摘要
数学物理中的很多问题都可以归结为带有Cauchy核的奇异积分与积分方程。因此,关于此类问题的文献比较多。但直接计算Cauchy积分和解积分方程有时显得非常困难。从而,对此类问题的讨论就转向了求其数值解。
本文首先叙述了Cauchy奇异积分与积分方程数值解的发展背景,以及现有的一些具有代表性的方法。其次,对于带有Cauchy核的奇异积分,我们给出了一种新型的求积公式和Euler-Maclaurin展开式,以及外推公式。同时,还给出了带有Hilbert核的奇异积分的求积公式。利用这些公式给出了具体带有Cauchy和Hilbert核的奇异积分算例的误差结果,并与已有的一些算法进行了数值结果比较,充分说明了这些公式是高精度公式。另外,本文还讨论了带有Cauchy核和Hilbert核的奇异积分方程的数值解法,也给出了具体算例的误差结果,且与其它方法的结果进行了比较,验证了本文求积公式的优越性。最后,对此问题进行了总结与展望。
A lot of problems in mathematics and physics can be boiled down to the integrals and integral equations with a Cauchy singular kernel. There are many literatures studying this kind of integrals and integral equations and the study becomes more and more important. But it is very difficult to calculate Cauchy integral and integral equations directly. So people switch from the study of these problems to their numerical solution of the integrals and integral equations.
Firstly, this paper mainly describes the development background of the methods to get the numerical solutions of the Cauchy integral equations. Secondly, For Cauchy singular integrals, we put forward a new style of integral formula, and Euler-Maclaurin expansion as well as extrapolation formula. At the same time, we give the integral formula of Hilbert singular integral. With these formulas, we give the error result of specific example of Cauchy and Hilbert singular integral quadrate. We show that these formulas are highly accurate by comparing with the numerical results of other algorithms. Thirdly, we discuss the numerical solutions of the integral equations with a Cauchy and Hilbert singular kernel and get the error result of specific examples. These also explain that the integral formula is highly accurate. At the end of this paper, It gives some conclusions and introduces the development direction of the future.
引文
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