Fahd Jarad

Caputo-type modification of the Hadamard fractional derivatives

Generalization of fractional differential operators was subjected to an intense debate in the last few years in order to contribute to a deep understanding of the behavior of complex systems with memory effect. In this article, a Caputo-type modification of Hadamard fractional derivatives is introduced. The properties of the modified derivatives are studied.

Fractional variational principles with delay

The fractional variational principles within Riemann–Liouville fractional derivatives in the presence of delay are analyzed. The corresponding Euler–Lagrange equations are obtained and one example is analyzed in detail.

On Caputo modification of the Hadamard fractional derivatives

This paper is devoted to the study of Caputo modification of the Hadamard fractional derivatives. From here and after, by Caputo-Hadamard derivative, we refer to this modified fractional derivative (Jarad et al. in Adv. Differ. Equ. 2012:142, 2012, p.7). We present the generalization of the fundamental theorem of fractional calculus (FTFC) in the Caputo-Hadamard setting. Also, several new related results are presented.

Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives

The existence and uniqueness theorems for functional right-left delay and left-right advanced fractional functional differential equations with bounded delay and advance, respectively, are proved. The continuity with respect to the initial function for these equations is also proved under some Lipschitz kind conditions. The Q-operator is used to transform the delay-type equation to an advanced one and vice versa. An example is given to clarify the results.

Razumikhin Stability Theorem for Fractional Systems with Delay

Fractional calculus techniques and methods started to be applied successfully during the last decades in several fields of science and engineering. In this paper we studied the stability of fractional-order nonlinear time-delay systems for Riemann-Liouville and Caputo derivatives and we extended Razumikhin theorem for the fractional nonlinear time-delay systems.

Fractional variational principles with delay within caputo derivatives

In this paper we investigate the fractional variational principles within Caputo derivatives in the presence of delay derivatives. The corresponding Euler-Lagrange equations are obtained for the case of one dependent variable. A generalization to n dependent variables is obtained. Physical example is analyzed in detail.

A semigroup-like Property for Discrete Mittag-Leffler Functions

AbstractDiscrete Mittag-Leffler function Eᾱ(λ,z) of order 0 < α ≤ 1, E1̄(λ,z)=(1-λ)-z, λ ≠ 1, satisfies the nabla Caputo fractional linear difference equation C∇0αx(t)=λx(t),x(0)=1,t∈ℕ1={1,2,3,…}. Computations can show that the semigroup identity Eᾱ(λ,z1)Eᾱ(λ,z2)=Eᾱ(λ,z1+z2) does not hold unless λ = 0 or α = 1. In this article we develop a semigroup property for the discrete Mittag-Leffler function Eᾱ(λ,z) in the case α ↑ 1 is just the above identity. The obtained semigroup identity will be useful to develop an operator theory for the discrete fractional Cauchy problem with order α ∈ (0, 1).

On the Stability of Some Discrete Fractional Nonautonomous Systems

Using the Lyapunov direct method, the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference is studied. The conditions for uniform stability, uniform asymptotic stability, and uniform global stability are discussed.

Fractional Sums and Differences with Binomial Coefficients

In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grunwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach.

Higher order fractional variational optimal control problems with delayed arguments

Abstract This article deals with higher order Caputo fractional variational problems in the presence of delay in the state variables and their integer higher order derivatives.