报告题目：A general collocation analysis for weakly singular Volterra integral equations with variable exponent
报 告 人：梁慧教授
所在单位：哈尔滨工业大学
报告时间：2023年9月14日，13:30-14:30
报告地点：#腾讯会议：481-991-835
联系人：zouyk@jlu.edu.cn
报告摘要: Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature, where integral kernels of the form $(t-s)^{-\alpha}$ for some constant $\alpha \in (0,1)$ are considered. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng and Wang SIAM J. Numer. Anal. 2020 such a problem is transformed to a weakly singular VIE whose kernel has the above form with variable $\alpha = \alpha(t)$, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed, then used to underpin an analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel — it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.
报告人简介：梁慧，哈尔滨工业大学（深圳）教授、博导。2008年7月获哈尔滨工业大学数学博士学位。2010.3.1-2011.9.31 在香港浸会大学担任客座研究学人，并多次访问香港浸会大学。2017.12.1-2018.11.30在加拿大纽芬兰纪念大学(Memorial University of Newfoundland) 担任访问学者。任期刊《Computational & Applied Mathematics》、《Communications on Analysis and Computation》和《中国理论数学前沿》的编委，中国仿真学会仿真算法专委会委员、中国仿真学会不确定性系统分析与仿真专业委员会秘书、广东省工业与应用数学学会理事。主要的研究方向为：延迟微分方程、Volterra积分方程的数值分析。主持国家自然科学基金面上项目、青年项目、深圳市杰出青年基金项目、深圳市基础研究计划等10余项科研项目，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、深圳市海外高层次人才认证。目前共被SCI收录文章40余篇，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20种不同的国际杂志上。
