On the asymptotic order of Filon-type methods for highly oscillatory integrals with an algebraic singularity

Filon-type methods for computing highly oscillatory integrals with an algebraic singularity of the form ∫01xαf(x)eiω/xβdx, where β > 0, α + β + 1 > 0 and f is a sufficiently smooth function on [0, 1] and ω ≫ 1, has been proposed by Hascelik [A.I. Hascelik, Suitable Gauss and Filon-type methods for oscillatory integrals with an algebraic singularity, Appl. Numer. Math 59 (2009) 101–118]. In this paper, we first expand such integrals into asymptotic series in inverse powers of ωβ and then give the asymptotic order of the Filon-type methods. Numerical examples are provided to confirm our analysis.