Shooting Methods for ODE's

Background for Boundary Value Problems

    Another type of differential equation has the form  

(1)        
     for   
  

with the boundary conditions  

         with  
  

This is called a boundary value problem.  The conditions that guarantee that a solution to (1) exists should be checked before any numerical scheme is applied; otherwise, a list of meaningless output may be generated. The general conditions are stated in the following theorem.

Theorem  (Boundary Value Problem).  Assume that  
  is continuous on the region  
  and  that  
  and  
  are continuous on  
.  If there exists a constant  
  for which  
  satisfy

    
  
and
    
,  

then the boundary value problem

    
  with  
  

has a unique solution   
.  

    The notation  
  has been used to distinguish the third variable of the function    
.   Finally, the special case of linear differential equations is worthy of mention.

Corollary (Linear Boundary Value Problem).  Assume that  
  in the theorem has the form  
  and that  f  and its partial derivatives  
  and  
are continuous on  
.  If there exists a constant  
  for which  p(t)  and   q(t)  satisfy

    
  
and
    
,  

then the linear boundary value problem

    
  with  
  

has a unique solution   
.  

    We are all familiar with the differential equation  
and its general solution  
.  The boundary conditions with  
  can only be solved if  
.  Unfortunately, because of this counter example, the "theory" which "guarantees" a solution must be phrased with
.  A careful reading of the "theory" reveals that this is a sufficient condition and not a necessary condition.  Indeed there are many problems that can be solved with the "shooting method" , all we ask is to be cautious with its implementation and take note that it might not apply sometimes.    

Reduction to Two I.V.P.s: Linear Shooting Method

    Finding the solution of a linear boundary problem is assisted by the linear structure of the equation and the use of two special initial value problems. Suppose that  u(t)  is the unique solution to the I.V.P.

    
  with  
.  

Furthermore, suppose that  v(t)  is the unique solution to the I.V.P.

    
  with  
.  

Then the linear combination

     
.   

is a solution to  
  with  
.  

Program (Linear Shooting Method).  To approximate the solution of the boundary value problem  

    
  with  
  

over the interval  [a,b]  by using the Runge-Kutta method of order n=4.

The method involves solving a two systems of equations over  
.  First solve     

    
                        with    
,  
    
    and    
.  
    
Then solve  

    
                        with    
,   
    
    and    
.  
    
Finally, the desired solution x(t) is the linear combination  

     
.   

The subroutine Runge2D will be used to construct the two solutions  
,  and  
.     

Theory of computation.  

    What should the "theory" really say?  "Existence theory" needs for numerical analysis needs to be "computational theory."  We really need to be guaranteed that two "linearly independent" solutions u(t) and v(t) given above can be computed.  In practice, if  
then we need to compute
.