed Solution.   The function      maps the quarter disk      

onto the upper half-disk   .   Furthermore,

The quarter-circle      is mapped onto the  semicircle ,

and the segment   .   is mapped onto the segment     .   

and the segment   .   is mapped onto the segment     .   

        This makes a new boundary value problem in the upper half-disk  H.  

Find the function    that is harmonic in the upper half-disk    that has the boundary values  

Use the result of Example 11.9 and observe that       

will be zero on the semicircle   

and    on the diameter  .   

Hence the solution is  .  

        Or, if more details for the construction of    then use the following construction.  

Exercise 4 in Section 10.2 showed that the transformation  

maps the upper half-disk onto the first quadrant .   Furthermore,

the upper semi-circle      is mapped onto the positive u-axis, i.e.  ,

and the diameter (or segment)    is mapped onto the positive v-axis, i.e.  .  

This makes a new boundary value problem in the first quadrant of the w-plane   

                    ,   for  ,   and      
            
                    ,   for  .        

Applying Example 11.2 in Section 11.1 we know that the form of the solution is  

                    .  

Use the values   ,  and     and write the system of equations  

                    

                    .  

Simplify them and obtain  

                     ,    and    .  

Solving we get      and the desired solution.

Thus,    .   

Since   ,   the solution    in the z-plane in the half-disk  H will be

                    .  

        Now use the intermediate mapping      and get  

                       

Therefore,   

                    .  

We are really really done.   

Aside.  We can let Mathematica double check our work.

We are really really really done.   

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function   

                     where      for   .  

We are really really really really done.   

                     A contour graph of the function   

                     where      for   .  

                    A graph of the function   

                       

                    A graph of the function   

                       

We are really really really really really done.   

Aside.  The Intermediate Solution

For illustration purposes we can graph the intermediate solution in the Z-plane   

                    .  

                    A graph of the intermediate solution in the Z-plane   

                    .  

                    A graph of the intermediate solution in the Z-plane   

                    .  

                    A graph of the intermediate solution in the Z-plane   

                    .  

                    A graph of the intermediate solution in the Z-plane   

                    .  

We are really really really really really really done.   

Alternative Sollution.  An alternative solution can be constructed

                    

                     A contour graph of the alternative solution   

                     where      for   .  

This solution is complements of the authors.

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