A quadrature formula for integrals with nearby singularities

Advances in Radio Science, 2006

Boundary Integral Equation formulations can be used to describe electromagnetic shielding problems. Yet, this approach frequently leads to integrals which contain a singularity and an oscillating part. Those integrals are difficult to handle when integrated naivly using standard integration techniques, and in some cases even a very high number of integration nodes will not lead to precise results.

This paper presents an automatic quadrature scheme (AQS) for the evaluation of hypersingular integrals (HSI)

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.

Quarterly of Applied Mathematics, 1998

This paper presents some explicit results concerning an extension of the mechanical quadrature technique, namely, the Gauss-Jacobi numerical integration scheme, to the class of integrals whose kernels exhibit second order of singularity (i.e., one degree more singular than Cauchy). In order to ascribe numerical values to these integrals they must be understood in Hadamard's finite-part sense. The quadrature formulae are derived from those for Cauchy singular integrals. The resulting discretizations are valid at a number of fixed points, determined as the zeroes of a certain Jacobi polynomial. As in all Gaussian quadratures, the final quadrature formulae involve fixed nodal points and provide exact results for polynomials of degree 2n-1, where n is the number of nodes. These properties make this approach rather attractive for applications to fracture mechanics problems, where often numerical solution of integral equations with strongly singular kernels is the objective. Numerical examples of the application of the Gauss-Chebyshev rule to some plane and axisymmetric crack problems are given.

Computers & Mathematics with Applications, 1990

Abatract--A group of quadrature formulae for end-point singular functions is presented generalizing classical end-point corrected trapezoidal quadrature rules. The actual values of the end-point corrections are obtained for each singularity as a solution of a system of linear algebraic equations. The algorithm is applicable to a wide class of monotonic singularities and does not require that an analytical expression for the singularity be known; only the knowledge of its first several moments and the ability to evaluate it on the interval of integration are needed.

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.

Jagannath University Journal of Science, 2021

Gauss quadrature formula has been modified in a non-symmetric form by introducing a small parameter and applied to evaluate singular integrals. The modified formula measures more accurate result than the original form presented by Gauss.

The numerical integration of polynomial functions is one of the most interesting processes for numerical calculus and analyses, and represents thus, a compulsory step especially in finite elements analyses. Via the Gauss quadrature, the users concluded a great inconvenience that is processing at certain points which not required the based in finite element method points for deducting the form polynomials constants. In this paper, the same accuracy and efficiency as the Gauss quadrature extends for the numerical integration of the polynomial functions, but as such at the same points and nods have chosen for the determination of the form polynomials. Not just to profit from the values of the polynomials at those points and nods, but also from their first derivatives, the chosen points positions are arbitrary and the resulted deducted formulas are therefore different, as will be presented bellow and implemented.

A mixed quadrature rule of blending Clenshaw-Curties five point rule and Gauss-Legendre 3 point rule is formed. The mixed rule has been tested and found to be more effective than that of its constituent Clenshaw-Curtis five point rule for the approximate evaluation of the integral of an analytic function over a line segment in complex plane. An asymptotic error estimate of the rule has been determined and the rule has been numerically verified.

IEEE Transactions on Antennas and Propagation, 2000