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02- Progress in Fractional Differentiation and Applications
An International Journal



Multidimensional Fractional Iyengar type inequalities for radial functions

George A. Anastassiou,
Abstract :
Here we derive a variety of multivariate fractional Iyengar type in- equalities for radial functions de…ned on the shell and ball. Our approach is based on the polar coordinates in RN, N  2, and the related multi- variate polar integration formula. Via this method we transfer author’s univariate fractional Iyengar type inequalities into multivariate fractional Iyengar inequalities.


Approximation of Fractional Order Conflict-Controlled Systems

Mikhail Gomoyunov,
Abstract :
We consider a conflict-controlled dynamical system described by a nonlinear ordinary fractional differential equation with the Caputo derivative of an order a ∈ (0,1). Basing on the finite-difference Gr¨unwald-Letnikov formulas, we propose an approximation of the considered system by a system described by a functional-differential equation of a retarded type. A mutual aiming procedure between the initial conflict-controlled system and the approximating system is given that guarantees the desired proximity between their motions. This procedure allows to apply, via the approximating system, the results obtained for functional-differential systems for solving control problems in fractional order systems. Examples are considered, results of numerical simulations are presented.


New Aspects of Caputo-Fabrizio Fractional Derivative

Mohammed Al-Refai, Kamal Pal,
Abstract :
In this paper, we consider classes of linear and nonlinear fractional differential equations involving the Caputo-Fabrizio fractional derivative of non-singular kernel. We transform the fractional problems to equivalent initial value problems with integer derivatives. We illustrate the obtained results by presenting two mathematical models of fractional differential equations and their equivalent initial value problems. We show that it is impossible to convert all types of linear fractional differential equations to the integer ones.The obtained results will lead to better understanding of fractional models, as the solutions of their equivalent models can be studied analytically and numerically using well-known techniques of differential equations.


On system of nonlinear fractional differential equations involving Hadamard fractional derivative with nonlocal integral boundary conditions

Muath Awadalla,
Abstract :
This article discusses the existence and uniqueness of solutions for a system of nonlinear fractional di erential equations involving Hadamard fractional derivative with nonlocal mixed boundary conditions with multiple orders. An example is given to demonstrate the application of our results.



Willy Zubiaga,
Abstract :
In this paper we consider the existence of solution for the following fractional differential equation \begin{eqnarray}\label{eq00} _{t}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) + L(t)u(t) = & \nabla W(t,u(t)) \end{eqnarray} where $\alpha \in (1/2, 1)$, $L\in C(\mathbb{R}, \mathbb{R}^{n^{2}})$ is a symmetric and positive definite matrix for all $t\in \mathbb{R}$, $W(t,u) = a(t)V(t)$ with $a\in C(\mathbb{R},\mathbb{R}^{+})$ and $V\in C^{1}(\mathbb{R}^{n}, \mathbb{R})$. The novelty of this paper is that, assuming $L$ is bounded from below in the sense that there is a constant $M>0$ such that $(L(t)u,u)\geq M|u|^{2}$ for all $(t,u)\in \mathbb{R}\times \mathbb{R}^{n}$, we establish a new compact embedding theorem. Subsequently, supposing that $V$ satisfies the global Ambrosetti-Rabinowitz condition, we obtain a new criterion to guarantee that (\ref{eq00}) has one nontrivial solution using the Mountain pass theorem.


Finite Difference Approximation Method for Two-Dimensional Space-Time Fractional Diffusion Equation Using Nonsingular Fractional Derivative

Norodin A. Rangaig, Alvanh Alem G. Pido,
Abstract :
In this work, we utilized the nonsingular kernel fractional derivative, known as Caputo-Fabrizio fractional derivative, to solve for the numerical solution of two-dimensional space-time fractional diffusion equation using finite difference approximation. Analysis for unconditional stability and convergence have been presented. Interestingly, by using the nonsingular kernel fractional derivative, it is found that the convergence generates a second order accuracy weighted by the memory kernel of the fractional derivative. In addition, fractional order dependency of the convergence have been discussed and compared to some previous works. Moreover, the obtained finite difference approximation method was employed to solve for a given example. Numerical test verified the analysis of this study.


Global Solution to a Nonlinear Fractional Differential Equation for the Caputo--Fabrizio Derivative

Sabrina Roscani,
Abstract :
This paper deals with the fractional Caputo-Fabrizio derivative and some basic properties related. A computation of this fractional derivative to power functions is given in terms of Mittag--Lefler functions. The inverse operator named the fractional Integral of Caputo--Fabrizio is also analyzed. The main result consists in the proof of existence and uniqueness of a global solution to a nonlinear fractional differential equation, which has been solved previously for short times by Lozada and Nieto (Progr. Fract. Differ. Appl., 1(2):87-92, 2015). The effects of memory as well as the convergence of the obtained results when $\al \nearrow 1$ (and the classical first derivative is recovered) are analyzed throughout the paper.


On Comparison of Two Reliable Techniques for the Riesz Fractional Complex Ginzburg-Landau-Schrödinger Equation in Modelling Superconductivity

Abstract :
In the present paper, the Complex Ginzburg-Landau-Schrödinger (CGLS) equation with the Riesz fractional derivative has been treated by a reliable implicit finite difference method (IFDM) of second order and furthermore for the purpose of a comparative study, and also for the investigation of the accuracy of the resulting solutions another effective spectral technique viz. time-splitting Fourier spectral (TSFS) technique has been utilized. In the case of the finite difference discretization, the Riesz fractional derivative is approximated by the fractional centered difference approach. Further the stability of the proposed methods has been analysed thoroughly and the TSFS technique is proved to be unconditionally stable. Moreover the absolute errors for the solutions of 2 \psi(x, t) obtained from both the techniques for various fractional order have been tabulated. Further the L^2 and L^inf error norms has been displayed for 2 \psi (x, t) and the results are also graphically depicted.


Mixed Conformable and Iterated fractional Approximation by Choquet integrals

George A. Anastassiou,
Abstract :
Here we study the quantitative mixed conformable and iterated frac- tional approximation of positive sublinear operators to the unit opera- tor. These are given a precise Choquet integral interpretation. Initially we start with the research of the mixed conformable and iterated frac- tional rate of the convergence of the well-known Bernstein-Kantorovich- Choquet and Bernstein-Durrweyer-Choquet polynomial Choquet-integral operators. Then we study the very general comonotonic positive sublinear operators based on the representation theorem of Schmeidler (1986) [13]. We …nish with the approximation by the very general direct Choquet- integral form positive sublinear operators. All mixed conformable and iterated fractional approximations are given via inequalities involving the modulus of continuity of the approximated function mixed conformable and iterated fractional order derivative.


Solving fractional Volterra-type differential equations by using artificial neural networks approach

Ahmad Jafarian, Fariba Rostami, Alireza Khalili Golmankhaneh,
Abstract :
Lately, there is the great concern in the applications of the artificial neural networks approach in modeling and mathematically analysis of various complex real-world phenomena. In this literature, one of the most successful and effective neural network architectures has been implemented to construct the numerical solution of the fractional Volterra-type equations. For this aim, one supervised back-propagation type learning algorithm which is planned on a three-layered feed-forward neural network is applied for approximating the mentioned problem as a convergent power series solution. To be more precise, we have also considered some numerical examples with the comparison to the results given by the Euler wavelet method. Obtained simulation and numerical results illustrated that the proposed iterative technique is globally convergent and specially efficient for solving this fractional problem.


New approximate solutions to fractional smoking model using the generalized Mittag-Leffler function method

H. M. Ali,
Abstract :
Smoking is one of the fundamental drivers of health problems and keeps on being one of the worlds most significant health challenges. In this paper, we consider the dynamics of a surrendering smoking model containing fractional derivatives. Generalized Mittag-Leffler function method (for short GMLFM) is applied to solve approximate and analytical of nonlinear fractional differential equation systems such as a smoking model of fractional order. The solution of the model will be acquired in the type of infinite series which converges quickly to its correct esteem. In addition, we compare our outcomes and the outcomes got by Runge-Kutta method. Some plots are introduced to demonstrate the dependability and effortlessness of the technique. Moreover, the solutions obtained are displayed graphically.


Efficient method for fractional Levy-Feller advection-dispersion equation using Jacobi polynomials

Nasser H. Sweilam,
Abstract :
In this paper, a practical formula expressing explicitly the fractional-order derivatives, in the sense of Riesz-Feller operator, of Jacobi polynomials is presented. Jacobi spectral collocation method together with the trapezoidal rule is used to reduce the fractional Levy- Feller advection-dispersion equation (LFADE) to a system of algebraic equations which greatly simpli es solving like this fractional di erential equation. Numerical simulations with some comparisons are introduced to con rm the effectiveness and reliability of the proposed technique for the Levy-Feller fractional partial differential equations.


Numerical method for space- and time-fractional telegraph equation with generalized Lagrange multipliers

Manoj Kumar,
Abstract :
In the present article, we applied a novel methodology to solve space-fractional and time-fractional telegraph equation with the help of new Lagrange multiplier. We compare the approximate and exact solution using graphs, find the analytic solution and plotted absolute error for different value of fractional order which shows that approximate solution obtained by proposed technique converges very fast to the exact solution.


A Robust Fractional Model Predictive Control (FMPC) Design

Abhaya Pal Singh,
Abstract :
This paper proposes a robust FMPC design for an under-actuated system. The system model is derived using Euler-Lagrange formulation. The response of FMPC and a traditional model predictive control (MPC) for the considered under-actuated system are compared and found that FMPC performs better to the existing controllers.


A numerical method for solving a class of first order fractional nonlinear Volterra integro-differential type of singularly perturbed problems

Muhammed Syam,
Abstract :
In this paper, we study a class of first order fractional nonlinear Volterra integro-differential type of singularly perturbed problems with fractional order. The method consists of two steps: The first step is to generate the reduced problem when ε=0. We solve it directly if it is possible or to use the finite difference method with Trapezoidal rule when it is not possible to solve it explicitly. The second step is fractional Volterra integro-differntial problem which we solve it by the reproducing kernel method. We prove the stably and the uniqueness of the solution for this class of problems. Numerical results are presented to show the efficiency of the proposed method.


Fractional control of a 7-DOF robot to behave like a human arm

Inés Tejado, André Ventura, Jorge Martins, Duarte Valério,
Abstract :
A key challenge of robotics is to endow robots with the capability to collaborate closely with humans. This requires systems that behave in a manner humans are comfortable with, and mimicking the way another human would behave is a good choice. This paper presents simulation results of how a KUKA LWR IV robot with 7 degrees of freedom (DOF) can be made to behave like a human arm, having surgical (e.g. orthopedic) applications in view. This performance is achieved using a fractional order transfer function model of the arm, which have been presented in previous papers. Results are satisfactory and could not be obtained using only the control features provided with the robot.


Fractional Copson and converses Copson - type inequalities via conformable calculus

Samir H. saker,
Abstract :
In this paper, we prove some fractional inequalities of Copson and converses Copsons type by using conformable fractional calculus and obtain some results of special cases of fractional orders. The main results will be proved by employing Hölders inequality for α-fractional differentiable functions and the integration by parts rule for α-fractional differentiable functions.


Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann-Liouville derivative

Owolabi Kolade Matthew,
Abstract :
Stability analysis and numerical treatment of chaotic fractional differential system in Riemann-Liouville sense are considered in this paper. Simulation results show that chaotic phenomena can only occur if the reaction or local dynamics of such system is coupled or nonlinear in nature. Illustrative examples that are still of current and recurring interest to economists, engineers, mathematicians and physicists are chosen, to describe the points and queries that may arise.


Numerical Simulation For The Variable Order Linear and Nonlinear Reaction Sub-diffusion Equation

Abstract :
Fractional reaction sub-diffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider the variable order linear and nonlinear reaction sub-diffusion equation. A numerical study for the variable order linear and nonlinear reaction sub-diffusion equations is introduced by using a class of finite difference methods. These methods are extension of the weighted average methods for ordinary (non-fractional) reaction sub-diffusion equation. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. The paper is ended by the results of a numerical examples supports the theoretical analysis


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.

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