几类高振荡奇异积分的高效数值算法研究

The computation of highly oscillatory integrals arises in many disciplines, such as mathematics, physics, etc. It is one of the significant research topics in computational mathematics and has been received significant attention of many well-known scholars. However, for the computations of highly oscillatory integrals on infinite interval with a Hankel function as kernel, which has a singularity at zero, and highly oscillatory Cauchy principal value integrals, Hadamard-Finite-Part integrals with a Bessel function as kernel, there is not enough study of these problems. This project is devoted to explore efficient numerical methods for a class of oscillatory integrals with Hankel function and Bessel function as kernels, including weakly singular integrals, Cauchy-singular integrals and hyper-singular integrals. In addition, we will give the error analyses for the proposed methods. The results of this project will provide important theoretical values for the research of high frequency electromagnetic acoustic scattering.

高振荡积分的计算广泛来源于数学和物理等诸多学科，是计算数学领域内研究的热点之一，得到了众多知名学者的极大关注。而关于在零点含有弱奇异性的无穷区间的Hankel型振荡积分, Bessel型高振荡Cauchy主值积分、Hadamard-Finite-Part积分的研究还不够完善。本项目主要研究振荡核函数为Hankel函数、Bessel函数的奇异（包括弱奇异、Cauchy奇异、超奇异）积分的数值方法。另外，针对上述方法给出误差析；本项目的成果将对高频电磁声波散射问题的研究提供重要的理论价值。

高振荡积分的计算广泛来源于数学和物理等诸多学科，是计算数学领域内研究的热点之一，得到了众多知名学者的极大关注。本项目主要利用快速傅里叶变换技术以及复积分理论，研究了有限区间上两端点具有弱奇异性的含双振荡子的高振荡积分、无穷区间上不含零点的高振荡Bessel变换的数值方法。其中，重点研究了含双振荡子的高振荡积分的Clenshaw-Curtis-Filon型方法所涉及的修正的矩的递推公式及其关于频率的渐进性、高振荡Bessel变换的Gauss型方法的构造。研究结果不仅使得高振荡奇异积分的计算方法得到进一步补充和完善，又为高频电磁声波散射问题的研究提供重要的理论价值。