Shooting method

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. The following exposition may be clarified by this illustration of the shooting method.

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 $y(t;a)$ 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 $y(t_{1};a)=y_{1}$, then $y(t;a)$ is also a solution of the boundary-value problem.

The shooting method is the process of solving the inital value problem for many different values of $a$ until one finds the solution $y(t;a)$ that satisfies the desired boundary condtions. 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 $a$ and find the root, one can employ standard root-finding algorithms like the bisection method or Newton's method.

Roots of $F$ and solutions to the boundary value problem are equivalent. If $a$ is a root of $F$, then $y(t;a)$ is a solution of the boundary value problem. Conversely, if the boundary value problem has a solution $y(t)$, it is also the unique solution $y(t;a)$ of the initial value problem where $a=y'(t_{0})$, so $a$ is a root of $F$.

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 $y(t_{0})=y_{0}$, then
• vary the direction $y'(t_{0})=a$ of the cannon, then
• fire the cannon until it hits the boundary value $y(t_{1})=y_{1}$.

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 — hence the name "shooting method".

Linear shooting method

The boundary value problem is linear if f has the form

$f(t,y(t),y'(t))=p(t)y'(t)+q(t)y(t)+r(t).\,$

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

$y(t)=y_{(1)}(t)+{\frac {y_{1}-y_{(1)}(t_{1})}{y_{(2)}(t_{1})}}y_{(2)}(t)$

where $y_{(1)}(t)$ is the solution to the initial value problem:

$y_{(1)}''(t)=p(t)y_{(1)}'(t)+q(t)y_{(1)}(t)+r(t),\quad y_{(1)}(t_{0})=y_{0},\quad y_{(1)}'(t_{0})=0,$

and $y_{(2)}(t)$ is the solution to the initial value problem:

$y_{(2)}''(t)=p(t)y_{(2)}'(t)+q(t)y_{(2)}(t),\quad y_{(2)}(t_{0})=0,\quad y_{(2)}'(t_{0})=1.$

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

Example

A boundary value problem is given as follows by Stoer and Bulirsch (Section 7.3.1).

$w''(t)={\frac {3}{2}}w^{2},\quad w(0)=4,\quad w(1)=1$

The initial value problem

$w''(t)={\frac {3}{2}}w^{2},\quad w(0)=4,\quad w'(0)=s$

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 Bulirsch 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).

The function F(s) = w(1;s) − 1.

Trajectories w(t;s) for s = w'(0) equal to −7, −8, −10, −36, and −40. The point (1,1) is marked with a circle.

See also

• Direct multiple shooting method
• Computation of radiowave attenuation in the atmosphere

References

• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 18.1. The Shooting Method". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.

External links

• Brief Description of ODEPACK (at Netlib; contains LSODE)
• Shooting method of solving boundary value problems – Notes, PPT, Maple, Mathcad, Matlab, Mathematica at Holistic Numerical Methods Institute [1]