Shooting method
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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
- <math> y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y(t_1) = y_1 <math>
be the boundary value problem. Let y(t1; a) denote the solution of the initial value problem
- <math> y''(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y'(t_0) = a <math>
Define the function F(a) as the difference between y(t1; a) and the specified boundary value y1.
- <math> F(a) = y(t_1; a) - y_1 \,<math>
If the boundary value problem has a solution, then F has a root, and that root is just the value of y'(t0) which yields a solution y(t) of the boundary value problem.
The usual methods for finding roots may be employed here, such as the bisection method or Newton's method.
References
- Josef Stoer and Roland Bulirsch. Introduction to Numerical Analysis. New York: Springer-Verlag, 1980. (See Section 7.3.)