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. 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 usual methods for finding roots may be employed here, such as the bisection method or Newton's method.
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:
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