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Title: Numerical approximations to hypersingular integrals and applications
Other Titles: Chao qi yi ji fen de shu zhi bi jin ji qi ying yong
Authors: Li, Buyang (李步揚)
Department: Department of Mathematics
Degree: Master of Philosophy
Issue Date: 2009
Publisher: City University of Hong Kong
Subjects: Integral equations -- Numerical solutions.
Singularities (Mathematics)
Approximation theory.
Notes: CityU Call Number: QA431 .L574 2009
i, 71 leaves 30 cm.
Thesis (M.Phil.)--City University of Hong Kong, 2009.
Includes bibliographical references (leaves [67]-71)
Type: thesis
Abstract: In this thesis, we investigate the composite Newton{Cotes rules in evaluating the following ¯nite-part (or hypersingular) integrals, I1(f; s) = F:P: Z b a f(x) (x ¡ s)2 dx; s 2 (a; b) ½ R1; I2(f; s; t) = F:P: ZZ ­ f(x; y) [(x ¡ s)2 + (y ¡ t)2]3=2 dxdy; (s; t) 2 ­ ½ R2; where s 2 (a; b) in 1D or (s; t) 2 ­ in 2D is called the singular point. The error estimate of the Newton{Cotes rules for hypersingular integrals was ¯rst studied by Linz, for the case of the singular point being an interior point of an element. Later, the composite trapezoidal rule and Simpson's rule for the singular point coincident with a mesh point were studied by Sun and Wu. Here we study the general composite Newton{Cotes rules in evaluating the hypersingular integrals, when the singular point coincides with a mesh point. We prove that the convergence rate of the kth order composite Newton{Cotes rule is of O(hkj ln hj) for the general case and O(hk+1j ln hj) when k is even and the singular point coincides with a point of element junction. Based on the analysis, we propose several modi¯ed Newton{ Cotes rules in terms of a local correction around the singular point. The modi¯ed methods improve the accuracy and has the convergence rate O(hk). Based on the composite Newton{Cotes rule and the superconvergence results presented by Sun and Wu, a class of collocation type methods are proposed for solving integral equations of hypersingular kernel. Finally, we generalize the Newton{Cotes rules and the modi¯ed methods to hypersingular integrals on a plane curve and on a 2D rectangular domain, respectively.
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