[Paper Review] Sur une généralisation de l'opérateur fractionnaire

Thomas M. Michelitsch, Gérard A. Maugin+3 more|arXiv (Cornell University)|Nov 8, 2011

Elasticity and Wave Propagation5 references3 citations

TL;DR

This paper introduces a generalized derivative operator based on linear operator calculus that unifies traditional and fractional derivatives—such as Riemann-Liouville, Caputo, and Marchaud—by defining them as limiting cases of a functional calculus on translation operators. The key contribution is a mathematically rigorous framework that extends fractional calculus to Hölder-continuous functions, enabling the treatment of non-differentiable functions in mechanics of scale-free materials.

ABSTRACT

The goal of this communication is to propose a generalized notion of the "traditional derivative". This generalization includes the fractional derivatives such as the Riemann-Liouville, Gruenwald-Letnikov, Weyl, Riesz, Caputo, Marchaud derivatives and other variants as special cases. The approach is useful to describe mechanical problems in material systems with microstructures without characteristic scale such as self-similar and fractal materials.

Motivation & Objective

• To unify traditional and fractional derivatives under a single mathematical framework using operator calculus.
• To extend fractional calculus to functions that are Hölder-continuous but not necessarily differentiable.
• To define new classes of generalized derivatives applicable to non-local and long-range interaction systems in mechanics.
• To demonstrate the framework's consistency with known fractional derivatives, particularly Marchaud's.
• To provide a foundation for modeling mechanical behavior in self-similar, fractal-like materials.

Proposed method

• The method employs the translation operator $ T(h)f(x) = f(x+h) $, which satisfies $ T(h_1)T(h_2) = T(h_1 + h_2) $, and is represented as $ T(h) = e^{hD_x} $ for smooth functions.
• A generalized derivative $ \mathcal{D}_g f(x) $ is defined as $ \lim_{h \to 0} \frac{g(T(h))}{g(1+h)} f(x) $, where $ g(\lambda) $ is a generating function (the 'constituent') satisfying $ g(1) = 0 $.
• The framework generalizes fractional derivatives by choosing $ g(\lambda) \propto (\lambda - 1)^\alpha $, leading to the Marchaud derivative in the limit.
• The derivation uses asymptotic analysis of discrete difference operators and integral representations involving the Gamma function for non-integer $ \alpha $.
• The method ensures $ \mathcal{D}_g(\text{const}) = 0 $, preserving a key property of derivatives.
• It establishes a duality: if $ g(1) = 0 $, then $ \mathcal{D}_g $ is a generalized derivative; if $ g(1) \to \infty $, it defines a generalized integral.

Experimental results

Research questions

• RQ1 How can fractional derivatives be systematically unified under a single operator-theoretic framework?
• RQ2 Can generalized derivatives be defined for non-differentiable, Hölder-continuous functions?
• RQ3 What conditions ensure the convergence and consistency of the generalized derivative in the limit $ h \to 0 $?
• RQ4 How does the generalized derivative recover known fractional derivatives like Marchaud’s?
• RQ5 What are the implications for modeling long-range interactions in self-similar mechanical systems?

Key findings

• The generalized derivative $ \mathcal{D}_g f(x) $ reproduces the Marchaud fractional derivative when $ g(\lambda) = (\lambda - 1)^\alpha $, yielding $ D_x^\alpha f(x) = \frac{(-\alpha)}{(-\alpha)!} \int_{-\infty}^x (x-t)^{-(\alpha+1)} (f(t) - f(x)) dt $.
• The operator is well-defined for functions satisfying Hölder continuity with exponent $ \delta > \alpha $, where $ 0 < \alpha < \delta \leq 1 $, including non-differentiable functions.
• The framework ensures $ D_x^\alpha(\text{const}) = 0 $, preserving the fundamental property of derivatives.
• The exponential function $ e^{\lambda x} $ is an eigenfunction of the generalized derivative with eigenvalue $ \lambda^\alpha $, confirming consistency with known fractional calculus.
• The method reveals that the order of limits $ h \to 0 $ and $ k_0 \to \infty $ affects the result, explaining the existence of multiple definitions of fractional derivatives in the literature.
• The approach provides a unified operator-theoretic basis for fractional calculus, enabling the treatment of non-local and long-range interactions in self-similar materials.