A general collocation analysis for weakly singular Volterra integral equations with variable exponent

主讲人：梁慧

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. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.

主讲人简介：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SCI期刊Computational & Applied Mathematics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。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、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。目前共被SCI收录文章30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18种不同的国际杂志上。

邀请人：黄乘明

时间：2022年6月28日（星期二）10:00-12:00

地点：腾讯会议室     ID:865221688