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湘潭大(da)学王冬岭教授(shou)的学术报告

发(fa)布时(shi)间: 2021年09月27日 作者: 王小捷   阅读次(ci)数: []

报告题目: Mittag-Leffler stability of numerical solutions to time fractional ODEs

报告人: 王冬岭(ling)教授 湘(xiang)潭大学 wdymath@nwu.edu.cn

报告时(shi)间: 2021年(nian)9月28日(ri) 9:50-12:50

报告地点: 腾讯会(hui)议(yi) 189 970 363

报告(gao)摘要(yao): The asymptotic stable region and long-time decay rate of solutions to linear homogeneous Caputo time fractional ordinary differential equations (F-ODEs) are known to be completely determined by the eigenvalues of the coefficient matrix. Very different from the exponential decay of solutions to classical ODEs, solutions of F-ODEs decay only polynomially, leading to the so-called Mittag-Leffler stability, which was already extended to semi-linear F-ODEs with small perturbations. This work is mainly devoted to the qualitative analysis of the long-time behavior of numerical solutions. By applying the singularity analysis of generating functions developed by Flajolet and Odlyzko (SIAM J. Disc. Math. 3 (1990), 216-240), we are able to prove that both L1 scheme and strong A-stable fractional linear multistep methods (F-LMMs) can preserve the numerical Mittag-Leffler stability for linear homogeneous F-ODEs exactly as in the continuous case. Through an improved estimate of the discrete fractional resolvent operator, we show that strong A-stable F-LMMs are also Mittag-Leffler stable for semi-linear F-ODEs under small perturbations. For the numerical schemes based on α-difference approximation to Caputo derivative, we establish the Mittag-Leffler stability for semi-linear problems by making use of properties of the Poisson transformation and the decay rate of the continuous fractional resolvent operator. Numerical experiments are presented for several typical time fractional evolutional equations, including time fractional sub-diffusion equations, fractional linear system and semi-linear F-ODEs. All the numerical results exhibit the typical long-time polynomial decay rate, which is fully consistent with our theoretical predictions.

Ref: Dongling Wang and Jun Zou. Mittag-Leffler stability of numerical solutions to time fractional ODEs. arXiv preprint arXiv:2108.09620, 2021.

王冬岭,副(fu)教授(shou),硕士生、博士生导师,博士毕业于湘潭大学(xue)计(ji)算(suan)数(shu)学(xue)系。主要从事(shi)动力系统保(bao)结构算(suan)法和分数(shu)阶(jie)微分方(fang)程(cheng)数(shu)值方(fang)法的研究。主持完成陕(shan)西(xi)(xi)省自(zi)科(ke)(ke)(ke)基金(jin)两(liang)项、主持完成国(guo)家(jia)自(zi)然科(ke)(ke)(ke)学(xue)基金(jin)天元基金(jin)和青年基金(jin);主持在研国(guo)家(jia)自(zi)然科(ke)(ke)(ke)学(xue)基金(jin)面上项目;参加(jia)在研国(guo)家(jia)自(zi)然科(ke)(ke)(ke)学(xue)基金(jin)重点项目。入选西(xi)(xi)北(bei)大学(xue)优秀(xiu)青年学(xue)术(shu)骨干计(ji)划(2015),陕(shan)西(xi)(xi)省科(ke)(ke)(ke)技新星(2018),获湖南省自(zi)然科(ke)(ke)(ke)学(xue)二(er)(er)等(deng)奖(2019),陕(shan)西(xi)(xi)省青年科(ke)(ke)(ke)技奖(2020)。多次到(dao)香港中(zhong)文大学(xue)做访问学(xue)者。已在SIAM J. Numer. Anal., Commun. Math. Sci., ESAIM: Math. Model. Numer. Anal., J. Comput. Phy., J. Sci. Comput.等(deng)国(guo)际主流的计(ji)算(suan)数(shu)学(xue)杂志发表论文二(er)(er)十余篇。



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