Chapter 9  Boundary-Value Problems for Ordinary Differential Equations


Boundary-Value Problems for ODE



Shooting method:

   The shooting technique for the nonlinear second-order boundary-value problem

,    ,   ,   ,      (1)

is using the solutions to a sequence of initial-value problems of the form

,    ,   ,   ,       (2)

involving a parameter , to approximate the solution to our boundary-value problem. We do this by choosing the parameters  in a manner to ensure that


where  denotes the solution to the initial-value problem (2) with  and  denotes the solution to the boundary-value problem (1).

This technique is called a ‘‘shooting’’ method, by analogy to the procedure of firing objects at a stationary target. We start with a parameter  that determines the initial elevation at which the object is fired from the point  and along the curve described by the solution to the initial-value problem:

,    ,   ,   .

If  is not sufficiently close to , we correct tour approximation by choosing elevations , , and so on, until  is sufficiently close to ‘‘hitting’’ .


Finite difference method:

   For the general nonlinear boundary-value problem

,    ,   ,  

to be solved by finite difference method, we first divide  into  equal subintervals whose endpoints are at  for . Assuming that the exact solution has a bounded fourth derivative allows us to replace  and  in each of the equations

by the appropriate centered-difference formula given below, ,    (3)

,              (4)

to obtain, for each ,


for some  and  in the interval .

    Then the difference method results when the error terms are deleted and the boundary conditions employed:




for each .

    The  nonlinear system obtained form this method,


is then solved by Newton’s method. The Jacobian matrix involved is tridiagonal.




【1】         R. L. Burden and J. D. Faires, Numerical Analysis, PWS, Boston, 1993.