Example 12.3.  Find the function    that is harmonic in the unit disk    

and takes on the boundary values   .  

Explore Solution 12.3.

Solution using Fourier series.

        Using the result of Example 12.1 in Section 12.1, we have the Fourier series for  ,   

                    .

Using Equation (12.11.1) for the extended Fourier series solution of the Dirichlet problem, we obtain

                    .  

An approximation using a partial sum.

        Summing up the first seven terms we get the approximations      and   .  

                    Figure 12.13.A. The functions    and   .  

                    Another 3-D graph and contour graph of the harmonic function  .  

The "true solution" using the infinite sum.

        Summing up all of the terms we get the boundary value function      on the unit circle  ,  
        
and the harmonic function      in the unit disk   .    

Aside.  The Maple commands are similar  

  

                                                               


  

                                                               

We can use Mathematica to plot the boundary function    and harmonic function   .  

                    Figure 12.13.B. The boundary function   ,  

                                      and the harmonic function   .  

                    Another 3-D graph and contour graph of the harmonic function  .  

Remark 1.  As we saw in Example 12.1 in Section 12.1, when we sum the infinite series we get solutions involving the logarithm function.  

Remark 2.  As we saw in Example 12.1 in Section 12.1, the other form of the solution will be similar.

This solution is complements of the authors.

(c) 2010 John H. Mathews, Russell W. Howell