Background for Elliptic Equations

    As examples of elliptic partial differential equations, we consider the Laplace equation, Poisson equation, and Helmholtz equation. Recall that the Laplacian of the function  u(x,y)  is  

            .  

With this notation, we can write the Laplace, Poisson, and Helmholtz equations in the following forms:  

It is often the case that the boundary values for the function  u(x,y)  are known at all points on the sides of a rectangular region  R  in the plane. In this case, each of these equations can be solved by the numerical technique known as the finite-difference method.

Proof  Elliptic PDE's  Elliptic PDE's  

Computer Programs  Elliptic PDE's  Elliptic PDE's  

Program (Dirichlet Method for Laplace's equation)  To approximate the solution of the Laplace's equation    over the rectangle    with  ,  ,  for  ,  and  ,  for  .  It is assumed that    and integers  n  and  m  exist so that  .  

Example 1.  Solve Laplace's equation over a  9 by 9  grid with boundary conditions
    Top:       180
    Left:         80
    Bottom:   20
    Right:        0
Solution 1.

Example 2.  Solve Laplace's equation over a  21 by 21  grid with boundary conditions
    Top:       180
    Left:         80
    Bottom:   20
    Right:        0
Solution 2.

Program (Neumann Method for Laplace's equation)  To approximate the solution of the Laplace's equation    over the rectangle    with  ,  ,  for  ,  and  ,  for  .  It is assumed that    and integers  n  and  m  exist so that  .  

Example 3.  Solve Laplace's equation with the Neuman boundary condition over a  9 by 9  grid with boundary conditions
    Top:            180
    Left:              80
    Bottom:     = 0
    Right:             0
Solution 3.

Improvements in the graphics.  Making better graphics which would include realistic labels is more complicated.  If you don't mind "editing pictures" then we can "patch up" the graph in Example 1 by using graphics commands which "label what ever we want."  

Example 4.  Solve Laplace's equation over the square where the boundary conditions are  
    Top:       180  
    Left:         80  
    Bottom:   20  
    Right :        0  
Solution 4.

Example 5.  Solve Laplace's equation over the square where the boundary conditions are  
    Top:       180
    Left:         80
    Bottom:   20
    Right:        0
Solution 5.

Example 6.  Solve Laplace's equation over the square where the boundary conditions are  

     ,  ,  for  ,  
and  
     ,  for  .
Solution 6.

Research Experience for Undergraduates

Elliptic PDE's  Elliptic PDE's  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Elliptic P.D.E.'s

(c) John H. Mathews 2004