报告题目 (Title)：Optimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations （非线性时间分数阶发展方程的完全保单调性格式解的最优长时间衰减率）

报告人 (Speaker)： Martin Stynes 教授（北京计算科学研究中心）

报告时间 (Time)：2023年5月8日(周一) 15：00-16：30

报告地点 (Place)：校本部F309

邀请人(Inviter)：蔡敏、李常品

主办部门：理学院数学系

报告摘要：The solution of the nonlinear initial-value problem $\mathcal{D}_{t}^{\alpha}y(t)=-\lambda y(t)^{\gamma}$ for $t>0$ with $y(0)>0$, where $\mathcal{D}_{t}^{\alpha}$ is the Caputo derivative of order $\alpha\in (0,1)$ and $\lambda, \gamma$ are positive parameters, is known to exhibit $O(t^{-\alpha/\gamma})$ decay as $t\to\infty$. No corresponding result for any discretisation of this problem has previously been proved. We shall show that for the class of complete monotonicity-preserving ($\mathcal{CM}$-preserving) schemes (which includes the L1 and Gr\"unwald-Letnikov schemes) on uniform meshes $\{t_n:=nh\}_{n=0}^\infty$, the discrete solution also has $O(t_{n}^{-\alpha/\gamma})$ decay as $t_{n}\to\infty$. For the L1 scheme, the $O(t_{n}^{-\alpha/\gamma})$ decay result is shown to remain valid on a very general class of nonuniform meshes. Our analysis uses a discrete comparison principle with discrete subsolutions and supersolutions that are carefully constructed to give tight bounds on the discrete solution. Numerical experiments are provided to confirm our theoretical analysis. This is joint work with Dongling Wang of Xiangtan University.