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

in an integral equation

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

(1)

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

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

(2)

where the are certain functions constructed from the kernel . 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