报告人: 梁慧 教授（哈尔滨工业大学（深圳））

报告题目: Integro-differential algebraic equations: theory of collocation methods

报告摘要: The notion of the tractability index based on the $\nu$-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It is used to decouple the given IDAE system of index 1 into the inherent system of regular Volterra integro-differential equations (VIDEs) and a system of second-kind Volterra integral equations (VIEs). This decoupling of the given general IDAE forms the basis for the convergence analysis of the two classes of piecewise polynomial collocation methods for solving the given index-1 IDAE system. The first one employs the same continuous piecewise polynomial space $S_m^{(0)}$ for both the VIDE part and the second-kind VIE part of the decoupled system. In the second one the VIDE part is discretized in $S_m^{(0)}$ but the second-kind VIE part employs the space of discontinuous piecewise polynomials $S_{m-1}^{(-1)}$. The optimal orders of convergence of these collocation methods are derived. For the first method, the collocation solution converges uniformly to the exact solution if and only if the collocation parameters satisfy a certain condition. This condition is no longer necessary for the second method: the collocation solution now converges to the exact solution for any choice of the collocation parameters. Numerical examples illustrate the theoretical results.

报告人简介：梁慧，哈尔滨工业大学（深圳）理学院教授。任SCI期刊Computational & Applied Mathematics编委、中国仿真学会仿真算法专委会委员、黑龙江省数学会理事。主要的研究方向为：延迟微分方程、Volterra积分方程、分数阶微分方程的数值分析。主持国家自然科学基金、青年基金、黑龙江省普通本科高等学校青年创新人才培养计划等9项科研项目，获中国系统仿真学会“2015年优秀论文”奖、2018第二届黑龙江省数学会优秀青年学术奖。目前共被SCI收录文章26篇，其中JCR一区15篇，发表在SIAM Journal on Numerical Analysis、IMA Journal of Numerical Analysis、Journal of Scientific Computing、BIT Numerical Mathematics、Advances in Computational Mathematics、Applied Numerical Mathematics 等15种不同的国际杂志上。

报告时间: 2019年6月15日（星期六）下午4:00

报告地点: 科技楼南楼702室