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No Nonlocality. No Fractional Derivative
Vasily E. Tarasov
TL;DR
The paper asks how to distinguish genuine non-integer fractional derivatives from operators that merely re-label integer-order differential operators. It proposes a nonlocality principle and shows that several named fractional derivatives are reducible to finite integer-order equations and thus are not fractional derivatives of non-integer order.
Problem
The paper addresses how to identify proposed operators as genuine fractional derivatives of non-integer order rather than re-designations of familiar differential operators.
Method
The paper formulates a nonlocality principle by testing whether equations containing proposed fractional derivatives can be represented using finitely many integer-order derivatives.
Results
The conformable, alternative, M-, local, and exponential-kernel Caputo–Fabrizio derivatives cannot be considered fractional derivatives of non-integer order under the proposed principle.
Takeaways & Limitations
For these operators, reported results can be derived using integer-order differential operators, so they add no new effects in spaces of differentiable functions beyond changed notation.
Takeaways & Limitations
Nonlocality is not sufficient to classify an operator as a fractional differential operator, since nonlocal fractional integrals also exist.
Abstract
from arXiv · showhide
The paper discusses the characteristic properties of fractional derivatives of non-integer order. It is known that derivatives of integer orders are determined by properties of differentiable functions only in an infinitely small neighborhood of the considered point. Therefore differential equation, which is considered for this point and contains a finite number of integer-order derivatives, cannot describe nonlocality in space and time. This allows us to propose a principle of nonlocality for fractional derivatives. We state that if the differential equation with fractional derivative can be presented as a differential equation with a finite number of integer-order derivatives, then this fractional derivative cannot be considered as a derivative of non-integer order. This means that all results obtained for this type of fractional derivatives can be derived by using differential operators with integer orders. To illustrate the application of the nonlocality principle, we prove that the conformable fractional derivative, the M-fractional derivative, the alternative fractional derivative, the local fractional derivative and the Caputo-Fabrizio fractional derivatives with exponential kernels cannot be considered as fractional derivatives of non-integer orders.
Introduction
The introduction proposes principles for determining whether proposed operators are genuine fractional derivatives of non-integer order. It centers nonlocality: genuine fractional derivatives cannot be represented by a finite set of integer-order derivatives and therefore can describe spatial nonlocality or dynamic memory.
- Motivation: The paper questions whether operators labeled fractional derivatives are genuinely non-integer-order derivatives or merely re-designations of familiar integer-order differential operators.The need for identification principles becomes especially important when the distinction is not obvious.
- Nonlocality: Integer-order derivatives depend only on an infinitesimally small neighborhood, so finite-order integer differential equations cannot describe spatial nonlocal effects.The passage contrasts local determination near the considered point with nonlocal effects.
- Nonlocality: Fractional derivatives are presented as tools for modeling spatial nonlocality and dynamic memory because they cannot be represented by a finite set of integer-order derivatives.Dynamic memory links present behavior to states or input changes over a finite or infinite interval of the past.
- Identification principle: The proposed test asks whether a differential equation containing a fractional derivative can be rewritten as an equivalent equation with finitely many integer-order derivatives.Equivalence is defined on a function space by coincidence of the equations’ solutions there.
2. Principles of nonlocality for fractional derivatives
The section defines a nonlocality principle: a tested fractional derivative is not genuinely non-integer-order if its differential equation can be represented using finitely many integer-order derivatives. Such an equation cannot describe nonlocality or memory.
- Principles of nonlocality for fractional derivatives: A tested fractional derivative cannot be considered non-integer-order when its equation is representable as a linear differential equation containing only finitely many integer-order derivatives.The equivalence is considered where all required integer-order derivatives exist, including C^n([t0,t1]) or analytic functions.
- Principles of nonlocality for fractional derivatives: Because finite integer-order differential equations are independent of initial time and finitely many initial values, they cannot describe nonlocality and memory.The equation must be independent of t0 and the initial values of X, Y, and their derivatives at t0.
- Principles of nonlocality for fractional derivatives: The linear principle likewise rejects a tested fractional derivative if its linear equation can be represented using a finite number of integer-order derivatives.This principle applies when X and Y are analytic on (t0,t1) or belong to C^n([t0,t1]), with n=max{nI,mI}.
- Principles of nonlocality for fractional derivatives: A linear equation with genuinely non-integer-order fractional derivatives cannot be represented using polynomial differential operators P_n(D) with finite n, even when coefficients are non-constant.Here D=d/dt and n<∞.
3. Examples of application of nonlocality principles.
The section applies the nonlocality principle to several proposed fractional derivatives and shows that, for differentiable functions, their equations can be represented using finite-order integer derivatives. Consequently, these operators cannot be regarded as non-integer-order fractional derivatives within the principle’s scope.
- Overview: The conformable, local, and Caputo-Fabrizio derivatives are shown not to be non-integer-order fractional derivatives because their equations reduce to finite-order integer differential equations.Results obtained with these operators can therefore be derived using integer-order differential operators.
- Conformable fractional derivative: The conformable derivative is a first-order differential operator with variable coefficient a(t) = t1−α, so its equations reduce to first-order integer equations for differentiable functions.The equivalence does not address applications to non-differentiable continuous functions, which lie outside the principle’s scope.
- Alternative and M-fractional derivatives: The alternative and M-fractional derivatives satisfy the standard Leibniz rule and are first-order operators with variable coefficients, making their equations equivalent to integer-order equations.For the M-fractional derivative, the coefficient is a(t) = t1−α Γ(β + 1) ⁄ .
- Local fractional derivative: The local fractional derivative acts as an integer-order derivative or zero operator on differentiable functions, and its equations can therefore be represented as first-order integer equations.Applications to non-differentiable continuous functions are outside the proposed principle because the comparison assumes differentiability.
4. Conclusion
The conclusion identifies nonlocality as the defining property of fractional differential operators of non-integer order and finds several named derivatives reducible to finite integer-order operators. It also distinguishes genuine fractional derivatives from fractional integrals, which are nonlocal but not fractional differential operators.
- 4. Conclusion: The conformable, alternative, M-fractional, Kolwankar–Gangal local fractional, and Caputo–Fabrizio exponential-kernel derivatives cannot be considered non-integer-order fractional derivatives.The paper states that results for these operators can instead be derived using integer-order differential operators.
- 4. Conclusion: For analytic functions, Riemann–Liouville, Caputo, Hadamard, and Marchaud derivatives can be expressed through infinite series of integer-order derivatives.This infinite-series representation generally prevents their equations from being represented by finite-order integer differential equations.
- 4. Conclusion: A fractional operator that cannot generally be specified as a finite sum of integer-order derivatives exhibits an important sign of nonlocality.The conclusion applies this criterion to linear differential equations involving non-integer-order fractional derivatives.
- 4. Conclusion: Nonlocality is the characteristic property of fractional differential operators of non-integer order, whereas locality means equations can be described by integer-order differential equations.The conclusion equates an operator’s locality with representation by integer-order differential equations.
- 4. Conclusion: Not all nonlocal operators are fractional differential operators, because fractional integrals of non-integer order also exist.This statement marks a limitation on identifying nonlocality alone with fractional differentiation.