This talk includes two parts. In the first part, we consider the high accuracy computation of Hadamard finite-part integrals. A general framework is constructed for efficiently and stably evaluating the Hadamard finite-part integrals by composite quadrature rules. Then a practical algorithm is designed for evaluating the Hadamard finite-part integrals by applying the Gauss-Legendre rule to the proposed framework. In the second part, we discuss the computation of singular nonlinear Volterra integral equations of the second kind. Firstly, we derive the general Puiseux expansion of the solution at the singularity by Picard iteration and series decomposition, which is the accurate measurement of the singular type and singular degree of the solution. Secondly, we use trapezoidal integration method to discretize the singular integral and derive the Euler-Maclaurin asymptotic expansion using the known Puiseux expansion of the solution. By accumulating some lower order error terms to the quadrature formula, we can obtain high accuracy evaluation to the nonlinear Volterra integral equations of the second kind.

王同科, 天津师范大学数学科学学院教授, 山东大学计算数学专业博士毕业, 主要从事偏微分方程数值解法和积分方程数值解法的教学和研究工作。目前关注奇异积分、奇异微分方程和奇异积分方程的高精度算法, 致力于刻画解在奇点的性质, 并由此设计高精度的数值算法。