Chapter 9 Boundary-Value Problems for Ordinary Differential Equations
Boundary-Value Problems for ODE
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)
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
R. L. Burden and J. D. Faires, Numerical Analysis, PWS,