On evaluation of Bessel transforms with oscillatory and algebraic singular integrands

In this paper, we study efficient methods for computing the integrals of the form ∫01xa(1−x)bf(x)Jv(ωx)dx, where a,b,v,ωa,b,v,ω are the given constants and ω≫1ω≫1, JvJv is the Bessel function of the first kind and of order vv, ff is a sufficiently smooth function on [0,1][0,1]. Firstly, we express the moments in a closed form with the aid of special functions. Secondly, we induce the Filon-type method based on the Taylor interpolation polynomial at two endpoints and the Hermite interpolation polynomial at Clenshaw–Curtis points on evaluating the highly oscillatory Bessel integrals with algebraic singularity. Theoretical results and numerical experiments perform that the methods are very efficient in obtaining very high precision approximations if ωω is sufficiently large.