In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to the system of an initial value problem. Roughly speaking, we 'shoot' out trajectories in different directions until we find a trajectory that has the desired boundary value. 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 y(t; a) denote the solution of the initial value problem
Define the function F(a) as the difference between y(t1; a) and the specified boundary value y1.
If F has a root a then the solution y(t; a) of the corresponding initial value problem is also a solution of the boundary value problem. Conversely, if the boundary value problem has a solution y(t), then y(t) is also the unique solution y(t; a) of the initial value problem where a = y'(t0), thus a is a root of F.
The term 'shooting method' has its origin in artillery. When firing a cannon towards a target, the first shot is fired in the general direction of the target. If the cannon ball hits too far to the right, the cannon is pointed a little to the left for the second shot, and vice versa. This way, the cannon balls will hit ever closer to the target.
The boundary value problem is linear if f 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.
was solved for s = -1, -2, -3, ..., -100, and F(s) = w(1;s) - 1 plotted in the first figure. Inspecting the plot of F, we see that there are roots near -8 and -36. Some trajectories of w(t;s) are shown in the second figure.
Stoer and Burlisch state that there are two solutions, which can be found by algebraic methods. These correspond to the initial conditions w?(0) = -8 and w?(0) = -35.9 (approximately).