J. F. Gómez-Aguilar

Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel

In this paper, the fractional equations of the mass-spring-damper system with Caputo and Caputo–Fabrizio derivatives are presented. The physical units of the system are preserved by introducing an auxiliary parameter σ. The input of the resulting equations is a constant and periodic source; for the Caputo case, we obtain the analytical solution, and the resulting equations are given in terms of the Mittag–Leffler function; for the Caputo–Fabrizio approach, the numerical solutions are obtained by the numerical Laplace transform algorithm. Our results show that the mechanical components exhibit viscoelastic behaviors producing temporal fractality at different scales and demonstrate the existence of Entropy 2015, 17 6290 material heterogeneities in the mechanical components. The Markovian nature of the model is recovered when the order of the fractional derivatives is equal to one.

Bateman–Feshbach Tikochinsky and Caldirola–Kanai Oscillators with New Fractional Differentiation

In this work, the study of the fractional behavior of the Bateman–Feshbach–Tikochinsky and Caldirola–Kanai oscillators by using different fractional derivatives is presented. We obtained the Euler–Lagrange and the Hamiltonian formalisms in order to represent the dynamic models based on the Liouville–Caputo, Caputo–Fabrizio–Caputo and the new fractional derivative based on the Mittag–Leffler kernel with arbitrary order α. Simulation results are presented in order to show the fractional behavior of the oscillators, and the classical behavior is recovered when α is equal to 1.

Analytical Solutions of the Electrical RLC Circuit via Liouville–Caputo Operators with Local and Non-Local Kernels

In this work we obtain analytical solutions for the electrical RLC circuit model defined with Liouville–Caputo, Caputo–Fabrizio and the new fractional derivative based in the Mittag-Leffler function. Numerical simulations of alternative models are presented for evaluating the effectiveness of these representations. Different source terms are considered in the fractional differential equations. The classical behaviors are recovered when the fractional order α is equal to 1.

Electrical circuits RC, LC, and RL described by Atangana–Baleanu fractional derivatives

Summary In this paper, the analytical solutions for the electrical series circuits RC, LC, and RL using novel fractional derivatives of type Atangana–Baleanu with non-singular and nonlocal kernel in Liouville–Caputo and Riemann–Liouville sense were obtained. The fractional equations in the time domain are considered derivatives in the range α∈(0;1]; analytical solutions are presented considering different source terms introduced in the fractional equation. We solved analytically the fractional equation using the properties of Laplace transform operator together with the convolution theorem. On the basis of the Mittag–Leffler function, new behaviors for the voltage and current were obtained; the classical cases are recovered when α=1. Copyright

Schrödinger equation involving fractional operators with non-singular kernel

An alternative model of fractional Schrödinger equation involving Caputo-Fabrizio fractional operator and the new fractional operator based on the Mittag–Leffler function is proposed. We obtain the eigenvalues and eigenfunctions for a free particle moving in the infinite potential well. Numerical simulations of alternative models are presented for evaluating the effectiveness of these representations. We showed that fractional Schrödinger equation via Caputo–Fabrizio operator is a particular case of fractional Schrödinger equation obtained with the new fractional operator based in the Mittag–Leffler function.

Chaos in a Cancer Model via Fractional Derivatives with Exponential Decay and Mittag-Leffler Law

José Francisco Gómez-Aguilar 1,* ID , María Guadalupe López-López 2 ID , Victor Manuel Alvarado-Martínez 2, Dumitru Baleanu 3,4 and Hasib Khan 5,6,* 1 CONACyT-Tecnológico Nacional de Mexico/CENIDET, Interior Internado Palmira s/n Col. Palmira C.P., Cuernavaca 62490, Mexico 2 Tecnológico Nacional de Mexico/CENIDET, Interior Internado Palmira s/n Col. Palmira C.P., Cuernavaca 62490, Mexico; guadalupe@cenidet.edu.mx (M.G.L.-L.); alvarado@cenidet.edu.mx (V.M.A.-M.) 3 Department of Mathematics, Faculty of Art and Sciences, Cankaya University, Ankara 06790, Turkey; dumitru@cankaya.edu.tr 4 Institute of Space Sciences, P.O. Box, MG-23, Magurele-Bucharest R 76900, Romania 5 College of Engineering, Mechanics and Materials, Hohai University, Nanjing 210098, China 6 Department of Mathematics, Shaheed Benazir Bhutto University Sheringal, Dir Upper, Sheringal 18000, Pakistan * Correspondence: jgomez@cenidet.edu.mx (J.F.G.-A.); hasibkhan@sbbu.edu.pk (H.K.); Tel.: +52-777-3627770 (J.F.G.-A.); +92-321-9760796 (H.K.)

Series Solution for the Time-Fractional Coupled mKdV Equation Using the Homotopy Analysis Method

We present new analytical approximated solutions for the space-time fractional nonlinear partial differential coupled mKdV equation. A homotopy analysis method is considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding exact solutions of the fractional equation proposed, for the special case when the limit of the integral order of the time derivative is considered. The comparison shows a precise agreement between these solutions.

Design of a state observer to approximate signals by using the concept of fractional variable-order derivative

Abstract This article proposes a state observer to find a model for a given signal, i.e. to approximate a treated signal. The design of the state observer is based on a dynamical system of equations which is generated from the increasing-order differentiation of a n-th order Fourier series. This dynamical system is set in state space representation by considering that the Fourier series is the first state and the rest of the states are the successive derivatives of the series. The purpose of the state observer is the recursive estimation of the states in order to recover the coefficients from them. This set of coefficients produces the best fit between the dynamical system and the signal. The dynamical system used for the observer conception shall be, together with the estimated coefficients, the model that will describe the signal behavior. The special feature of the proposed observer is the order of the differential equations of the model on which it is based, d α ( t ) ν ( t ) / d t α ( t ) , which can take integer and non-integer values, i.e. α ( t ) ∈ ( 0 , 1 ] . Even more important, α ( t ) can be a smooth function such that α ( t ) ∈ ( 0 , 1 ] in the interval t ∈ [ 0 , T ] . The procedure to design the state observer of variable-order as well as some examples of its use in engineering applications are presented.

On the Possibility of the Jerk Derivative in Electrical Circuits

A subclass of dynamical systems with a time rate of change of acceleration are called Newtonian jerky dynamics. Some mechanical and acoustic systems can be interpreted as jerky dynamics. In this paper we show that the jerk dynamics are naturally obtained for electrical circuits using the fractional calculus approach with order . We consider fractional LC and RL electrical circuits with for different source terms. The LC circuit has a frequency dependent on the order of the fractional differential equation , since it is defined as , where is the fundamental frequency. For , the system is described by a third-order differential equation with frequency , and assuming the dynamics are described by a fourth differential equation for jerk dynamics with frequency .

Chaotic Attractors with Fractional Conformable Derivatives in the Liouville–Caputo Sense and Its Dynamical Behaviors

This paper deals with a numerical simulation of fractional conformable attractors of type Rabinovich–Fabrikant, Thomas’ cyclically symmetric attractor and Newton–Leipnik. Fractional conformable and β-conformable derivatives of Liouville–Caputo type are considered to solve the proposed systems. A numerical method based on the Adams–Moulton algorithm is employed to approximate the numerical simulations of the fractional-order conformable attractors. The results of the new type of fractional conformable and β-conformable attractors are provided to illustrate the effectiveness of the proposed method.