报告学者：Martin Stynes

报告者单位：北京计算科学研究中心

报告时间：7月3日14:00

报告地点：SX106

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 using novel techniques. These results then underpin an error 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.

报告学者简介：Martin Stynes，在美国俄勒冈州立大学获得博士学位，曾在爱尔兰科克大学任教，现为北京计算科学研究中心教授，入选国家级人才项目。Martin Stynes教授的主要研究领域是分数阶微分方程及奇异摄动微分方程的数值解法。他与H.-G.Roos及L.Tobiska共同编撰的关于奇异摄动微分方程的数值方法的图书（1996年第1版，2008年第2版），是这一领域的主要参考资料。Martin Stynes教授曾担任工业与应用数学学会（SIAM）英国及爱尔兰分会主席及SIAM Journal on Numerical Analysis期刊编委。目前为Advances in Computational Mathematics等期刊编委。