Shooting method explained

In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to an initial value problem. It involves finding solutions to the initial value problem for different initial conditions until one finds the solution that also satisfies the boundary conditions of the boundary value problem. In layman's terms, one "shoots" out trajectories in different directions from one boundary until one finds the trajectory that "hits" the other boundary condition.

Mathematical description

Suppose one wants to solve the boundary-value problem$$y$$(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y(t_1) = y_1. Let

solve the initial-value problem$$y$$(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y'(t_0) = a. If

y(t_1;a)=y₁

, then

y(t;a)

is also a solution of the boundary-value problem.

The shooting method is the process of solving the initial value problem for many different values of

until one finds the solution that satisfies the desired boundary conditions. Typically, one does so numerically. The solution(s) correspond to root(s) of $$F(a)\; =\; y(t\_1;\; a)\; -\; y\_1.$$To systematically vary the shooting parameter and find the root, one can employ standard root-finding algorithms like the bisection method or Newton's method.

Roots of

and solutions to the boundary value problem are equivalent. If is a root of , then is a solution of the boundary value problem. Conversely, if the boundary value problem has a solution , it is also the unique solution of the initial value problem where , so is a root of .

Etymology and intuition

The term "shooting method" has its origin in artillery. An analogy for the shooting method is to

• place a cannon at the position

, then

• vary the angle

of the cannon, then

• fire the cannon until it hits the boundary value

.

Between each shot, the direction of the cannon is adjusted based on the previous shot, so every shot hits closer than the previous one. The trajectory that "hits" the desired boundary value is the solution to the boundary value problem — hence the name "shooting method".

Linear shooting method

The boundary value problem is linear if f has the form$$f(t,\; y(t),\; y\text{'}(t))\; =\; p(t)\; y\text{'}(t)\; +\; q(t)y(t)\; +\; r(t).$$In this case, the solution to the boundary value problem is usually given by:$$y(t)\; =\; y\_(t)\; +\; \backslash frac\; y\_(t)$$where

is the solution to the initial value problem:$$y\_$$(t) = p(t) y_'(t) + q(t) y_(t) + r(t),\quad y_(t_0) = y_0, \quad y_'(t_0) = 0, and

y₍₂₎(t)

is the solution to the initial value problem:$$y\_$$(t) = p(t) y_'(t) + q(t) y_(t),\quad y_(t_0) = 0, \quad y_'(t_0) = 1. See the proof for the precise condition under which this result holds.^[1]

Examples

Standard boundary value problem

Notes and References

1. Book: Mathews . John H. . Fink . Kurtis K. . Numerical methods using MATLAB . 2004 . Pearson . Upper Saddle River, N.J. . 0-13-065248-2 . 4th . https://web.archive.org/web/20061209234620/http://math.fullerton.edu/mathews/n2003/shootingmethod/ShootingProof.pdf . 9 December 2006 . 9.8 Boundary Value Problems .