谱分析中的加权魏尔分数算子：衰老、无限记忆与遗忘效应
Spectral Analysis of Weighted Weyl Fractional Operators: Aging, Infinite Memory, and the Amnesia Effect
AI总结：
本文提出了一种加权魏尔分数算子的谱分析方法，用于建模具有衰老和无限记忆的非局部系统，并揭示了快速衰老导致记忆丧失的机制。
摘要：
本文建立了一个严格的谱框架，用于加权魏尔分数微积分，旨在建模表现出衰老和主观时间尺度的非局部系统。通过构造一个涉及时间依赖权重ω(t)和尺度函数ψ(t)的共轭映射，我们定义了一类新的分数算子，这些算子保持了时间不变系统的谱可处理性。我们推导了这些算子的谱映射定理，并证明加权 Mittag-Leffler 函数是其基本特征函数，证明加权傅里叶变换自然对角化了相关的演化方程。作为物理应用，我们为具有无限记忆的衰老黏弹性材料制定了一种本构定律。关键的是，我们分析地证明了“遗忘现象”：我们证明快速衰老调节系统的记忆，有效地将遗传幂律衰减转化为短程指数松弛。这一结果为快速衰老介质中的记忆丧失提供了闭合形式的解释，克服了标准离散化方法的计算瓶颈。
英文摘要：
This paper establishes a rigorous spectral framework for the Weighted Weyl Fractional Calculus, designed to model non-local systems exhibiting aging and subjective time scales. By constructing a conjugation map involving a time-dependent weight $ω(t)$ and a scale function $ψ(t)$, we define a new class of fractional operators that preserve the spectral tractability of time-invariant systems. We derive the Spectral Mapping Theorem for these operators and prove that Weighted Mittag-Leffler functions act as their fundamental eigenfunctions, demonstrating that the Weighted Fourier Transform naturally diagonalizes the associated evolution equations. As a physical application, we formulate a constitutive law for aging viscoelastic materials with infinite memory. Crucially, we analytically demonstrate the "Amnesia Phenomenon": we prove that rapid aging modulates the system's history, effectively transforming the hereditary power-law decay into a short-range exponential relaxation. This result provides a closed-form explanation for the loss of memory in fast-aging media, overcoming the computational bottlenecks of standard discretization methods.
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