Quadrature-sum method

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

References

[a1]	C.T.H. Baker, "The numerical treatment of integral equations" , Clarendon Press (1977) pp. Chapt. 4 MR0467215 Zbl 0373.65060