On some new properties of fractional derivatives with Mittag-Leffler kernel

Abstract

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.

arXiv comment22 pages; accepted for publication in Communications in Nonlinear Science and Numerical Simulation

References in corpus (5)

• Fractional Vector Calculus and Fractional Maxwell's Equations
• No Violation of the Leibniz Rule. No Fractional Derivative
• On a differential equation with Caputo-Fabrizio fractional derivative of order $1<\beta \le 2$ and application to mass-spring-damper system
• A Note on Non-Locality and Ostrogradski's Construction
• Exponential quadrature rules for linear fractional differential equations

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• On some analytic properties of tempered fractional calculus
• A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators
• Analysis of Nonlinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative
• A complex analysis approach to Atangana-Baleanu fractional calculus
• A class of fractional differential equations via power non-local and non-singular kernels: existence, uniqueness and numerical approximations
• An elliptic regularity theorem for fractional partial differential operators
• On a new definition of fractional differintegrals with Mittag-Leffler kernel
• Lyapunov Functions and Stability Analysis of Fractional-Order Systems
• A note on fractional Askey--Wilson integrals
• Generalized Taylor's formula for power fractional derivatives
• Properties of fractional integral operators involving the three-parameters Mittag-Leffler function in the kernels with respect to another function