数值计算与计算机应用
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数值计算与计算机应用  2017, Vol. 38 Issue (3): 215-224    DOI:
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一类含奇异积分核的高振荡积分数值估计
王现辉1, 乔慧2
1. 河南理工大学机械与动力工程学院, 焦作 454003;
2. 河南理工大学数学与信息科学学院, 焦作 454003
NUMERICAL EVALUATION OF HIGHLY OSCILLATORY INTERGRALS WITH WEAK SINGULARITIES
Wang Xianhui1, Qiao Hui2
1. School of Mechanical and Power Engineering, Henan Polytechnic University, Jiaozuo 454003, China;
2. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China
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摘要 本文针对一类含弱奇异积分核的高振荡积分
ablnx-alnb-x)/(x-aαx-η)(b-xβfxeiωxdx
其中0 < α < 1, 0 < β < 1, η∈(a,b), fx)在区间[a, b]中解析,提出一种数值积分方法.在该方法中,高振荡积分处理主要分为两部分,一部分采用渐进展开方法处理,另一部分使用n点Guass-Laguerre积分计算.在渐进展开中,每次展开产生的弱奇异性采用构造函数来处理,而渐进误差是频率ω幂次方的倒数.数值算例验证了该方法的有效性.
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关键词振荡函数   数值积分   渐进展开   Guass-Laguerre积分   弱奇异性     
Abstract: In this paper, we explore quadrature methods for highly oscillatory integral
ab(ln(x-a)ln(b-x)/(x-a)α(x-η)(b-x)β)f(x)eiωxdx,
where 0 < α < 1, 0 < β < 1, η∈(a,b), f(x) is an analytic function in [a, b]. In the presented method, the highly oscillatory integrals can be separated to two parts. One part is evaluated by the asymptotic method, and another part is evaluated by the n points GuassLaguerre quadrature. A new constructed function is used to treat the weak singularity in each expansion of asymptotic method. The asymptotic error is given in inverse powers of the frequency ω. The validity of the presented method has been demonstrated by the numerical examples.
Key wordsoscillatory function   numerical quadrature   asymptotic method   GuassLaguerre quadrature   weak singularity   
收稿日期: 2016-06-14;
基金资助:

河南省高校基本科研业务费专项资金资助(NSFRF140122)和河南理工大学科学研究基金(B2014-38).

引用本文:   
. 一类含奇异积分核的高振荡积分数值估计[J]. 数值计算与计算机应用, 2017, 38(3): 215-224.
. NUMERICAL EVALUATION OF HIGHLY OSCILLATORY INTERGRALS WITH WEAK SINGULARITIES[J]. Journal of Numerical Methods and Computer Applicat, 2017, 38(3): 215-224.
 
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