本文選題：Hadamard奇異積分 + 三次樣條插值　； 參考：《華北電力大學(xué)》2017年碩士論文

【摘要】：聲學(xué)、電磁散射學(xué)、斷裂力學(xué)等諸多物理問題中都會廣泛涉及到Hadamard奇異積分計算問題。但是Hadamard奇異積分在普遍意義和主值意義下是發(fā)散的,這增加了研究的難度。多年來,人們致力于超奇異積分研究并給出了一些有效計算方法,如牛頓科茨型公式、高斯型求積公式、復(fù)合埃爾米特插值型公式等。通常,高斯積分需要被積函數(shù)有較好光滑性,并需要配置高斯節(jié)點;牛頓科茨公式由于靈活方便的網(wǎng)格而具有吸引力,不過要得到較高收斂階需要更多的插值節(jié)點。因此,針對不同的實際問題需要探尋不同的近似計算方法。本文介紹了Hadamard奇異積分的研究現(xiàn)狀,在此基礎(chǔ)上討論了基于三次樣條插值逼近的Hadamard奇異積分的計算公式及誤差分析,數(shù)值算例說明了該算法的可行性和有效性。全文共分四章。第一章,介紹了超奇異積分的研究狀況、研究意義及國內(nèi)外發(fā)展的一些動態(tài);第二章,介紹Hadamard奇異積分的概念及常見的插值求積分公式;第三章,研究基于三次樣條函數(shù)插值的Hadamard奇異積分計算公式和誤差分析,理論證明該方法的超收斂性,實例驗證了該方法的可行性和有效性;第四章,是全文的總結(jié)和今后的工作目標(biāo)。
[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.
【學(xué)位授予單位】：華北電力大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.83

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本文編號：1732343