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Progress in Fractional Differentiation and Applications
An International Journal



Vectorial Fractional Approximation by Linear Operators

George A. Anastassiou,
Abstract :
In this article we study quantitatively with rates the convergence of sequences of linear operators applied on Banach space valued functions. The results are pointwise and uniform estimates. To prove our main results we use an elegant boundedness property of our linear operators by their companion positive linear operators. Our inequalities are fractional involving the right and left vector Caputo type fractional derivatives, built in vector moduli of continuity. We treat very general classes of Banach space valued functions. We give applications to vectorial Bernstein operators.


Technical note on a new definition of fractional derivative.

Vincenzo Ciancio,
Abstract :
In this article, we have discovered a connection between the new fractional time derivative of Caputo-Fabrizio and his associated ordinary derivative. We will prove this connection, giving an evident physical meaning.


On the Fractional Legendre Equation and Fractional Legendre Functions

Mohammed Al-Refai, Muhammed Syam, Qasem Al-Mdallal,
Abstract :
In this paper we propose a fractional generalization of the well-known Legendre equation. We obtain a solution in the form of absolutely convergent power series with radius of convergence 1. We then truncate the power series to obtain the even and odd fractional Legendre functions in closed forms. These functions converge to the Legendre polynomials as the fractional derivative approaches 1, and new explicit formulas of the even and odd Legendre polynomials have been derived


On a fractional oscillator equation with natural boundary conditions

A. Guezane-Lakoud, R. Khaldi, D. F. M. Torres,
Abstract :
We prove existence of solutions for a nonlinear fractional oscillator equation with both left Riemann-Liouville and right Caputo fractional derivatives subject to natural boundary conditions. The proof is based on a transformation of the problem into an equivalent lower order fractional boundary value problem followed by the use of an upper and lower solutions method. To succeed with such approach, we first prove a result on the monotonicity of the right Caputo derivative.


Perturbation-Iteration Algorithm for Systems of Fractional Differential Equations and Convergence Analysis

Hamed Daei Kasmaei, Mehmet Şenol,
Abstract :
In this study, a perturbation-iteration algorithm, namely PIA, is applied to solve some types of systems of fractional differential equations (FDEs) and also the convergence analysis of the method is presented for the first time.To illustrate the efficiency of the method, numerical solutions are compared with the results published in the literature by considering a systems of FDEs. The results confirm that the PIA is powerful, simple and reliable method for solving system of nonlinear fractional differential equations.


Certain New Integral Inequalities Involving Erdelyi-Kober ` Operators

Naresh Menaria, Faruk Uc¸ar, Sunil Dutt Purohit,
Abstract :
In this article, the Erdelyi-Kober fractional integral operator is employed to generate certain new classes of integral ` inequalities using a family of n positive functions, (n ∈ N). Certain special cases and consequent results of the main results are also pointed out.


A coupled method to solve reaction-diffusion-convection equation with the time fractional derivative without singular kernel

Hossein Jafari, Alireza ‎Mohammadpour,
Abstract :
‎In this article, ‎we find the solution of a class of time fractional reaction-diffusion-convection equations. ‎The time fractional derivatives are described in new definition of fractional derivative without singular kernel which has been recently introduced by Caputo and Fabrizio‎. For obtaining the solution, we apply a coupled method based on the Laplace and the differential transformations. Finally, some test problems are discussed to show ability and utility of the proposed method.


Global convergence of successive approximations for partial fractional differential equations and inclusions

Yong Zhou, Said Abbas, Mouffak Benchohra,
Abstract :
This paper deals with the global convergence of successive approximations as well as the uniqueness of solutions for some classes of partial functional differential equations and inclusions involving the Caputo fractional derivative. We prove a theorem on the global convergence of successive approximations to the unique solution of our problems.


Existence of solutions of multi-point boundary value problems for fractional differential systems with impulse effects

Yuji Liu,
Abstract :
In this article, we present a new method for converting the considered impulsive systems to integral systems. As application of this method, we establish existence results for solutions of boundary value problems for nonlinear impulsive fractional differential systems. Our analysis relies on the well known Schauders fixed point theorem. The mistakes in [Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance, Electron. J. Qual. Theory Differ. Equ. 89(2011), 1-19] and [Existence result for boundary value problem of nonlinear impulsive fractional differential equation at resonance, J, Appl, Math, Comput., 39(2012) 421-443] are corrected, see Remark 2.1.


On Chebyshev type Inequalities using Generalized k-Fractional Integral Operator

Vaijanath L Chinchane, Vaijanath L Chinchane,
Abstract :
In this paper, using generalized k-fractional integral operator (in terms of the Gauss hypergeometric function), we establish new results on generalized k-fractional integral inequalities by considering the extended Chebyshev functional in case of synchronous function and some other inequalities.



Abstract :
In this paper, we have defined the new generalized Elzaki – Tarig transformation and find out its relations with other transformations. Furthermore we have derived the inversion formula, convolution theorem for it. Also as an application we have solve fractional differential equation with non – singular kernel.


Experimental Analysis of Fractional PID Controller Parameters on Time Domain Specifications

Sudhir Agashe,
Abstract :
A fractional PID controller is an extension of the classical PID controller, employing five tuning parameters rather than just three. General guidelines are available for the effect of classical controller parameters on the time domain specification. However, no guidelines are available for fractional PID controllers, particularly for the order of differentiation and integration. To assist with fine tuning, the effect of the order of differentiation and integration parameters on the time domain specifications for various plants are investigated. The relationship with the time domain specification serves as general guideline for manual tuning, and the effect of parameters will also assist with auto-tuning. In this paper, five plants covering integer order as well as non-integer order are simulated. The relationship between time domain specifications is plotted by varying the order of differentiation and integration between 0 and 2. Simulation results have revealed an association between the order of differentiation and the maximum overshoot for all plants. No other particular behavior was observed with other time domain specifications. However, some remarks on time domains specifications are made from the simulation results. Simulation results were validated using an experimental set up of the quadruple tank system.


On the existence and uniqueness of solutions for nonlinear fractional differential equations with fractional boundary conditions

Brahim Tellab,
Abstract :
This paper concerns some new existence and uniqueness results obtained by applying classical fixed point theorems for a class of Riemann-Liouville fractional differential equations with fractional boundary conditions.


On a discretized fractional-order SIR model for Influenza A viruses

Sanaa Moussa Salman,
Abstract :
In this paper, a discretized fractional-order SIR model for an Influenza A viruses is derived. The basic reproductive number $\mathfrak{R}_0$ is defined and the dynamic behavior of the model is studied. Local stability of the disease free equilibrium and the endemic equilibrium is investigated. Equations and inequalities of critical bifurcation surfaces at the disease free equilibrium are given. Numerical simulations are performed not only to show the theoretical results but also to demonstrate the complicated dynamics of the model.


Non-Instantaneous Impulsive Fractional Neutral Di erential Equations with State-dependent Delay

Abstract :
We prove the existence and uniqueness of fractional neutral differential equations with state-dependent delay subject to non-instantaneous impulsive conditions. The results are proved by using fixed point theorem for condensing map and justified by a suitable application.


Design and Analysis of Fractional Order Controller for 2-D Gantry Crane System

Abhaya Pal Singh,
Abstract :
This paper demonstrate that fractional order calculus could provide better description and model various real dynamic systems more adequately than the integer order ones. The proposed under-actuated dynamical system is 2-D Gantry crane that has two degrees of freedom and one control input. Precise control of a crane is essential as its failure may cause accidents and may harm the people around it. In this paper we design an integer order PID (IOPID) controller and a fractional order PID (FOPID) controller for the 2-D Gantry crane system and then compared their performances. The purpose of the controllers is to control the position of the trolley and the swing angle of the cable through which the load is suspended.

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