摘要
本文针对高频振荡函数的积分提出了复合的moment-free的数值积分公式.当被积函数没有奇点、振子没有驻点时,复合的数值积分公式是基于对积分区间按振子导数值和波数剖分的原则设计,利用moment-freeFilon型数值积分公式计算子区间上的振荡积分.当被积函数有奇点、振子存在驻点时,根据函数的奇性和波数对积分区间剖分,使其于每个子区间上为非剧烈振荡函数的奇异积分,或光滑函数没有驻点的振荡积分.然后,采用经典的数值奇异积分公式计算非剧烈振荡函数的奇异积分,采用修改的moment-free Filon型数值积分公式计算振荡积分.与计算振荡积分的已知方法相比,本文发展的复合moment-free的数值积分方法,既不需要计算振子的反函数,也不必借助特殊函数的积分.
In this paper, we develop composite moment-free numerical quadratures for computing highly os- cillatory integrals with singularities and stationary points. The composite quadrature rules for computing highly oscillatory integrals with a smooth integrand and without a stationary point are developed based on partitioning the integration domain according to the values of the derivative of the oscillator and the wave number. The moment-free Filon-type quadrature is used for each of the oscillatory integrals defined on the subintervals. The composite quadrature rules for computing highly oscillatory integrals with singularities and stationary points are developed by partitioning the integration domain according to the singularity of the integrand and the wave number, such that the integral defined on a subinterval has either a weak singularity without rapid oscillation or oscillation without a singularity or stationary point. The classical quadrature rules for weakly singular integrals using graded points are employed for computing the singular integral without rapid oscillation and the modified moment-free Filon-type method is used for computing the oscillatory integrals. Unlike the existing methods, the proposed methods do not have to compute the inverse of the oscillator or to utilize the integral of special functions.
出处
《中国科学：数学》
CSCD
北大核心
2015年第8期1133-1152,共20页
Scientia Sinica：Mathematica
基金
国家自然科学基金(批准号:91130009和11471013)
广东省引进计算科学创新科研团队计划(批准号:11071286)资助项目
