On the implementation of the asymptotic Filon-type method for infinite integrals with oscillatory Bessel kernels

Abstract

In this paper we consider the implementation of the asymptotic Filon-type method for the semi-infinite highly oscillatory Bessel integrals of the form ∫1∞f(x)Cv(ωx)dx, where Cv(ωx) denotes Bessel function Jv(ωx) of the first kind, Yv(ωx) of the second kind, Hv(1)(ωx) and Hv(2)(ωx) of the third kind, and modified Bessel function Kv(ωx) of the second kind, respectively, f is a smooth function on [1,∞), limx→∞f(k)(x)=0(k=0,1,2,…) and ω is large. By approximating f by a linear combination of negative integer powers so that the moments can be expressed by some special functions, we complete the implementation of the method. Furthermore, we give the error analysis of the method for computing the integrals. The method is very efficient in obtaining very high precision approximations if ω is sufficiently large. Numerical examples are provided to confirm our analysis.