On the numerical evaluation of singular integrals (original) (raw)

On Generalized Gaussian Quadrature Rules for Singular and Nearly Singular Integrals

SIAM Journal on Numerical Analysis, 2009

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 and we show that the convergence rate is unaffected by the singularity of the integrand. We characterize the quadrature rules in terms of two families of functions that share many properties with orthogonal polynomials, but that are orthogonal with respect to a discrete scalar product that in most cases is not known a priori.

A mixed scheme of product integration rules in (−1,1)

Applied Numerical Mathematics, 2019

The paper deals with the numerical approximation of integrals of the type I(f, y) := 1 −1 f (x)k(x, y)dx, y ∈ S ⊂ IR where f is a smooth function and the kernel k(x, y) involves some kinds of "pathologies" (for instance, weak singularities, high oscillations and/or endpoint algebraic singularities). We introduce and study a product integration rule obtained by interpolating f by an extended Lagrange polynomial based on Jacobi zeros. We prove that the rule is stable and convergent with the order of the best polynomial approximation of f in suitable function spaces. Moreover, we derive a general recurrence relation for the new modified moments appearing in the coefficients of the rule, just using the knowledge of the usual modified moments. The new quadrature sequence, suitable combined with the ordinary product rule, allows to obtain a "mixed " quadrature scheme, significantly reducing the number of involved samples of f. Numerical examples are provided in order to support the theoretical results and to show the efficiency of the procedure.

A New Straightforward Method for Evaluating Singular Integrals

Applied and Computational Mathematics, 2015

A new more accurate straightforward method is presented for evaluating the singular integrals. A few methods in numerical analysis is useful for evaluating the integral where singularities arises, most of them uses extrapolation technique at singular point. This new method uses directly and gives better results and the Romberg integration of this formula converses faster than others previous methods.

A More Accurate and Straightforward Method for Evaluating Singular Integrals

Universal Journal of Applied Mathematics, 2015

Recently, a straightforward formula has been presented for evaluating singular integrals. Earlier extrapolation technique was used to guess the functional values at the singular points since most of the classical formulae contain both ends points. In this article a more accurate straightforward formula is presented for evaluating singular integrals. The new formula converges faster than others existing formulae. The Romberg integration scheme of this method also converges faster.

Numerical analysis of singular weighted integrals

Computing, 1994

Numerical Analysis of Singular Weighted Integrals. In this article we investigate the numerical aspects of integrals of the form) f(x)O(x)dx (1) where f is an unobjectionable function and ~ is singular, i.e. ~ is oscillating with high frequency, is discontinuous or unbounded. Suitable integration algorithms are presented.

Numerical indefinite integration of functions with singularities

Mathematics of Computation - Math. Comput., 2001

We derive an indenite quadrature formula, based on a theorem of Ganelius, for Hp functions, for p> 1, over the interval ( 1; 1). The main factor in the error of our indenite quadrature formula is O(e p N=q ), with 2N nodes and 1p + 1 q = 1. The convergence rate of our formula is better than that of the Stenger-type formulas by a factor of p 2 in the constant of the exponential. We conjecture that our formula has the best possible value for that constant. The results of numerical examples show that our indenite quadrature formula is better than Haber's indenite quadrature formula for Hp-functions.

Quadrature rules for evaluation of hyper singular integrals

Applied Mathematical Sciences, 2014

Some Newton-Cote's type of quadrature rules have been formulated for the numerical evaluation of hyper singular integrals, which are interpreted as of Hadamard finite part type. Rules have been tested numerically by some standard test integrals and their respective error bounds have been determined.

A quadrature formula for integrals with nearby singularities

International Journal for Numerical Methods in Engineering, 2006

The purpose of this paper is to propose a new quadrature formula for integrals with nearby singularities. In the boundary element method, the integrands of nearby singular boundary integrals vary drastically with the distance between the field and the source point. Especially, field variables and their derivatives at a field point near a boundary cannot be computed accurately. In the present paper a quadrature formula for l-isolated singularities near the integration interval, based on Lagrange interpolatory polynomials, is obtained. The error estimation of the proposed formula is also given. Quadrature formulas for regular and singular integrals with conjugate poles are derived. Numerical examples are given and the proposed quadrature rules present the expected polynomial accuracy.