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Quadrature-sum method

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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

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

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=51749
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article