反常扩散记忆分解：特征时间尺度及其在逆问题中的应用
Anomalous diffusion memory factorization: Characteristic timescales and application to inverse problem
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中文总结 AI 辅助
本文针对描述复杂介质等中反常扩散的分数阶微分方程，提出解的时空分量分解方法，推导Mittag-Leffler函数特征时间尺度，将其应用于梳状结构热传导逆问题。
中文摘要 AI 辅助
复杂介质、分形及粘弹性材料中的输运模型产生的记忆效应与反常扩散可由分数阶微分方程描述，例如时间分数阶扩散方程 $D^\alpha_t u - \Delta_x u = 0$。本文将这类方程的解分解为空间分量与时间分量的乘积，对应长时间下的“冻结”特性。对于带Caputo导数的分数阶扩散，空间因子为初始数据的逆拉普拉斯算子 $(-\Delta)^{-1}$；对于带Riemann-Liouville导数的分数阶扩散，空间因子为初始数据的逆双拉普拉斯算子 $(-\Delta)^{-2}$。两种情况下，解的时间因子均为时间的缩放负幂次。该记忆分解显式编码了初始数据，为利用长时间测得的解值重构初始条件的时间逆问题提供了简单且鲁棒的方法。我们推导了对应反常扩散分解的Mittag-Leffler函数的特征时间尺度，这也能精确近似Mittag-Leffler函数的实零点数量。我们将上述结果应用于梳状结构上的热传导，该模型因梳状的分形结构表现出亚扩散特性。
英文摘要
Memory effects and anomalous diffusion arising in models of transport in complex media, fractals, and viscoelastic materials can be described by fractional-differential equations, such as the time-fractional diffusion equation $D^α_t u - Δ_x u = 0$. The paper develops a decomposition of solutions of these equations into a product of spatial and temporal components, corresponding to a freeze-out at long times. For fractional diffusion with the Caputo derivative the spatial factor is the inverse-Laplacian $(-Δ)^{-1}$ of the initial data, while for the Riemann-Liouville derivative it is the inverse bi-Laplacian $ (-Δ)^{-2}$. In both cases, the temporal factor of the solution is a scaled negative power of time. This memory artifact explicitly encodes the initial data, which gives a simple and robust way to reconstruct the initial conditions in the backward-in-time inverse problem with solution values measured at long times. We derive characteristic timescales for Mittag-Leffler functions, which correspond to such factorization in anomalous diffusion. This also enables an accurate approximation of the number of real zeros of the Mittag-Leffler function. We apply these results to heat transfer on a comb, a model which manifests subdiffusion arising from the comb's fractal structure.
