Hadamard奇异积分的计算方法

发布时间：2018-04-10 18:12

  本文选题：Hadamard奇异积分 + 三次样条插值　； 参考：《华北电力大学》2017年硕士论文

【摘要】：声学、电磁散射学、断裂力学等诸多物理问题中都会广泛涉及到Hadamard奇异积分计算问题。但是Hadamard奇异积分在普遍意义和主值意义下是发散的,这增加了研究的难度。多年来,人们致力于超奇异积分研究并给出了一些有效计算方法,如牛顿科茨型公式、高斯型求积公式、复合埃尔米特插值型公式等。通常,高斯积分需要被积函数有较好光滑性,并需要配置高斯节点;牛顿科茨公式由于灵活方便的网格而具有吸引力,不过要得到较高收敛阶需要更多的插值节点。因此,针对不同的实际问题需要探寻不同的近似计算方法。本文介绍了Hadamard奇异积分的研究现状,在此基础上讨论了基于三次样条插值逼近的Hadamard奇异积分的计算公式及误差分析,数值算例说明了该算法的可行性和有效性。全文共分四章。第一章,介绍了超奇异积分的研究状况、研究意义及国内外发展的一些动态;第二章,介绍Hadamard奇异积分的概念及常见的插值求积分公式;第三章,研究基于三次样条函数插值的Hadamard奇异积分计算公式和误差分析,理论证明该方法的超收敛性,实例验证了该方法的可行性和有效性;第四章,是全文的总结和今后的工作目标。
[Abstract]:Acoustics, electromagnetic scattering, fracture mechanics and many other physical problems are widely related to the Hadamard singular integral calculation.However, the Hadamard singular integral is divergent in the sense of universal meaning and principal value, which makes it more difficult to study.Over the years, people have devoted themselves to the study of hypersingular integrals and given some effective calculation methods, such as Newton-Coates formula, Gao Si quadrature formula, composite Hermitian interpolation formula and so on.In general, Gao Si integral needs to have better smoothness of the integrable function and need to configure Gao Si nodes; Newtonkotz formula is attractive because of its flexible and convenient grid, but more interpolation nodes are needed to obtain higher convergence order.Therefore, different approximate calculation methods need to be explored for different practical problems.In this paper, the research status of Hadamard singular integral is introduced. On this basis, the calculation formula and error analysis of Hadamard singular integral based on cubic spline interpolation approximation are discussed. Numerical examples show the feasibility and effectiveness of the algorithm.The full text is divided into four chapters.The first chapter introduces the research status of hypersingular integral, the significance of the research and some developments at home and abroad, the second chapter, introduces the concept of Hadamard singular integral and the common interpolation formula.The calculation formula and error analysis of Hadamard singular integral based on cubic spline function interpolation are studied. The superconvergence of the method is proved theoretically, and the feasibility and effectiveness of the method are verified by an example.
【学位授予单位】：华北电力大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.83

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