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A method for approximating an [[Integral operator|integral operator]] by constructing numerical methods for the solution of integral equations. The simplest version of a quadrature-sum method consists in replacing an integral operator, for instance of the form
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<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/q/q076/q076210/q0762101.png" /></td> </tr></table>
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in an [[Integral equation|integral equation]]
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A method for approximating an [[integral operator]] by constructing numerical methods for the solution of integral equations. The simplest version of a quadrature-sum method consists in replacing an integral operator, for instance of the form
  
<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/q/q076/q076210/q0762102.png" /></td> </tr></table>
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$$
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\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s ,
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$$
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in an [[integral equation]]
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$$
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\lambda \phi ( x) +
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\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  = f ( s)
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$$
  
 
by an operator with finite-dimensional range, according to the rule
 
by an operator with finite-dimensional range, according to the rule
  
<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/q/q076/q076210/q0762103.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
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\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  \approx \
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\sum _ { i=1 } ^ { N }  a _ {i}  ^ {(N)} K ( x , s _ {i} ) \phi ( s _ {i} ) .
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$$
  
 
The integral equation, in turn, is approximated by the linear algebraic equation
 
The integral equation, in turn, is approximated by the linear algebraic 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/q/q076/q076210/q0762104.png" /></td> </tr></table>
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$$
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\lambda \widetilde \phi  ( s _ {j} ) +
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\sum _ { i=1 } ^ { N }
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a _ {i}  ^ {(N)} K ( s _ {j} , s _ {i} )
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\widetilde \phi  ( s _ {i} )  = f ( s _ {j} ) ,\ \
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j = 1 \dots N .
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$$
  
On the right-hand side of the approximate equation (1) is a [[Quadrature formula|quadrature formula]] for the integral with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076210/q0762105.png" />. Various generalizations of (1) are possible:
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On the right-hand side of the approximate equation (1) is a [[quadrature formula]] for the integral with respect to $  s $.  
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Various generalizations of (1) are possible:
  
<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/q/q076/q076210/q0762106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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$$ \tag{2 }
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\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  \approx \
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\sum _ { i=1 } ^ { N }  a _ {i}  ^ {(N)} ( x) \phi ( s _ {i} ) ,
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$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076210/q0762107.png" /> are certain functions constructed from the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076210/q0762108.png" />. The quadrature-sum method as generalized in the form (2) can be applied for the approximation of integral operators with singularities in the kernel and even of singular integral operators.
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where the $  a _ {i}  ^ {(N)} ( x) $ are certain functions constructed from the kernel $  K ( x , s ) $.  
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The quadrature-sum method as generalized in the form (2) can be applied for the approximation of integral operators with singularities in the kernel and even of singular integral operators.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.V. Kantorovich,   V.I. Krylov,   "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) {{MR|0106537}} {{ZBL|0083.35301}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.T.H. Baker,   "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4 {{MR|0467215}} {{ZBL|0373.65060}} </TD></TR></table>

Latest revision as of 16:32, 13 July 2021


A method for approximating an integral operator by constructing numerical methods for the solution of integral equations. The simplest version of a quadrature-sum method consists in replacing an integral operator, for instance of the form

$$ \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) d s , $$

in an integral equation

$$ \lambda \phi ( x) + \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) d s = f ( s) $$

by an operator with finite-dimensional range, according to the rule

$$ \tag{1 } \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) d s \approx \ \sum _ { i=1 } ^ { N } a _ {i} ^ {(N)} K ( x , s _ {i} ) \phi ( s _ {i} ) . $$

The integral equation, in turn, is approximated by the linear algebraic equation

$$ \lambda \widetilde \phi ( s _ {j} ) + \sum _ { i=1 } ^ { N } a _ {i} ^ {(N)} K ( s _ {j} , s _ {i} ) \widetilde \phi ( s _ {i} ) = f ( s _ {j} ) ,\ \ j = 1 \dots N . $$

On the right-hand side of the approximate equation (1) is a quadrature formula for the integral with respect to $ s $. Various generalizations of (1) are possible:

$$ \tag{2 } \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) d s \approx \ \sum _ { i=1 } ^ { N } a _ {i} ^ {(N)} ( x) \phi ( s _ {i} ) , $$

where the $ a _ {i} ^ {(N)} ( x) $ are certain functions constructed from the kernel $ K ( x , s ) $. The quadrature-sum method as generalized in the form (2) can be applied for the approximation of integral operators with singularities in the kernel and even of singular integral operators.

References

[1] L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian) MR0106537 Zbl 0083.35301

Comments

References

[a1] C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4 MR0467215 Zbl 0373.65060
How to Cite This Entry:
Quadrature-sum method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quadrature-sum_method&oldid=16202
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article