含时间依赖系数的次扩散方程的完全单调保持格式的 Mittag-Leffler 稳定性
Mittag-Leffler stability of complete monotonicity-preserving schemes for time-dependent coefficients sub-diffusion equations
AI总结：
本文针对变系数幂非线性次扩散模型，证明了其解具有 Mittag-Leffler 稳定性，并利用完全单调保持格式及离散比较原理，通过构造上下解获得了与理论解一致的数值解长时间最优代数衰减率。
摘要：
反常次解方程的一个关键特征是解在长时间区间上呈现代数衰减率，这通常被称为 Mittag-Leffler 型稳定性。对于一类具有变系数的幂非线性次扩散模型，我们证明当源函数满足自然衰减假设时，其解具有 Mittag-Leffler 稳定性。即解具有衰减率 $\\|u(t)\\|_{L^{s}(Ω)}=O\left( t^{-(α+β)/γ} \right)$，其中 $t\rightarrow\infty$，$α$、$γ$ 为正的常数，$β\in(-α,\infty)$，$s\in (1,\infty)$。然后我们为这类模型开发了结构保持算法。对于 Li 和 Wang（Commun. Math. Sci., 19(5):1301-1336, 2021）提出的完全单调保持（$\mathcal{CM}$-preserving）格式，我们证明了它们满足变系数时间分数阶微分方程的离散比较原理。进而，通过精心构造离散上解和下解，我们得到了数值解的长时间最优衰减率 $\\|u_{n}\\|_{L^{s}(Ω)}=O\left( t_n^{-(α+β)/γ} \right)$，其中 $t_{n}\rightarrow\infty$，这与理论解完全一致。最后，我们通过数值实验验证了分析结果。
英文摘要：
A key characteristic of the anomalous sub-solution equation is that the solution exhibits algebraic decay rate over long time intervals, which is often refered to the Mittag-Leffler type stability. For a class of power nonlinear sub-diffusion models with variable coefficients, we prove that their solutions have Mittag-Leffler stability when the source functions satisfy natural decay assumptions. That is the solutions have the decay rate $\|u(t)\|_{L^{s}(Ω)}=O\left( t^{-(α+β)/γ} \right)$ as $t\rightarrow\infty$, where $α$, $γ$ are positive constants, $β\in(-α,\infty)$ and $s\in (1,\infty)$. Then we develop the structure preserving algorithm for this type of model. For the complete monotonicity-preserving ($\mathcal{CM}$-preserving) schemes developed by Li and Wang (Commun. Math. Sci., 19(5):1301-1336, 2021), we prove that they satisfy the discrete comparison principle for time fractional differential equations with variable coefficients. Then, by carefully constructing the fine the discrete supsolution and subsolution, we obtain the long time optimal decay rate of the numerical solution $\|u_{n}\|_{L^{s}(Ω)}=O\left( t_n^{-(α+β)/γ} \right)$ as $t_{n}\rightarrow\infty$, which is fully agree with the theoretical solution. Finally, we validated the analysis results through numerical experiments.
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