Efficient quadrature of highly oscillatory integrals with algebraic singularities

In this paper we are concerned with the numerical evaluation of a class of highly oscillatory integrals containing algebraic singularities. First, we expand such integrals derived by two transformations t=x−β,β>0t=x−β,β>0,t=21+z,−1≤z≤1, into asymptotic series in inverse powers of the frequency ωω. Then, based the asymptotic series, two methods are presented. One is the Filon-type method. The other is the Clenshaw–Curtis–Filon-type method which is based on a special Hermite interpolation polynomial in the Clenshaw–Curtis points and can be evaluated efficiently in O(NlogN)O(NlogN) operations, where N+1N+1 is the number of Clenshaw–Curtis points in the interval of integration. Some error and convergence analysis and robust numerical examples are used to demonstrate the accuracy and effectiveness of the proposed approaches for approximating the class of highly oscillatory singular integrals.