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Difference between revisions of "Quadrature-sum method"

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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
+
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
  
 
$$  
 
$$  
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$$
 
$$
  
in an [[Integral equation|integral equation]]
+
in an [[integral equation]]
  
 
$$  
 
$$  
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$$ \tag{1 }
 
$$ \tag{1 }
 
\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  \approx \  
 
\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} ) .
+
\sum _ { i=1 } ^ { N }  a _ {i}  ^ {(N)} K ( x , s _ {i} ) \phi ( s _ {i} ) .
 
$$
 
$$
  
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$$  
 
$$  
 
\lambda \widetilde \phi  ( s _ {j} ) +
 
\lambda \widetilde \phi  ( s _ {j} ) +
\sum _ { i= } 1 ^ { N }  
+
\sum _ { i=1 } ^ { N }  
a _ {i}  ^ {(} N) K ( s _ {j} , s _ {i} )
+
a _ {i}  ^ {(N)} K ( s _ {j} , s _ {i} )
 
\widetilde \phi  ( s _ {i} )  =  f ( s _ {j} ) ,\ \  
 
\widetilde \phi  ( s _ {i} )  =  f ( s _ {j} ) ,\ \  
 
j = 1 \dots N .
 
j = 1 \dots N .
 
$$
 
$$
  
On the right-hand side of the approximate equation (1) is a [[Quadrature formula|quadrature formula]] for the integral with respect to  $  s $.  
+
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:
 
Various generalizations of (1) are possible:
  
 
$$ \tag{2 }
 
$$ \tag{2 }
 
\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  \approx \  
 
\int\limits _ { a } ^ { b }  K ( x , s ) \phi ( s)  d s  \approx \  
\sum _ { i= } 1 ^ { N }  a _ {i}  ^ {(} N) ( x) \phi ( s _ {i} ) ,
+
\sum _ { i=1 } ^ { N }  a _ {i}  ^ {(N)} ( x) \phi ( s _ {i} ) ,
 
$$
 
$$
  
where the  $  a _ {i}  ^ {(} N) ( x) $
+
where the  $  a _ {i}  ^ {(N)} ( x) $ are certain functions constructed from the kernel  $  K ( x , s ) $.  
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.
 
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.
  

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=48363
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article