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Shooting method - Wikipedia, the free encyclopedia

Shooting method - Wikipedia, the free encyclopediaShooting methodFrom Wikipedia,
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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

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 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.
Contents [hide]
1 Linear shooting method
2 Example
3 See also
4 References
5 External links


[edit] Linear shooting methodThe 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 y(1)(t) is the solution to the initial value problem:

and y(2)(t) is the solution to the initial value problem:

See the proof for the precise condition under which this result holds.
[edit] ExampleA boundary value problem is given as follows by Stoer and Burlisch
(Section 7.3.1).

The initial value problem

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.
Solutions of the initial value problem were computed by using the LSODE
algorithm, as implemented in the mathematics package GNU Octave.
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 (red,
green, blue, cyan, and magenta, respectively). The point (1,1) is marked with a
red diamond.

[edit] See alsoDirect multiple shooting method
Computation of radiowave attenuation in the atmosphere
[edit] ReferencesJosef Stoer and Roland Bulirsch. Introduction to Numerical
Analysis. New York: Springer-Verlag, 1980. (See Section 7.3.)
[edit] External linksBrief 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]
Shooting Method for Boundary Value Problems
Boundary value problems: the shootin

g method
edited by: Yang Jiao(yj215)
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