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A method for solving initial and boundary value problems for ordinary differential equations. It consists of introducing control variables (parameters) and subsequently determining them from the system of equations, where this choice of parameters has a decisive influence on the acceleration of the solution of the system.
 
A method for solving initial and boundary value problems for ordinary differential equations. It consists of introducing control variables (parameters) and subsequently determining them from the system of equations, where this choice of parameters has a decisive influence on the acceleration of the solution of the system.
  
Suppose, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s0849501.png" />, one is given the differential equation
+
Suppose, for $  a \leq  x \leq  b $,  
 +
one is given the differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s0849502.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
y  ^  \prime  = \
 +
F ( x, y)
 +
$$
  
 
with boundary condition
 
with boundary condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s0849503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
g ( y ( a), y ( b)) =  h,
 +
$$
  
where the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s0849504.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s0849505.png" /> is to be determined, the vector functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s0849506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s0849507.png" /> are known, and the numerical vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s0849508.png" /> is given.
+
where the vector function $  y = ( y _ {1} \dots y _ {n} )  ^ {T} $
 +
of $  x $
 +
is to be determined, the vector functions $  F = ( F _ {1} \dots F _ {n} )  ^ {T} $
 +
and $  g = ( g _ {1} \dots g _ {n} )  ^ {T} $
 +
are known, and the numerical vector $  h = ( h _ {1} \dots h _ {n} )  ^ {T} $
 +
is given.
  
 
Suppose that the [[Cauchy problem|Cauchy problem]]
 
Suppose that the [[Cauchy problem|Cauchy problem]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s0849509.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
 
 +
\frac{\partial  Z }{\partial  x }
 +
  = \
 +
F ( x, Z),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
Z ( a, r) =  r,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495012.png" />, has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495013.png" />, defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495015.png" />. Upon substituting in (2) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495016.png" /> the given value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495017.png" /> and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495018.png" /> the calculated value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495019.png" />, one obtains the equation
+
where $  Z = ( Z _ {1} \dots Z _ {n} )  ^ {T} $
 +
and $  r = ( r _ {1} \dots r _ {n} )  ^ {T} $,  
 +
has a unique solution $  Z ( x, r) $,  
 +
defined for $  a \leq  x \leq  b $,  
 +
$  r \in \mathbf R  ^ {n} $.  
 +
Upon substituting in (2) for $  y ( a) $
 +
the given value $  Z ( a, r) = r $
 +
and for $  y ( b) $
 +
the calculated value $  Z ( b, r) $,  
 +
one obtains the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
g ( r, Z ( b, r)) =  h
 +
$$
  
with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495021.png" />.
+
with respect to the parameter $  r $.
  
The algorithm of the shooting method is as follows. First one finds a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495022.png" /> of (5), and subsequently the required solution of the boundary value problem (1)–(2) as the solution of the Cauchy problem
+
The algorithm of the shooting method is as follows. First one finds a solution $  r = r  ^ {*} $
 +
of (5), and subsequently the required solution of the boundary value problem (1)–(2) as the solution of the Cauchy problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495023.png" /></td> </tr></table>
+
$$
 +
y  ^  \prime  = F ( x, y),\ \
 +
y ( a)  = r  ^ {*} .
 +
$$
  
 
This problem can be solved using numerical methods. In order to solve (5) it is usually necessary to choose some iteration method.
 
This problem can be solved using numerical methods. In order to solve (5) it is usually necessary to choose some iteration method.
  
If some of the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495024.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495025.png" /> alone and the others on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084950/s08495026.png" /> alone, it is useful to make another choice of parameters (cf. [[#References|[1]]], as well as [[Non-linear boundary value problem, numerical methods|Non-linear boundary value problem, numerical methods]]). There are other versions of the shooting method (cf. [[#References|[4]]]). The shooting method is also used in solving grid boundary value problems.
+
If some of the components of $  y $
 +
depend on $  y ( a) $
 +
alone and the others on $  y ( b) $
 +
alone, it is useful to make another choice of parameters (cf. [[#References|[1]]], as well as [[Non-linear boundary value problem, numerical methods|Non-linear boundary value problem, numerical methods]]). There are other versions of the shooting method (cf. [[#References|[4]]]). The shooting method is also used in solving grid boundary value problems.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.K. Godunov,  V.S. Ryaben'kii,  "The theory of difference schemes" , North-Holland  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Krylov,  V.V. Bobkov,  P.I. Monastyrnyi,  "Numerical methods" , '''2''' , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Hall (ed.)  J.M. Watt (ed.) , ''Modern numerical methods for ordinary differential equations'' , Clarendon Press  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.S. Bakhvalov,  "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.K. Godunov,  V.S. Ryaben'kii,  "The theory of difference schemes" , North-Holland  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.I. Krylov,  V.V. Bobkov,  P.I. Monastyrnyi,  "Numerical methods" , '''2''' , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Hall (ed.)  J.M. Watt (ed.) , ''Modern numerical methods for ordinary differential equations'' , Clarendon Press  (1976)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U.M. Ascher,  R.M.M. Mattheij,  R.D. Russell,  "Numerical solution for boundary value problems for ordinary differential equations" , Prentice-Hall  (1988)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U.M. Ascher,  R.M.M. Mattheij,  R.D. Russell,  "Numerical solution for boundary value problems for ordinary differential equations" , Prentice-Hall  (1988)</TD></TR></table>

Latest revision as of 08:13, 6 June 2020


A method for solving initial and boundary value problems for ordinary differential equations. It consists of introducing control variables (parameters) and subsequently determining them from the system of equations, where this choice of parameters has a decisive influence on the acceleration of the solution of the system.

Suppose, for $ a \leq x \leq b $, one is given the differential equation

$$ \tag{1 } y ^ \prime = \ F ( x, y) $$

with boundary condition

$$ \tag{2 } g ( y ( a), y ( b)) = h, $$

where the vector function $ y = ( y _ {1} \dots y _ {n} ) ^ {T} $ of $ x $ is to be determined, the vector functions $ F = ( F _ {1} \dots F _ {n} ) ^ {T} $ and $ g = ( g _ {1} \dots g _ {n} ) ^ {T} $ are known, and the numerical vector $ h = ( h _ {1} \dots h _ {n} ) ^ {T} $ is given.

Suppose that the Cauchy problem

$$ \tag{3 } \frac{\partial Z }{\partial x } = \ F ( x, Z), $$

$$ \tag{4 } Z ( a, r) = r, $$

where $ Z = ( Z _ {1} \dots Z _ {n} ) ^ {T} $ and $ r = ( r _ {1} \dots r _ {n} ) ^ {T} $, has a unique solution $ Z ( x, r) $, defined for $ a \leq x \leq b $, $ r \in \mathbf R ^ {n} $. Upon substituting in (2) for $ y ( a) $ the given value $ Z ( a, r) = r $ and for $ y ( b) $ the calculated value $ Z ( b, r) $, one obtains the equation

$$ \tag{5 } g ( r, Z ( b, r)) = h $$

with respect to the parameter $ r $.

The algorithm of the shooting method is as follows. First one finds a solution $ r = r ^ {*} $ of (5), and subsequently the required solution of the boundary value problem (1)–(2) as the solution of the Cauchy problem

$$ y ^ \prime = F ( x, y),\ \ y ( a) = r ^ {*} . $$

This problem can be solved using numerical methods. In order to solve (5) it is usually necessary to choose some iteration method.

If some of the components of $ y $ depend on $ y ( a) $ alone and the others on $ y ( b) $ alone, it is useful to make another choice of parameters (cf. [1], as well as Non-linear boundary value problem, numerical methods). There are other versions of the shooting method (cf. [4]). The shooting method is also used in solving grid boundary value problems.

References

[1] N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian)
[2] S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian)
[3] V.I. Krylov, V.V. Bobkov, P.I. Monastyrnyi, "Numerical methods" , 2 , Moscow (1977) (In Russian)
[4] G. Hall (ed.) J.M. Watt (ed.) , Modern numerical methods for ordinary differential equations , Clarendon Press (1976)

Comments

References

[a1] U.M. Ascher, R.M.M. Mattheij, R.D. Russell, "Numerical solution for boundary value problems for ordinary differential equations" , Prentice-Hall (1988)
How to Cite This Entry:
Shooting method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shooting_method&oldid=48690
This article was adapted from an original article by A.F. Shapkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article