# Ngô Quốc Anh

## July 24, 2009

### Linear shooting method

In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to the solution of an initial value problem. The following exposition may be clarified by this illustration of the shooting method.

For a boundary value problem of a second-order ordinary differential equation, the method is stated as follows. Let

be the boundary value problem. Let denote the solution of the initial value problem

Define the function as the difference between and the specified boundary value .

If the boundary value problem has a solution, then has a root, and that root is just the value of which yields a solution of the boundary value problem. The usual methods for finding roots may be employed here, such as the bisection method or Newton’s method.

Linear shooting method

The boundary value problem is linear if has the form

In this case, the solution to the boundary value problem is usually given by

where is the solution to the initial value problem

and is the solution to the initial value problem

See the proof for the precise condition under which this result holds.

Remark

One can easily see that if then for all . Thus for all . Besides, the formula

comes from the fact that we need to find as a combination of and . In this manner, should be of the form

for all .

At ,

which implies that . This is self-satisfied.
At ,

which implies that

.

Therefore,

Example

A boundary value problem is given as follows by Stoer and Bulirsch.

.

The initial value problem

was solved for , and plotted in the first figure. Inspecting the plot of , we see that there are roots near and . Some trajectories of are shown in the second figure.

Solutions of the initial value problem were computed by using the LSODE algorithm, as implemented in the mathematics package GNU Octave. Stoer and Bulirsch state that there are two solutions, which can be found by algebraic methods. These correspond to the initial conditions and (approximately).

The function .

Trajectories for equal to , , , , and (red, green, blue, cyan, and magenta, respectively). The point is marked with a red diamond.