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Progress in Fractional Differentiation and Applications
An International Journal
               
 
 
 
 
 
 
 
 
 
 
 
 

Forthcoming
 

 

Algorithmic Convergence on Banach Space Valued Functions in Abstract g-Fractional Calculus

George A. Anastassiou, Ioannis K. Argyros,
Abstract :
The novelty of this paper is the design of suitable algorithms for solving equations on Banach spaces. Some applications of the semi-local convergence are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochner-type.

 

Derivation of the Fractional Dodson Equation and Beyond: Transient diffusion with a non-singular memory and exponentially fading-out diffusivity

Jordan Hristov,
Abstract :
Starting from the Cattaneo constitutive relation with exponential kernel applied to mass diffusion the derivation of a new form the diffusion equation with a relaxation term expressed through the Caputo-Fabrizio time fractional derivative has been developed. The developed equation reduces to the fractional Dodson equation for large relaxation times corresponding to low fractional order of the Caputo-Fabrizio derivative. The approach separates large time effects resulting in the classical Dodson equation with exponentially decaying in time diffusivity and the short time relaxation process modelled by Caputo-Fabrizio time fractional derivative. The solution developed allows seeing a new physical background of the Caputo-Fabrizio time fractional derivative and to demonstrate a new interpretation of the non-linear Dodson equation incorporating fading memory effects. Moreover a new model with two memories corresponding to large and short time relation effects has been conceived. Defining the diffusion process parameters then the fractional order of the Caputo-Fabrizio time fractional derivative can be determined in a straightforward manner as a function of the Deborah number calculated as a ratio of the relaxation time to the characteristic diffusion time of the process.

 

Iterative Methods and their Applications to Banach Space Valued Functions in Abstract Fractional Calculus

George A. Anastassiou, Ioannis K. Argyros,
Abstract :
Explicit iterative methods have been used extensively to generate a sequence approximating a solution of an equation on a Banach space setting. However, little attention has been given to the study of implicit iterative methods. We present a semi-local convergence analysis for a some general implicit and explicit iterative methods. Some applications are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochner-type

 

Vector Fractional Trigonometric Korovkin Approximation

George A. Anastassiou,
Abstract :
n this paper we study quantitatively with rates the trigonometric fractional convergence of sequences of linear operators applied on Banach space valued functions. We derive pointwise and uniform estimates. To establish our main results we apply an elegant boundedness property of our linear operators by their companion positive linear operators. Our inequalities are trigonometric fractional involving the right and left vector Caputo type fractional derivatives, built in vector moduli of continuity. We consider very general classes of Banach space valued functions. Finally we present applications to vector Bernstein operators.

 

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.

 

Lyapunov Inequality for a Boundary Value Problem Involving Conformable Derivative

Guezane-Lakoud Assia, Rabah Khaldi,
Abstract :
We consider a boundary value problem involving conformable derivative of order α , 1 < α < 2 and Dirichlet conditions. To prove the existence of solutions, we apply the method of upper and lower solutions together with Schauder’s fixed-point theorem. Futhermore, we give the Lyapunov inequality for the corresponding problem.

 

Design of Controller for a Higher Order System Without Using Model Reduction Methods

Sudhir Agashe,
Abstract :
In the industry, many plants are described by higher order systems. Most of the time, higher order systems are approximated with the lower order system using model reduction method, and then the appropriate controllers are designed. In this paper, the controller for higher order system is designed without using model reduction methods. Instead, a fractional PID (FPID) controller is designed for higher order system. In simulations, ten different plants were examined, ranging from order 3 to order 7, with and without delay. The time-domain optimal tuning of higher order systems was carried out using integrated squared error (ISE) as the performance index. Results indicate that the controller for higher order system can be designed without model reduction methods by using FPID controller. The results of FPID controllers are also compared with classical PID controller. The FPID controller displayed robust performance; better gain and phase margin. The complementary sensitivity and sensitivity functions are better achieved with FPID controller. The FPID controller exhibits an iso-damping property (flat response around gain crossover frequency) for higher order systems.

 

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.

 

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.

 

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.

 

GENERALIZED ELZAKI – TARIG TRANSFORMATION ANDITS APPLICATIONSTO NEW FRACTIONAL DERIVATIVE WITH NON SINGULAR KERNEL

SHRINATH DILIP MANJAREKAR,
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.

 

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.

 

Backward bifurcation in a fractional order epidemiological model

A.M.A. El-Sayed,
Abstract :
An epidemiological fractional order model which exhibits backward bifurcation for certain values of the parameters is studied in this paper. The reason for using fractional order model is that the integer order model does not carry any information about the memory and learning mechanism of the human population which influences disease transmission while the fractional order derivative can be considered as the index of memory. The aim of this paper is to study the impact of introducing the fractional order derivative on the phenomenon of backward bifurcation and on the basic reproduction number R_0. The added fractional-order parameter enhances the system performance through adding a new degree of freedom to the system and to R_0. Fractional order derivative is an attractive tool for describing memory phenomena in biology and epidemiology.

 

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

KANJANADEVI S,
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.

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