论文标题
非线性时间分节性进化方程的完整单调性保留方案的最佳长期衰减率
Optimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations
论文作者
论文摘要
非线性初始值问题的解决方案$ \ MATHCAL {d} _ {t}^αy（t）= - λy（t）^γ$ for $ y（0）> 0 $，$ y（0）> 0 $，其中$ \ Mathcal {d} _ {d} _ {t} _ {t}^a $ caputo $ caputo $ caputo $ caputo $ and $ caputivity $ capta γ$是正参数，已知$ o（t^{α/γ}）$衰减为$ t \ to \ infty $。以前没有证明对此问题的任何离散化的相应结果。在本文中，显示出，对于一系列完整的单调性能（$ \ Mathcal {cm} $ - 保存）方案（包括均匀网状网格的L1和Grünwald-LeTnikov计划） $ O（t_ {n}^{ - α/γ}）$衰减为$ t_ {n} \ to \ infty $。然后将此结果扩展到$ \ MATHCAL {CM} $ - 保留某些时间折痕非线性亚扩散问题的离散性，例如时间折痕多孔介质和$ p $ -laplace方程。对于L1方案，$ O（t_ {n}^{ - α/γ}）$衰减结果显示在非常一般的非均匀网格类别上保持有效。我们的分析使用了离散的比较原理与离散的下市和超溶液，这些原理经过精心构造，以在离散解决方案上构成紧密的界限。提供数值实验以确认我们的理论分析。
The solution of the nonlinear initial-value problem $\mathcal{D}_{t}^αy(t)=-λy(t)^γ$ for $t>0$ with $y(0)>0$, where $\mathcal{D}_{t}^α$ is a Caputo derivative of order $α\in (0,1)$ and $λ, γ$ are positive parameters, is known to exhibit $O(t^{α/γ})$ decay as $t\to\infty$. No corresponding result for any discretisation of this problem has previously been proved. In the present paper it is shown that for the class of complete monotonicity-preserving ($\mathcal{CM}$-preserving) schemes (which includes the L1 and Grünwald-Letnikov schemes) on uniform meshes $\{t_n:=nh\}_{n=0}^\infty$, the discrete solution also has $O(t_{n}^{-α/γ})$ decay as $t_{n}\to\infty$. This result is then extended to $\mathcal{CM}$-preserving discretisations of certain time-fractional nonlinear subdiffusion problems such as the time-fractional porous media and $p$-Laplace equations. For the L1 scheme, the $O(t_{n}^{-α/γ})$ 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.
