In this paper, it is considered that chaos synchronization of Chen fractional order time-delay chaotic systems and its application in secure communication. By use of the fractional calculus techniques, we find that chaos exists in the fractional order Chen system with time delays when the order is less than 3. Firstly, based on the Laplace transform theory, the conditions for achieving synchron...

Traditional continuous-time filters are of integer order. However, using fractional calculus, filters may also be represented by the more general fractional-order differential equations in which case integer-order filters are only a tight subset of fractional-order filters. In this work, we show that low-pass, high-pass, band-pass, and all-pass filters can be realized with circuits incorporatin...

Abstract Fractional calculus is a branch of classical mathematics, which deals with the generalization of operations of differentiation and integration to fractional order. Such a generalization is not merely a mathematical curiosity but has found applications in various fields of physical sciences. In this paper, we review the definitions and properties of fractional derivatives and integrals,...

Abstract The fractional wave equation is presented as a generalization of the wave equation when arbitrary fractional order derivatives are involved. We have considered variable dielectric environments for the wave propagation phenomena. The Jumarie’s modified Riemann-Liouville derivative has been introduced and the solutions of the fractional Riccati differential equation have been applied to ...

Aiming at the characteristics of permanent magnet synchronous motor (PMSM) such as strong coupling and nonlinearity, we propose a fractional-order control scheme for the speed control of PMSM system based on the analysis of the fractional calculus theory. The simulation is conducted by using Matlab / Simulink. Simulation results indicate that the fractional-order PI controller can improve the d...

We numerically investigate the dynamical behavior of the fractional-order Chua’s system. By utilizing the multistep generalized differential transform method MSGDTM , we find that the fractional-order Chua’s system with “effective dimension” less than three can exhibit chaos as well as other nonlinear behavior. Numerical results are presented graphically and reveal that the multistep generalize...

We obtain approximation formulas for fractional integrals and derivatives of Riemann-Liouville and Marchaud types with a variable fractional order. The approximations involve integer-order derivatives only. An estimation for the error is given. The efficiency of the approximation method is illustrated with examples. As applications, we show how the obtained results are useful to solve different...

We apply the tools of the fractional calculus to introduce new fractional-order beam solutions to the paraxial wave equation that can be regarded as intermediate solutions between the known integral-order solutions. We restrict our attention to the fractionalization of the elegant and standard Hermite-Gaussian beams.

This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional-order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons in biology. Special attention is given to numerical computation of fractional derivatives and integ...

In this study, we introduce conditions for the existence of solutions for an iterative functional differential equation of fractional order. We prove that the solutions of the above class of fractional differential equations are bounded by Tsallis entropy. The method depends on the concept of Hyers-Ulam stability. The arbitrary order is suggested in the sense of Riemann-Liouville calculus.