Efficient computation of highly oscillatory finite-part integrals

Abstract

This study is devoted to the numerical evaluation of finite-part integrals given by⨎0+∞xαf(x)(x−τ)m+1S(ωx)dx,τ>0,α>−1,m∈N, where S(ωx) represents highly oscillatory functions of the form S(ωx)=Hν(1)(ωx) and S(ωx)=eiωx with ω≫1. Through singularity separation, we introduce a uniform approximation scheme utilizing the complex integration method. The study includes an analysis of error estimates and convergence rates, demonstrating the superior effectiveness of the proposed algorithm for handling oscillatory integrands. Plentiful numerical results highlight the stability and efficiency of the developed schemes.