Computation of integrals with oscillatory singular factors of algebraic and logarithmic type

Abstract

In this paper, we present the Clenshaw–Curtis–Filon methods and the higher order methods for computing many classes of oscillatory integrals with algebraic or logarithmic singularities at the two endpoints of the interval of integration. The methods first require an interpolant of the nonoscillatory and nonsingular parts of the integrands in N+1 Clenshaw–Curtis points. Then, the required modified moments, can be accurately and efficiently computed by constructing some recurrence relations. Moreover, for these quadrature rules, their absolute errors in inverse powers of the frequency ω, are given. The presented methods share the advantageous property that the accuracy improves greatly, for fixed N, as ω increases. Numerical examples show the accuracy and efficiency of the proposed methods.