Study on the numerical method for solving singular integral equation = 특이적분 방정식의 해를 구하는 수치적 방법에 관한 연구

The thesis is devoted to develop numerical methods for computing the Cauchy principal value integrals. It is concerned an integral Q(f;t) defined by \begin{displaymath} Q(f;t)=\int_{-1}^1\frac{f(\tau)}{\tau-t}d\tau= \lim_{\epsilon\to0}\big\{\int_{-1}^{t-\epsilon}+ \int_{t+\epsilon}^1\big\} \frac{f(\tau)}{\tau-t}d\tau , \qquad ltl<1 \end{displaymath} with a smooth function f. By making use of the change of variables $\tau$=cos y and t=cos x, we prove that the Cauchy integral Q(f;t) can be transformed by a standard integral of the form \begin{displaymath} Q(f(cos\cdot);cosx)=\int_0^{\pi} \frac{h(y)siny-h(x)sinx}{cosy-cosx}dy := Q(h;x), \quad say, \end{displaymath} where h(x)=f(cosx). Three algorithms for evaluating the integral Q(h;x) are described. One algorithm is based on a knowledge of the Fourier series expansion of h on [0, π], the other two algorithm on polynomial interpolations to h at the zeros of Nx and sin x sin Nx, respectively. Convergence theorems are given for each algorithm and Stability analysis for the two methods based on the polynomial interpolations are given. We prove that the last method has a uniform error bound independent of the set of pole values. This fact enables us to construct an automatic algorithm for evaluating the set of approximations {$Q_N$} for the integral $Q(h;x)$. In order to check potentialities of the proposed quadrature rules, we apply the methods to a solution of some singular integral equations occurred in analyzing a center cracked panel subjected to both normal and shear tractions.
Advisors
Choi, U-Jinresearcher최우진researcher
Publisher
한국과학기술원
Issue Date
2001
Identifier
166368/325007 / 000945123
Language
eng
Description

학위논문(박사) - 한국과학기술원 : 수학전공, 2001.2, [ vii, 75 p. ]

Keywords

trigonometric interpolation; quadrature rule; Cauchy singular integral; crack problem; 삼각함수 내삽법; 균열; 구분구적법; 특이적분

URI
http://hdl.handle.net/10203/41835
Link
http://library.kaist.ac.kr/search/detail/view.do?bibCtrlNo=166368&flag=t
Appears in Collection
MA-Theses_Ph.D.(박사논문)
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