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

Abstract: Filon-type methods for computing highly oscillatory integrals with an algebraic singularity of the form , 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. [Copyright &y& Elsevier]