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High-Order Quadratures for Integral Operators with Singular Kernels

Summary: High-Order Quadratures for Integral Operators with
Singular Kernels
Bradley K. Alpert
A numerical integration method that has rapid convergence for integrands with
known singularities is presented. Based on endpoint corrections to the trapezoidal
rule, the quadratures are suited for the discretization of a variety of integral equations
encountered in mathematical physics. The quadratures are based on a technique in-
troduced by Rokhlin (Computers Math. Applic. 20, pp. 51-62, 1990). The present
modi cation controls the growth of the quadrature weights and permits higher-order
rules in practice. Several numerical examples are included.
Abbreviated Title. Quadratures for Integral Equations
Key Words. numerical integration, singular kernels, quadrature rules, corrected trape-
zoidal rules
AMS(MOS) subject classi cations. 65D30, 65B15, 65R20
1 Introduction
The discretization of a linear Fredholm integral equation of the second kind,
Z b


Source: Alpert, Bradley K. - Mathematical and Computational Sciences Division, National Institute of Standards and Technology (NIST)


Collections: Mathematics; Computer Technologies and Information Sciences