A Unified Spectral Framework for Aging, Heterogeneous, and Distributed Order Systems via Weighted Weyl-Sonine Operators

TL;DR

This paper introduces a comprehensive spectral framework extending fractional calculus to the entire real line, enabling unified analysis of anomalous transport, wave mechanics, and aging processes in complex media.

Contribution

It develops the Weighted Weyl-Sonine framework and a spectral mapping theorem, unifying diverse physical regimes under a single harmonic analysis formalism.

Findings

Derived exact solutions for diffusive relaxation, inertial wave propagation, and retarded aging.

Unified description of anomalous transport and wave mechanics.

Established the Weighted Fourier Transform as a unitary diagonalization map.

Abstract

While General Fractional Calculus has successfully expanded the scope of memory operators beyond power-laws, standard formulations remain predominantly restricted to the half-line via Riemann-Liouville or Caputo definitions. This constraint artificially truncates the system's history, limiting the thermodynamic consistency required for modeling processes on unbounded domains. To overcome these barriers, we construct the \textbf{Weighted Weyl-Sonine Framework}, a generalized formalism that extends non-local theory to the entire real line without history truncation. Unlike recent algebraic approaches based on conjugation for finite intervals, we develop a rigorous harmonic analysis framework. Our central contribution is the \textbf{Generalized Spectral Mapping Theorem}, which establishes the Weighted Fourier Transform as a unitary diagonalization map for these operators. This result allows us to rigorously classify and solve distinct physical regimes under a single algebraic structure. We explicitly derive exact solutions for \textit{diffusive relaxation} (governed by Complete Bernstein Functions), \textit{inertial wave propagation} (exhibiting oscillatory dynamics), and \textit{retarded aging} (via distributed order), proving that our framework unifies the description of anomalous transport and wave mechanics in complex, time-deformed media.