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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 [[#Example|illustration of the shooting method]].
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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''(''t''; ''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''(''t''<sub>1</sub>; ''a'') and the specified boundary value ''y''<sub>1</sub>.
 
:<math> F(a) = y(t_1; a) - y_1 \,</math>
 
If the boundary value problem has a solution, then ''F'' has a [[root of a function|root]],
and that root is just the value of ''y''<nowiki>'</nowiki>(''t''<sub>0</sub>) 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]].
 
== Linear shooting method ==
The boundary value problem is linear if ''f'' has the form
:<math> f(t, y(t), y'(t))=p(t)y'(t)+q(t)y(t)+r(t). \, </math>
In this case, the solution to the boundary value problem is usually given by:
:<math>y(t) = y_{(1)}(t)+\frac{y_1-y_{(1)}(t_1)}{y_{(2)}(t_1)}y_{(2)}(t)</math>
where <math>y_{(1)}(t)</math> is the solution to the initial value problem:
:<math>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, </math>
and <math>y_{(2)}(t)</math> is the solution to the initial value problem:
:<math>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. </math>
See [http://math.fullerton.edu/mathews/n2003/shootingmethod/ShootingProof.pdf the proof] for the precise condition under which this result holds.
 
== Example ==
 
A [[boundary value problem]] is given as follows by Stoer and Burlisch (Section 7.3.1).
 
:<math> w''(t) = \frac{3}{2} w^2, \quad w(0) = 4, \quad w(1) = 1 </math>
 
The [[initial value problem]]
 
:<math> w''(t) = \frac{3}{2} w^2, \quad w(0) = 4, \quad w'(0) = s</math>
 
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).
 
[[Image:Shooting_method_error.png|thumb|center|400px|The function ''F''(''s'') = ''w''(1;''s'') − 1.]]
[[Image:Shooting_method_trajectories.png|thumb|center|400px|Trajectories ''w''(''t'';''s'') for ''s'' = ''w''<nowiki>'</nowiki>(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.]]
<br clear="all" />
 
== See also ==
* [[Direct multiple shooting method]]
* [[Computation of radiowave attenuation in the atmosphere]]
 
== References ==
 
* Josef Stoer and Roland Bulirsch. ''Introduction to Numerical Analysis''. New York: Springer-Verlag, 1980. ''(See Section 7.3.)''
*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press |  publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 18.1. The Shooting Method | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=959}}
 
== External links ==
* [http://www.netlib.org/odepack/opks-sum Brief Description of ODEPACK] ''(at [[Netlib]]; contains LSODE)''
* [http://numericalmethods.eng.usf.edu/topics/shooting_method.html Shooting method of solving boundary value problems – Notes, PPT, Maple, Mathcad, Matlab, Mathematica] at ''Holistic Numerical Methods Institute'' [http://numericalmethods.eng.usf.edu]
* [http://math.fullerton.edu/mathews/n2003/ShootingMod.html Shooting Method for Boundary Value Problems]
 
[[Category:Numerical differential equations]]

Latest revision as of 15:09, 4 November 2014

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