关键词: Cauchy型积分, 矩阵压缩策略, 数值积分公式

Abstract:

The Petrov-Galerkin method based on Jacobi polynomials is the conventional and standard numerical method for solving the Cauchy singular integral equations with constant coeffcients. This conventional numerical method leads to a linear system with a full coeffcient matrix. When the order of the linear system is large, the computational cost for obtaining and then solving the fully discrete linear system is huge. So in this paper the author develops a fast fully discrete Petrov-Galerkinmethod for solving this kind of integral equations. First compress this full coe?cient matrix into a sparse matrix. Then apply the numerical integration scheme to obtain the fully discrete truncated linear system with a nearly linear computational cost. At last, the fully discrete truncated linear system is solved. It is established that the optimal convergence order of the approximation solution remains optimal.

Key words: Cauchy singular integral equation, Matrix truncation strategy, Numerical inte-gration method, Multilevel augmentation method