»ã±¨±êÌâ (Title)£ºOptimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations £¨·ÇÏßÐÔ¹¦·ò·ÖÊý½×·¢Õ¹·½³ÌµÄÆëÈ«±£µ¥µ÷ÐÔÌåʽ½âµÄ×îÓų¤¹¦·òË¥¼õÂÊ£©

»ã±¨ÈË (Speaker)£º Martin Stynes ½ÌÊÚ£¨±±¾©ÍÆËã¿Æѧ×êÑÐÖÐÐÄ£©

»ã±¨¹¦·ò (Time)£º2023Äê5ÔÂ8ÈÕ(ÖÜÒ») 15£º00-16£º30

»ã±¨µØÖ· (Place)£ºÐ£±¾²¿F309

Ô¼ÇëÈË(Inviter)£º²ÌÃô¡¢ÀƷ

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»ã±¨ÌáÒª£ºThe 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.