Quadrature formulae for Cauchy principal value integrals of oscillatory kind

Abstract

The problem considered is that of evaluating numerically an integral of the form /-1 e"*xf(x) dx, where / has one simple pole in the interval [-1,1). Modified forms of the Lagrangian interpolation formula, taking account of the simple pole are obtained, and form the bases for the numerical quadrature rules obtained. Further modification to deal with the case when an abscissa in the interpolation formula is coincident with the pole is also considered. An error bound is provided and some numerical examples are given to illustrate the formulae developed.

Using modified moments for numerical solution of oscillatory integrals

“…The exact integral value is I =−28.6384665450831 +i×17.1789068416939. In this example, we compute the integral by using Equation (10) and compare the present rule (10) with the formula proposed by Okecha [15] and the rule proposed by Wang and Xiang [12,13]. …”

Numerical implementation of the integral-transform solution to Lamb's point-load problem

“…Davis and Rabinowitz 1984;Okecha 1987). Instead, we follow an adaptive technique (based essentially on ideas of Brock, Georgiadis and Charalambakis 1994) according to which one ®rst evaluates the abscissa of the Rayleigh peak (this is accomplished here through the Newton-Raphson method) and then considers small sub-intervals to the left and right of that abscissa.…”

Approximate Solution Technique for Singular Fredholm Integral Equations of the First Kind with Oscillatory Kernels

“…where f is a given continuous function, and an unknown function, have wide applications in mathematics, physics, engineering and other applied and computational sciences [1]. If is large, the integrand is highly oscillatory and in most cases the integral equation cannot be solved analytically and so, there is need for numerical methods.…”

“…Ting and Luke [7] approximated integrals whose integrands are oscillatory and contain singularities at the end points of the interval of integration by expanding the function in series of orthogonal polynomials over the interval of integration with respect to the weight function. Okecha [1] developed algorithms based on the modified Lagrange interpolation formula, Legendre polynomial and the Christoffel-Darboux formula to evaluate Cauchy principal value integrals of oscillatory kind. Different numerical techniques like collocation and Galerkin's methods [8,9], asymptotic method [10], generalized quadrature rule [11], and modified Clenshaw-Curtis method [12] have also been developed.…”