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:

,

and

,      

for each .

    The  nonlinear system obtained form this method,

 

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

 

 

References:

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