Source-linked AI summary
No Violation of the Leibniz Rule. No Fractional Derivative
Vasily E. Tarasov
TL;DR
The paper addresses whether fractional derivatives of non-integer order can satisfy the ordinary Leibniz rule. Using an algebraic approach based on linearity and the Leibniz rule, it proves that any such operator coincides with first-order differentiation, while noting limitations in general fractional-derivative properties.
Problem
Fractional derivatives have useful applications but exhibit unusual properties, including failure of the ordinary Leibniz rule, motivating attempts to construct Leibniz-rule-preserving alternatives.
Method
The paper uses an algebraic approach that considers fractional derivatives through linearity and the Leibniz rule, supported by Hadamard’s theorem.
Results
A linear operator of order α satisfying the Leibniz rule coincides with the derivative of first order, so fractional derivatives of non-integer orders cannot satisfy it.
Takeaways & Limitations
A derivative satisfying the ordinary Leibniz rule must have integer order, and non-integer fractional derivatives should instead obey a generalized Leibniz rule.
Takeaways & Limitations
The algebraic argument considers linearity and the Leibniz rule, while the paper notes that the relevant property is not satisfied by all types of fractional derivatives.
Abstract
from arXiv · showhide
We demonstrate that a violation of the Leibniz rule is a characteristic property of derivatives of non-integer orders. We prove that all fractional derivatives D^a, which satisfy the Leibniz rule D^(fg)=(D^a f) g + f (D^a g), should have the integer order a=1, i.e. fractional derivatives of non-integer orders cannot satisfy the Leibniz rule.
1 Introduction
Fractional derivatives model power-law nonlocality, long-term memory, and fractal behavior, but non-integer derivatives generally violate the Leibniz rule. The paper argues that this violation is characteristic and rules out non-integer order for any linear operator satisfying the ordinary Leibniz rule.
- Fractional derivatives and integrals are used to investigate systems with power-law nonlocality, long-term memory, or fractal properties.
- Several fractional-derivative definitions exist, including Riemann-Liouville, Riesz, Caputo, Grünwald-Letnikov, Marchaud, Weyl, and Sonin-Letnikov forms.
- For non-integer orders, the usual Leibniz rule is not satisfied; the Riemann-Liouville product formula instead involves an infinite series.For sufficiently large k, the series contains fractional-order integrals.
- These unusual properties create difficulties in physics and mechanics and have motivated attempts to define fractional derivatives that obey the Leibniz rule.
- The paper proves that a linear operator on C2(U) satisfying the Leibniz rule cannot have non-integer order, so such a derivative must have integer order.
2 Hadamard’s theorem
This section introduces continuously differentiable function spaces, the ordinary first derivative, and Hadamard’s theorem before establishing its stated representation result.
- Cm(U) denotes functions on U that are m times continuously differentiable.
- The usual first-order derivative is defined as d/dx: Cm(U) → Cm−1(U).
- Hadamard’s theorem states that any function f(x) in C1(U) near a point x0 has the representation introduced in the section.
- The proof considers an auxiliary function and concludes by establishing representation (2).
3 Algebraic approach to fractional derivatives
The paper studies linear operators of fractional order using only algebraic properties, especially R-linearity and the Leibniz rule. It proves that any such operator satisfying these conditions acts as the first derivative rather than a non-integer-order derivative.
- Assumptions: The algebraic approach assumes R-linearity, the Leibniz rule, and vanishing action on the unit for operators defined on C2(U).The argument does not depend on a special fractional-derivative definition.
- Theorem: The theorem concludes that an operator satisfying the assumptions is the derivative D^1_x of integer first order.Thus, under these conditions, the operator cannot retain a non-integer order.
- Proof strategy: The resulting operator has the form D^α_x = a(x)D^1_x, with a(x) a function on R1.The coefficient function is introduced through the algebraic derivation.
- Proof strategy: The proof applies Hadamard’s theorem and successive operator manipulations to derive the operator’s action on functions near x0.The derivation decomposes functions locally and uses the stated linearity conditions.
- Scope and qualification: The paper notes that the unit-function condition is not satisfied by every fractional-derivative type, including the Riemann-Liouville derivative in general.For the Riemann-Liouville derivative, D^α applied to x is not generally equal to one.
- Implications: The authors state that proposed fractional derivatives obeying the Leibniz rule cannot be fractional derivatives of non-integer order.They recommend a generalized Leibniz rule, generally represented by an infinite series, for fractional differentiation and integration.