报告题目：Mittag-Leffler stability of numerical solutions to time fractional ODEs
报告人：王冬岭教授 湘潭大学 firstname.lastname@example.org
报告地点：腾讯会议 189 970 363
报告摘要：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，副教授，硕士生、博士生导师，博士毕业于湘潭大学计算数学系。主要从事动力系统保结构算法和分数阶微分方程数值方法的研究。主持完成陕西省自科基金两项、主持完成国家自然科学基金天元基金和青年基金；主持在研国家自然科学基金面上项目；参加在研国家自然科学基金重点项目。入选西北大学优秀青年学术骨干计划(2015)，陕西省科技新星(2018)，获湖南省自然科学二等奖(2019)，陕西省青年科技奖(2020)。多次到香港中文大学做访问学者。已在SIAM J. Numer. Anal., Commun. Math. Sci., ESAIM: Math. Model. Numer. Anal., J. Comput. Phy., J. Sci. Comput.等国际主流的计算数学杂志发表论文二十余篇。