報告題目：A general collocation analysis for weakly singular Volterra integral equations with variable exponent
報 告 人：梁慧教授
所在單位：哈爾濱工業大學
報告時間：2023年9月14日，13:30-14:30
報告地點：#騰訊會議：481-991-835
聯系人：zouyk@jlu.edu.cn
報告摘要: 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.
報告人簡介：梁慧，哈爾濱工業大學（深圳）教授、博導。2008年7月獲哈爾濱工業大學數學博士學位。2010.3.1-2011.9.31 在香港浸會大學擔任客座研究學人，并多次訪問香港浸會大學。2017.12.1-2018.11.30在加拿大紐芬蘭紀念大學(Memorial University of Newfoundland) 擔任訪問學者。任期刊《Computational & Applied Mathematics》、《Communications on Analysis and Computation》和《中國理論數學前沿》的編委，中國仿真學會仿真算法專委會委員、中國仿真學會不确定性系統分析與仿真專業委員會秘書、廣東省工業與應用數學學會理事。主要的研究方向為：延遲微分方程、Volterra積分方程的數值分析。主持國家自然科學基金面上項目、青年項目、深圳市傑出青年基金項目、深圳市基礎研究計劃等10餘項科研項目，獲中國系統仿真學會“2015年優秀論文”獎、2018第二屆黑龍江省數學會優秀青年學術獎、深圳市海外高層次人才認證。目前共被SCI收錄文章40餘篇，發表在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種不同的國際雜志上。
