**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

References:

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