Spectral Analysis of Weighted Weyl Fractional Operators: Aging, Infinite Memory, and the Amnesia Effect

TL;DR

This paper develops a spectral framework for Weighted Weyl Fractional Calculus to model aging systems with infinite memory, revealing how rapid aging causes a loss of historical memory through an exponential relaxation effect.

Contribution

It introduces a new class of fractional operators with a spectral analysis framework, including a spectral mapping theorem and eigenfunctions, and explains the Amnesia Phenomenon in aging media.

Findings

Weighted Mittag-Leffler functions are eigenfunctions of the operators.

The Fourier Transform diagonalizes the evolution equations.

Rapid aging transforms power-law decay into exponential relaxation.

Abstract

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 $\omega(t)$ and a scale function $\psi(t)$, 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.