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

Gustavo Dorrego

TL;DR

The paper develops a unified spectral framework to model aging, heterogeneity, and multi-scale memory by introducing Weighted Weyl-Sonine Operators on the real line. It proves a Generalized Spectral Mapping Theorem showing that the Weighted Fourier Transform diagonalizes these operators, enabling direct algebraic solutions of nonlocal evolution equations via a spectral symbol $\Phi(s)$, with kernels in the Complete Bernstein Function class. The framework yields a generalized Marchaud representation and a robust connection between algebraic Sonine definitions and analytical Lévy-measure formulations, and it applies to generalized relaxation and weighted distributed-order diffusion, including ultra-slow/logarithmic diffusion and retarded aging. This approach provides explicit Green’s functions and spectral solvability results, offering a powerful tool for analyzing aging, heterogeneous media, and multi-scale memory in applications across physics and materials science.

Abstract

Standard fractional calculus has successfully modeled systems with power-law memory. However, complex phenomena in heterogeneous media often exhibit multi-scale memory effects and aging properties that classical operators cannot capture. In this work, we construct a unified framework by defining the \textit{Weighted Weyl-Sonine Operators}. This formalism offers a fundamental generalization of fractional calculus, freeing the theory from the constraints of power-law memory (via Sonine kernels), time-translation invariance (via scale and weight functions), and artificial history truncation (via Weyl integration). The main result is a Generalized Spectral Mapping Theorem, proving that the Weighted Fourier Transform acts as a universal diagonalization map for these operators. We rigorously characterize the admissible memory kernels through the class of \textit{Complete Bernstein Functions}, ensuring that the resulting operators preserve the fundamental properties of positivity and monotonicity. Furthermore, we establish a theoretical bridge between the algebraic Sonine definition and the analytical Marchaud representation involving Lévy measures. Finally, we apply this theory to solve generalized relaxation equations and \textit{Weighted Distributed Order} evolution problems, demonstrating that phenomena of ultra-slow diffusion and retarded aging can be treated explicitly within this unified spectral framework.

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

TL;DR

The paper develops a unified spectral framework to model aging, heterogeneity, and multi-scale memory by introducing Weighted Weyl-Sonine Operators on the real line. It proves a Generalized Spectral Mapping Theorem showing that the Weighted Fourier Transform diagonalizes these operators, enabling direct algebraic solutions of nonlocal evolution equations via a spectral symbol , with kernels in the Complete Bernstein Function class. The framework yields a generalized Marchaud representation and a robust connection between algebraic Sonine definitions and analytical Lévy-measure formulations, and it applies to generalized relaxation and weighted distributed-order diffusion, including ultra-slow/logarithmic diffusion and retarded aging. This approach provides explicit Green’s functions and spectral solvability results, offering a powerful tool for analyzing aging, heterogeneous media, and multi-scale memory in applications across physics and materials science.

Abstract

Standard fractional calculus has successfully modeled systems with power-law memory. However, complex phenomena in heterogeneous media often exhibit multi-scale memory effects and aging properties that classical operators cannot capture. In this work, we construct a unified framework by defining the \textit{Weighted Weyl-Sonine Operators}. This formalism offers a fundamental generalization of fractional calculus, freeing the theory from the constraints of power-law memory (via Sonine kernels), time-translation invariance (via scale and weight functions), and artificial history truncation (via Weyl integration). The main result is a Generalized Spectral Mapping Theorem, proving that the Weighted Fourier Transform acts as a universal diagonalization map for these operators. We rigorously characterize the admissible memory kernels through the class of \textit{Complete Bernstein Functions}, ensuring that the resulting operators preserve the fundamental properties of positivity and monotonicity. Furthermore, we establish a theoretical bridge between the algebraic Sonine definition and the analytical Marchaud representation involving Lévy measures. Finally, we apply this theory to solve generalized relaxation equations and \textit{Weighted Distributed Order} evolution problems, demonstrating that phenomena of ultra-slow diffusion and retarded aging can be treated explicitly within this unified spectral framework.
Paper Structure (18 sections, 7 theorems, 57 equations, 1 table)

This paper contains 18 sections, 7 theorems, 57 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and Functional Framework
  3. General Fractional Calculus with Sonine Kernels
  4. Weighted Spaces and Conjugation
  5. The Weighted Fourier Transform
  6. The Weighted Weyl-Sonine Framework and Spectral Analysis
  7. Admissible Kernels and Operators
  8. Spectral Analysis
  9. The Generalized Marchaud Representation
  10. Applications to Anomalous Relaxation and Diffusion
  11. Spectral Solvability of Generalized Relaxation Models
  12. Application: Weighted Distributed Order and Ultra-Slow Transitions
  13. Example: Retarded Aging via Logarithmic Diffusion
  14. Conclusions and Perspectives
  15. Well-posedness of Convolution Integrals
  16. ...and 3 more sections

Key Result

Theorem 2.8

Let $f$ be a function such that both $f$ and $\mathcal{F}_{\psi,\omega}f$ are integrable in the appropriate weighted sense. Then, the inverse transform is given pointwise by:

Theorems & Definitions (42)

  • Definition 2.1: Space $C_{-1}$
  • Definition 2.2: The Sonine Condition
  • Definition 2.3: General Fractional Operators
  • Definition 2.4: Generalized Sonine Pairs of Arbitrary Order
  • Remark 2.5
  • Remark 2.6: Transition to Weighted Weyl Operators
  • Definition 2.7
  • Theorem 2.8: Inversion Formula
  • proof
  • Remark 2.9
  • ...and 32 more