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, ${\int }_0^b{x}^{\alpha }{(b-x)}^{\beta }f\left(x\right)e^{i\omega x^r}\text{d}x,-1<\alpha ,\beta \le 0,0<b<+\infty ,\omega \in \mathbb{R},r\in {\mathbb{N}}^{+}$. 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 $f\left(x\right)$. 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 $\omega$. 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.

最近的文章（J. Math. Anal. Appl. 494 (2021)，文章编号：124448）提出了一种渐近 Filon 型方法，用于计算具有特殊振荡器和弱奇点的振荡积分，${\int }_0^b{X}^{\alpha }{(b-X)}^{\beta }F\left(X\right)e^{\mathrm{一世}\omega X^r}\text{d}X,-1<\alpha ,\beta \le 0,0<b<+\infty ,\omega \in \mathbb{R},r\in {\mathbb{ñ}}^{+}$. 在本文中，我们针对这种奇异振荡积分提出并分析了两种不同的高效且准确的求积方法。首先，我们给出了一种两点泰勒插值方法，使用两点泰勒多项式代替$F\left(X\right)$. 此外，我们提出了一种更有效的轮廓积分方法。通过利用函数的泰勒多项式$F$在$X=0$，然后基于积分区间的可加性，我们将考虑的积分变为两个积分。基于柯西残差定理和广义高斯-拉盖尔求积法则的轮廓积分方法可以有效地计算一个积分。另一个积分可以通过特殊函数显式计算。具体来说，我们对所提出的方法进行严格的误差分析，并以频率参数的逆幂获得渐近误差估计$\omega$. 最后，将所提出的方法与本文中给出的渐近 Filon 型方法（J. Math. Anal. Appl. 494 (2021)，文章编号：124448）和改进的 Filon 型方法进行了比较。在计算成本相同的情况下，两点泰勒插值法和渐近菲隆型法具有非常接近的精度水平。它们的精度高于改进的Filon型方法，轮廓积分法的精度远高于渐近Filon型方法、改进的Filon型方法和两点Taylor插值法. 我们通过实验结果验证了所提出方法的误差分析。数值实验也可以验证所提出方法的效率和精度。