scholarly journals On the Fractional Derivative of Dirac Delta Function and Its Application

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Zaiyong Feng ◽  
Linghua Ye ◽  
Yi Zhang
Keyword(s):  
Integral Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Delta Function ◽  
Dirac Delta Function ◽  
Fractional Order System ◽  
Order System ◽  
Dirac Delta ◽  
Integer Order

The Dirac delta function and its integer-order derivative are widely used to solve integer-order differential/integral equation and integer-order system in related fields. On the other hand, the fractional-order system gets more and more attention. This paper investigates the fractional derivative of the Dirac delta function and its Laplace transform to explore the solution for fractional-order system. The paper presents the Riemann-Liouville and the Caputo fractional derivative of the Dirac delta function, and their analytic expression. The Laplace transform of the fractional derivative of the Dirac delta function is given later. The proposed fractional derivative of the Dirac delta function and its Laplace transform are effectively used to solve fractional-order integral equation and fractional-order system, the correctness of each solution is also verified.

scholarly journals The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

2021 ◽  
Vol 5 (1) ◽  
pp. 18
Author(s):  
Nicos Makris
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Memory Function ◽  
Recurrence Formula ◽  
Delta Function ◽  
Dirac Delta Function ◽  
Inverse Laplace Transform ◽  
Complex Materials ◽  
Dirac Delta ◽  
Anomalous Relaxation

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α<q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.

scholarly journals Some Fundamental Properties on the Sampling Free Nabla Laplace Transform

2019 ◽  
Author(s):  
Yiheng Wei ◽  
Yuquan Chen ◽  
Yong Wang ◽  
YangQuan Chen
Keyword(s):  
System Theory ◽  
Laplace Transform ◽  
Fractional Order ◽  
Fractional Order System ◽  
The Other ◽  
Order System ◽  
Fractional Order Systems ◽  
Fundamental Properties ◽  
Other Hand

Abstract Discrete fractional order systems have attracted more and more attention in recent years. Nabla Laplace transform is an important tool to deal with the problem of nabla discrete fractional order systems, but there is still much room for its development. In this paper, 14 lemmas are listed to conclude the existing properties and 14 theorems are developed to describe the innovative features. On one hand, these properties make the Ntransform more effective and efficient. On the other hand, they enrich the discrete fractional order system theory.

The Synchronization of a Fractional-Order Chaotic System

2013 ◽  
Vol 655-657 ◽  
pp. 1488-1491
Author(s):  
Fan Di Zhang
Keyword(s):  
Numerical Simulations ◽  
Fractional Order ◽  
Chaotic System ◽  
Stability Theory ◽  
Fractional Order System ◽  
Order System ◽  
Integer Order

In this paper, the synchronization of fractional-orderchaotic system is studied. Based on the fractional stability theory, suitable controller is designed to realize the synchronization between fractional-order system and a integer-order system. Numerical simulations show that the effectiveness and feasibility of the controllers .

scholarly journals Period-Doubling Bifurcation of Stochastic Fractional-Order Duffing System via Chebyshev Polynomial Approximation

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Youming Lei ◽  
Yanyan Wang
Keyword(s):  
Fractional Order ◽  
Chebyshev Polynomial ◽  
Polynomial Approximation ◽  
Period Doubling ◽  
Random Parameter ◽  
Fractional Order System ◽  
Period Doubling Bifurcation ◽  
Order System ◽  
Integer Order ◽  
Chebyshev Polynomial Approximation

Fractional-order calculus is more competent than integer-order one when modeling systems with properties of nonlocality and memory effect. And many real world problems related to uncertainties can be modeled with stochastic fractional-order systems with random parameters. Therefore, it is necessary to analyze the dynamical behaviors in those systems concerning both memory and uncertainties. The period-doubling bifurcation of stochastic fractional-order Duffing (SFOD for short) system with a bounded random parameter subject to harmonic excitation is studied in this paper. Firstly, Chebyshev polynomial approximation in conjunction with the predictor-corrector approach is used to numerically solve the SFOD system that can be reduced to the equivalent deterministic system. Then, the global and local analysis of period-doubling bifurcation are presented, respectively. It is shown that both the fractional-order and the intensity of the random parameter can be taken as bifurcation parameters, which are peculiar to the stochastic fractional-order system, comparing with the stochastic integer-order system or the deterministic fractional-order system. Moreover, the Chebyshev polynomial approximation is proved to be an effective approach for studying the period-doubling bifurcation of the SFOD system.

scholarly journals Coexisting Three-Scroll and Four-Scroll Chaotic Attractors in a Fractional-Order System by a Three-Scroll Integer-Order Memristive Chaotic System and Chaos Control

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
Xikui Hu ◽  
Ping Zhou
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Chaos Control ◽  
Initial Conditions ◽  
Fractional Order System ◽  
Order System ◽  
Chaotic Attractors ◽  
Integer Order ◽  
Memristive System ◽  
Fractional Orders

Based on the integer-order memristive system that can generate two-scroll, three-scroll, and four-scroll chaotic attractors, in this paper, we found other phenomena that two kinds of three-scroll chaotic attractors coexist in this system with different initial conditions. Furthermore, we proposed a coexisting fractional-order system based on the three-scroll chaotic attractors system, in which the three-scroll or four-scroll chaotic attractors emerged with different fractional-orders q. Meanwhile, with fractional-order q=0.965 and different initial conditions, coexistence of two kinds of three-scroll and four-scroll chaotic attractors is found simultaneously. Finally, we discussed controlling chaos for the fractional-order memristive chaotic system.

scholarly journals Fractional Order Control Methodology for the Self-Balance Robot.

2020 ◽  
Vol 9 (4) ◽  
pp. 276-381
Keyword(s):  
Fractional Order ◽  
Control Algorithm ◽  
Research Paper ◽  
Fractional Order System ◽  
The Self ◽  
Order System ◽  
Control Algorithms ◽  
Linear Quadratic ◽  
Integer Order ◽  
Fractional Order Controller

In this research paper the control algorithms like LQR and PID has been proposed for the integer and fractional order system. In this research paper the modeling of the selfbalance robot system has been carried out in integer domain and fractional domain. This research paper presents the simulation analysis of control algorithms for two wheel self-balancing robot using Linear Quadratic Regulator, Proportional-IntegralDerivative and Fractional order Proportional-Integral-Derivative control algorithm. These all control algorithm are applied on the integer order system and the fractional order system and comparative analysis has been done. The comparison between integer order PID against the fractional order PID is also been made for the self-balance robot. It has been demonstrated through simulation that fractional order controller gives better response as compared to integer order controller. Further it has been found out that fractional order controller gives better results when applied to fractional order system compared to its integer order counterpart.

Initialization Energy in Fractional-Order Systems

2015 ◽  
Author(s):  
Tom T. Hartley ◽  
Jean-Claude Trigeassou ◽  
Carl F. Lorenzo ◽  
Nezha Maamri
Keyword(s):  
Laplace Transform ◽  
Fractional Order ◽  
Caputo Derivative ◽  
Fractional Order System ◽  
Order System ◽  
Time Varying ◽  
Fractional Order Systems ◽  
History Of ◽  
Infinite Energy ◽  
Order Element

This paper seeks a deeper understanding of the need for time-varying initialization of fractional-order systems. Specifically, the paper determines the energy stored in a fractional-order element based on the history of the element, and shows how this initialization energy is manifest into the future as an initialization function. Further, it is shown that infinite energy is required to initialize a fractional-order system when using the Caputo derivative Laplace transform.

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