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On some new properties of fractional derivatives with Mittag-Leffler kernel
Dumitru Baleanu, Arran Fernandez
TL;DR
Fractional derivatives with Mittag-Leffler kernels require a more developed theory for non-local systems and applications. The paper derives series representations in Riemann–Liouville integrals, studies fractional ODEs and operator rules, and finds that the semigroup property almost never holds. It also identifies extensions toward more complicated differential equations and higher-order derivatives.
Problem
The paper addresses the need to develop the ground-level theory of Mittag-Leffler-kernel fractional calculus for modeling systems with memory and applications such as control and variational principles.
Method
The paper derives series formulas for AB derivatives and applies them to fractional ODEs, semigroup analysis, and extensions of product and chain rules.
Results
The semigroup property for Mittag-Leffler-kernel fractional integrals and derivatives almost never holds under the new definition.
Takeaways & Limitations
The series formula clarifies non-locality, supports direct operator proofs, and can facilitate numerical treatment through finite truncation and Riemann–Liouville methods.
Takeaways & Limitations
The paper leaves extensions to more complicated differential equations and higher-order derivatives α > 1 for future work.
Abstract
from arXiv · showhide
We establish a new formula for the fractional derivative with Mittag-Leffler kernel, in the form of a series of Riemann-Liouville fractional integrals, which brings out more clearly the non-locality of fractional derivatives and is easier to handle for certain computational purposes. We also prove existence and uniqueness results for certain families of linear and nonlinear fractional ODEs defined using this fractional derivative. We consider the possibility of a semigroup property for these derivatives, and establish extensions of the product rule and chain rule, with an application to fractional mechanics.
1 Introduction
The paper introduces Mittag-Leffler-kernel fractional derivatives as non-local, non-singular operators and develops their foundational theory, including a series representation and results for fractional ODEs.
- Fractional calculus extends differentiation and integration beyond integer orders to real and complex orders.
- Existing applications of fractional calculus include control, variational principles, viscoelasticity, chaotic systems, thermoelasticity, vibration, diffusion, and bioengineering.
- Fractional operators model systems with memory because they are non-local, unlike ordinary derivatives, which are local operators.
- Earlier Caputo–Fabrizio derivatives addressed nonsingular kernels and had applications including diffusion modelling and mass-spring-damper systems.
- The Mittag-Leffler-kernel calculus uses a non-local, non-singular kernel to describe dynamics of non-local complex systems.
- The paper addresses incomplete foundational theory by developing a series formula, solving linear and nonlinear fractional ODEs, examining semigroup properties, and extending product and chain rules.
2 A new formula for the fractional derivative with Mittag-Leffler kernel
The paper develops series representations for ABR and ABC fractional derivatives using Riemann–Liouville integrals, then uses them to establish operator properties and computationally relevant consequences.
- Theorem 2.1 expresses the ABR fractional derivative as a locally uniformly convergent series of Riemann–Liouville fractional integrals.
- The series formulation makes the operators’ non-locality more explicit and avoids direct manipulation of the transcendental Mittag-Leffler function.
- Finite truncation combined with standard Riemann–Liouville numerical methods provides an approximation route for AB derivatives.
- The series formula supports direct proofs of Laplace-transform identities, inverse relations, commutativity, and a Newton–Leibniz formula under stated assumptions.
- The ABR semigroup property almost never holds, although conditions for its validity can be identified.
- Theorem 2.2 gives an analogous series expression for the ABC fractional derivative.
3 Some ordinary differential equations
The paper applies its AB series and transform methods to establish unique solutions for broad linear fractional ODE families and explicit solutions for selected nonlinear equations.
- Linear ODEs of Riemann–Liouville type: Laplace-transform methods yield unique Laplace-transformable solutions for basic ABR fractional ODEs.The solution can be expressed through auxiliary functions depending linearly on the forcing function and initial values.
- Linear ODEs of Riemann–Liouville type: Repeated application extends the construction to sequential linear fractional ODEs with constant coefficients, producing nested integral formulas.The forcing function must be Laplace-transformable, with fractional orders in (0, 1).
- Linear ODEs of Caputo type: For ABC derivatives, analogous linear ODEs also have unique Laplace-transformable solutions, with the main difference appearing in the initial-condition terms.The paper states that the ABC and ABR linear results differ only in how initial conditions enter the solutions.
- Nonlinear ODEs by Laplace methods: Laplace methods solve certain nonlinear ODEs by reducing transformed equations to algebraic problems and then inverting the resulting expressions.For one class, the transformed equation becomes quadratic in the transformed unknown, while the right-hand side depends on the forcing function and initial value.
- Nonlinear ODEs using the series formula: The series formula enables a convergent-series solution for a fractional Riccati equation, extending earlier Caputo-context results to the AB model.The derivative and nonlinear right-hand side become matching double series, allowing coefficient identities to determine the solution.
4 The semigroup property
The AB fractional differintegrals generally lack the semigroup property, but the paper derives conditions under which it can hold and reduces those conditions to fractional equations.
- Failure of the semigroup property: The semigroup property for AB fractional differintegrals is not satisfied in general.With B(α) = 1, composition leads to a nontrivial fractional differential equation that characterizes the relevant functions.
- Reduction to fractional integrals: It is sufficient to study AB fractional integrals, because functions satisfying the derivative semigroup property correspond to functions satisfying the integral property.The integral formulation is described as simpler and easier to work with than ABR derivatives.
- Conditions for validity: The integral semigroup property is equivalent to conditions involving AB and Riemann–Liouville fractional integrals.Under B(α)B(β) = B(α + β), these conditions simplify to a Riemann–Liouville fractional integral equation.
- Conditions for validity: Using composition properties, the paper derives a necessary condition for the semigroup property as a Riemann–Liouville fractional differential equation.The condition can be rewritten in a more elegant form when the normalization function itself has the semigroup property.
- Explicit special cases: For equal fractional orders, the indicial polynomial becomes easier to solve, allowing solutions to be constructed from incomplete gamma functions.The general rational-order case instead requires finding roots of the indicial polynomial and may not have neat solutions.
5 The product rule
The paper extends the Leibniz product rule to ABR fractional derivatives using a convergent series representation, expanding the class of functions whose derivatives can be computed.
- 5 The product rule: The ABR product rule generalizes the classical Leibniz rule for suitable functions and fractional orders.The identity applies for α ∈ (0, 1) under the stated analyticity and integrability conditions.
- 5 The product rule: The product-rule series is rigorously justified by decomposing it into finite and remainder terms whose remainder vanishes.Local uniform convergence permits interchange of summations in the proof.
- 5 The product rule: An example verifies the generalized identity using u(t) = t^2 and v(t) = t with a = 0.
- 5 The product rule: The product rule is a key test for fractional-calculus models and is important for their applications.Its role is connected to the broader significance of Leibniz-type identities in fractional calculus.
- 5 The product rule: The generalized rule enables computation of AB derivatives for products of functions whose individual AB derivatives are already known.Although cumbersome, the resulting expression remains computationally manageable.
6 The chain rule
The paper derives a generalized chain rule for AB fractional derivatives from the series representation, establishes convergence, and applies it to fractional dynamical systems and mechanics.
- 6 The chain rule: The chain-rule identity extends Osler’s Riemann–Liouville result to ABR fractional derivatives for suitable composite functions.The corresponding ABR identity applies when f(g(t)) is L1 and the stated regularity conditions hold.
- 6 The chain rule: The resulting series is well-defined because the inner series converge and the outer series converges locally uniformly.These convergence properties justify interchanging the sums.
- 6 The chain rule: For f(t) = t^2 and g(t) = e^t, the composite function is e^(2t), providing an explicit verification example.Only the first two derivatives of f with respect to g(t) are nonzero in this example.
- 6 The chain rule: The chain rule expands the class of functions with readily computable AB derivatives from known derivatives under composition.The resulting expression is more cumbersome than the product-rule expression but remains feasible to compute.
- 6 The chain rule: The chain rule generalizes earlier fractional mechanics constructions to a larger domain and supports a new fractional mechanics.The application uses fractionalized Lagrangians and Euler–Lagrange equations involving non-local dynamics.
7 Conclusions
The paper develops foundational theory for AB differintegrals with Mittag–Leffler kernels, including a series formula, ODE solutions, semigroup conditions, and product and chain rules.
- 7 Conclusions: The new ABR and ABC derivative formula expresses the operators as infinite series of Riemann–Liouville fractional integrals.This avoids direct manipulation of the transcendental Mittag–Leffler function and clarifies non-locality.
- 7 Conclusions: The paper constructs solutions for certain relatively simple linear and nonlinear fractional ordinary differential equations.These results provide an initial basis for broader equations defined with Mittag–Leffler kernels.
- 7 Conclusions: The semigroup property almost never holds for the new Mittag–Leffler-kernel definition, despite conditions under which it can hold.
- 7 Conclusions: Extensions of the product and chain rules support numerical computation of many specific AB derivatives and an application to fractional dynamical systems.The chain rule is also used to construct a new fractional mechanics.
- 7 Conclusions: These rigorously proved results form part of the foundations for AB differintegrals and may support later theory and more advanced fractional ODEs.
- 7 Conclusions: The results currently leave higher-order derivatives with α > 1 and more complicated differential equations as directions for extension.The paper identifies exact analytic solutions for more complicated equations as a desired future direction.