Mittag-Leffler stability of complete monotonicity-preserving schemes for time-dependent coefficients sub-diffusion equations

TL;DR

The paper analyzes nonlinear time-fractional sub-diffusion with time-dependent coefficients, proving Mittag-Leffler stability for continuous solutions under natural decay assumptions on the source and a structural condition on the nonlinear operator. It then develops complete monotonicity-preserving ($\mathcal{CM}$-preserving) numerical schemes that retain the discrete comparison principle and match the continuous long-time decay rates. By constructing discrete sub- and super-solutions and employing a discrete energy framework, the authors establish numerical Mittag-Leffler stability for homogeneous, non-homogeneous, and time-fractional PDE cases, and validate the results with comprehensive numerical experiments. The work extends existing constant-coefficient and homogeneous analyses to variable coefficients and non-homogeneous sources, offering a robust, structure-preserving approach for accurately capturing long-time sub-diffusive dynamics in complex media.

Abstract

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.

Figures (2)

• Figure 1: Logarithm of numerical solutions $\log(U_{n})$ for $\gamma=1/4,1/2,1,2,4$ with $\beta=0.3$, $\alpha=0.4$ (left) and $\beta=-0.3$, $\alpha=0.8$ (right).
• Figure 2: Logarithm of numerical solutions $\log(V_{n})$ for $\beta=-0.3,-0.1,~0.1,~0.3,~0.6$, $\alpha=0.5$ (left) and $\beta=0.3$, $\alpha=0.1,~0.3,~0.5,~0.7,~0.9$ (right).

Theorems & Definitions (17)

• Lemma 2.1
• Lemma 2.2: Fractional comparison principle
• Theorem 2.1
• proof
• Corollary 2.1
• proof
• Example 3.1
• Example 3.2
• Lemma 4.1
• proof