主讲人：梁慧 哈尔滨工业大学（深圳）教授
时间：2025年7月10日14：00
地点：徐汇校区三号楼332室
举办单位：数理学院
主讲人介绍：梁慧，哈尔滨工业大学（深圳）理学院副院长、教授、博导。入选首届“深圳市优秀科技创新人才培养项目（杰出青年基础研究）”，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、广东省计算数学学会常务理事、广东省工业与应用数学学会理事、深圳市数学学会常务理事。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、深圳市杰出青年基金、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“优秀论文”奖、黑龙江省数学会优秀青年学术奖、深圳市海外高层次人才。目前已被SCI收录文章40余篇，发表在SIAM J. Numer. Anal.、IMA J. Numer. Anal.、J. Sci. Comput.、BIT、Adv. Comput. Math.等20余种不同的国际杂志上。
内容介绍：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.
