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Progress in Fractional Differentiation and Applications
An International Journal



Deformations in Elasto-Plastic Media with Memory: the Inverse Problem

Mauro Fabrizio,
Abstract :
We consider anelastic media governed by constitutive equations with memory behavior, which depend on the physical properties of the medium itself. In this note we use a model of elasto-plastic media with two un- specified memory formalisms, which are determined by performing a single virtual experiment on a sample of the medium. As an application, using a mathematical memory formally mimicking the Caputo-Fabrizio fractional derivative, we show that, when the applied stress is asymptotically vanish- ing, then a shorter memory in the constitutive equation and/or a slower decay of the applied stress, generate larger asymptotic plastic residual strain. In the last part of the paper, we present a non-linear stress-strain constitutive equation, which is suitable for describing hysteresis loops with discontinuity in the first derivative of the cycle.


An Inverse Problem for the Caputo Fractional Derivative by means of the Wavelet Transform

Fabio Marcela,
Abstract :
In this article, we build an approximate solution to an Inverse Problem that consist in finding a function whose Caputo Fractional Derivative is given. We decompose and project the data in appropriate wavelet subspaces and, by a Galerkin scheme, we calculate the coefficients of the unknown function in the chosen wavelet basis. Based on properties of the operator and of the basis, the scheme is efficient and the errors introduced by the approximation can be handled and controlled. We illustrate the results with an example.


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.


New Analytical Solutions and Approximate Solution of The Conformable Space-Time Fractional Sharma-Tasso-Olver Equation

O. Tasbozan,
Abstract :
The main purpose of this article is to find the exact and approximate solutions of conformable space-time fractional Sharma-Tasso-Olver equation using first integral method (FIM) and q-homotopy analysis method (q-HAM) respectively. The obtained exact and numerical solutions are compared with each other. Also, the numerical results obtained by q-HAM are compatible with the exact solutions obtained by FIM. Hence, it is clearly seen that these techniques are powerful and efficient in finding approximate and exact solutions for nonlinear conformable fractional PDEs.


On Applications of the Fractional Calculus for Some Singular Differential Equations

Resat Yilmazer,
Abstract :
Generalized Leibniz rule and some theorems in the calculus of the fractional derivatives and integrals was used to obtain the fractional solutions of the radial Schrödinger equation transformed into a singular differential equation and, these solutions were also exhibited as hypergeometric notations.


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.


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.


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.

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