Spline product quadrature rules for cauchy singular integrals

Quadrature Rules for Fuzzy Integrals

Gauss Type Quadrature Rules for Cauchy Principal Value Integrals

Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integ...

Quadrature rules for singular integrals on unbounded intervals

The importance of singular and hypersingular integral transforms, coming from their many applications, justifies some interest in their numerical approximation. The literature about the numerical evaluation of such integrals on bounded intervals is wide and quite satisfactory; instead only few papers deal with the numerical evaluation of such integral transforms on half-infinite intervals or on...

On Generalized Gaussian Quadrature Rules for Singular and Nearly Singular Integrals

We construct and analyze generalized Gaussian quadrature rules for integrands with endpoint singularities or near endpoint singularities. The rules have quadrature points inside the interval of integration and the weights are all strictly positive. Such rules date back to the study of Chebyshev sets, but their use in applications has only recently been appreciated. We provide error estimates an...

Quadrature methods for highly oscillatory singular integrals

We study asymptotic expansions, Filon-type methods and complex-valued Gaussian quadrature for highly oscillatory integrals with power-law and logarithmic singularities. We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the integral. A truncated asymptotic expansion...