主講人：梁慧 哈爾濱工業(yè)大學(xué)（深圳）教授
時間：2025年7月10日14：00
地點：徐匯校區(qū)三號樓332室
舉辦單位：數(shù)理學(xué)院
主講人介紹：梁慧，哈爾濱工業(yè)大學(xué)（深圳）理學(xué)院副院長、教授、博導(dǎo)。入選首屆“深圳市優(yōu)秀科技創(chuàng)新人才培養(yǎng)項目（杰出青年基礎(chǔ)研究）”，任期刊《Computational ＆ Applied Mathematics》《Communications on Analysis and Computation》和《中國理論數(shù)學(xué)前沿》的編委，中國仿真學(xué)會仿真算法專委會委員、中國仿真學(xué)會不確定性系統(tǒng)分析與仿真專業(yè)委員會秘書、廣東省計算數(shù)學(xué)學(xué)會常務(wù)理事、廣東省工業(yè)與應(yīng)用數(shù)學(xué)學(xué)會理事、深圳市數(shù)學(xué)學(xué)會常務(wù)理事。主要的研究方向為：延遲微分方程、Volterra積分方程的數(shù)值分析。主持國家自然科學(xué)基金、深圳市杰出青年基金、深圳市基礎(chǔ)研究計劃等10余項科研項目，獲中國系統(tǒng)仿真學(xué)會“優(yōu)秀論文”獎、黑龍江省數(shù)學(xué)會優(yōu)秀青年學(xué)術(shù)獎、深圳市海外高層次人才。目前已被SCI收錄文章40余篇，發(fā)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(nèi)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.