In this paper we investigate the calculation and analysis for a class of the highly oscillatory generalized Bessel transform with endpoint singularities of algebraic type. First, when the oscillator has either zeros or stationary points, we give many asymptotic expansions for the transform. On the basis of these results, we construct a new and good modified Filon-type method. Moreover, based on the resulting asymptotic expansions we can perform the error analysis in inverse powers of ω. In addition, the proposed methods share the property that the precision improves greatly, for fixed multiplicities and number of nodes, as ω increases. Our theoretical analysis can be verified by some numerical examples. Finally, by numerical comparison we can see that the accuracy of the proposed modified Filon-type methods is much higher than those of the existing Filon method and the method proposed by Tripathi etc. in 2014 at the same computational cost.

中文翻译：

---

振荡和弱奇异广义贝塞尔变换的快速算法和分析


在本文中，我们研究了一类具有代数类型端点奇点的高度振荡广义贝塞尔变换的计算和分析。首先，当振荡器有零点或静止点时，我们为变换给出许多渐近展开。基于这些结果，我们构建了一种新的、良好的改进 Filon 型方法。此外，基于得到的渐近展开，我们可以在 ω 的逆幂中进行误差分析。此外，所提出的方法具有一个特性，即随着 ω 的增加，对于固定的多重性和节点数，精度会大大提高。我们的理论分析可以通过一些数值示例来验证。最后，通过数值对比可以看出，在相同的计算成本下，所提出的改进 Filon 型方法的精度远高于现有的 Filon 方法和 Tripathi 等人在 2014 年提出的方法。