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

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 and a scale function , 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,…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsThermoelastic and Magnetoelastic Phenomena · Fractional Differential Equations Solutions · Stability and Controllability of Differential Equations
