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   .  

Proof  Shooting Methods for O.D.E.'s  Shooting Methods for O.D.E.'s  

Footnote. The significance of the theory.  

    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  .  

Proof  Shooting Methods for O.D.E.'s  Shooting Methods for O.D.E.'s  

Computer Programs  Shooting Methods for O.D.E.'s  Shooting Methods for O.D.E.'s  

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 .  

Example 1.  Consider the D. E.    over    with    and  .
Is this a "linear differential equation" ?  Why ?
Identify the functions p(t), q(t) and r(t).  
Solution 1.

Example 2.  Consider the D. E.    over    with    and  .
Use the "linear shooting" method and solve for the first function u(t).  
Solution 2.

Example 3.  Use the "linear shooting" method and solve for the second function v(t).  
Solution 3.

Example 4.  Neither of the solutions in 2 and 3 solve the given boundary values.  Graph them to verify this.
However, you should be able to "see the initial conditions"  u(1) = 1, u'(1) = 0  and  v(1) = 0, v'(1) = 1. for u and v, respectively.
Solution 4.

Example 5.  Form the desired solution to Example 1.
Solution 5.

Example 6.  Find the analytic solution to    over    with    and  .
Solution 6.

Research Experience for Undergraduates

Shooting Methods for ODE's  Shooting Methods for ODE's  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Shooting Method for Boundary Value Problems

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(c) John H. Mathews 2004