Efficient and accurate quadrature methods of Fourier integrals with a special oscillator and weak singularities

Abstract

The recent article (J. Math. Anal. Appl. 494 (2021), Article number: 124448) presented an asymptotic Filon-type method for computing the oscillatory integral with a special oscillator and weak singularities, ∫0bxα(b−x)βf(x)eiωxrdx,−1<α,β≤0,0<b<+∞,ω∈R,r∈N+. In this article, we propose and analyze two different efficient and accurate quadrature methods for this singularly oscillatory integral. First, we give a two-point Taylor interpolation method by using a two-point Taylor polynomial instead of f(x). In addition, we propose a more efficient contour integration method. By exploiting the Taylor polynomial of the function f at x=0, and then based on the additivity of the integration interval, we change the considered integral into two integrals. One integral can be efficiently computed by the contour integration method based on Cauchy Residue Theorem and generalized Gaussian-Laguerre quadrature rule. The other integral can be explicitly calculated by special functions. Specifically, we perform the rigorous error analysis of the proposed methods and obtain asymptotic error estimates in inverse powers of the frequency parameter ω. Ultimately, the proposed methods are compared with the asymptotic Filon-type method given in this work (J. Math. Anal. Appl. 494 (2021), Article number: 124448) and the modified Filon-type method. At the same computational cost, the two-point Taylor interpolation method and the asymptotic Filon-type method have a very close accuracy level. Their accuracy is higher than that of the modified Filon-type method, and the precision of the contour integration method is much higher than that of the asymptotic Filon-type method, the modified Filon-type method, and the two-point Taylor interpolation method. We verify error analyses of the proposed methods by experimental results. Numerical experiments can also verify the efficiency and precision of the proposed methods.