主講人：梁慧 哈爾濱工業大學（深圳）教授

時間：2025年7月10日14：00

地點：徐匯校區三號樓332室

舉辦單位：數理學院

主講人介紹：梁慧，哈爾濱工業大學（深圳）理學院副院長、教授、博導。入選首屆“深圳市優秀科技創新人才培養項目（杰出青年基礎研究）”，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、中國仿真學會不確定性系統分析與仿真專業委員會秘書、廣東省計算數學學會常務理事、廣東省工業與應用數學學會理事、深圳市數學學會常務理事。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、深圳市杰出青年基金、深圳市基礎研究計劃等10余項科研項目，獲中國系統仿真學會“優秀論文”獎、黑龍江省數學會優秀青年學術獎、深圳市海外高層次人才。目前已被SCI收錄文章40余篇，發表在SIAM J. Numer. Anal.、IMA J. Numer. Anal.、J. Sci. Comput.、BIT、Adv. Comput. Math.等20余種不同的國際雜志上。

內容介紹：The piecewise polynomial collocation method does not always work for Caputo fractional differential equations (FDEs), since it is related to the well-known Conjecture 6.3.5 in Brunner’s 2004 monograph on the convergence of the collocation solution for weakly singular Volterra integral equations (VIEs) of the first kind, and this is the reason why in the existing literature, the collocation method is not used directly to solve FDEs, but rather indirectly to solve the reformulated VIEs. The Bagley-Torvik equation is a typical representative of a class of FDEs, whose highest order derivative of the unknown function is an integer, and a Caputo derivative is also involved, and the characteristic with dominant integer order derivative allows us to use collocation methods directly to numerically solve the Bagley-Torvik equation. In this paper, the existence, uniqueness and regularity of the exact solution for the initial value problem of the Bagley-Torvik equation are given by virtue of the theory of VIEs, but the piecewise polynomial collocation method is used directly to solve the Bagley-Torvik equation, and the global convergence is derived on graded meshes and the pointwise error estimate is obtained on uniform meshes. Moreover, the global superconvergence of the collocation solution is also obtained without any postprocessing techniques. Unlike the indirect reformulated numerical methods, one has to resort to the iterated numerical solution to improve the numerical accuracy. Some numerical examples are given to illustrate the theoretical results, and it also shows that our analysis for the Bagley-Torvik equation can be extended to more general integer order derivative dominant FDEs, even for time fractional partial differential equation with this characteristic.