Study on the numerical method for solving singular integral equation

Abstract

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.