




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 ConflictControlled Systems


Mikhail Gomoyunov,


Abstract
: 

We consider a conflictcontrolled dynamical system described by a nonlinear ordinary fractional differential equation with
the Caputo derivative of an order a ∈ (0,1). Basing on the finitedifference Gr¨unwaldLetnikov formulas, we propose an approximation
of the considered system by a system described by a functionaldifferential equation of a retarded type. A mutual aiming procedure
between the initial conflictcontrolled 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 functionaldifferential systems for
solving control problems in fractional order systems. Examples are considered, results of numerical simulations are presented. 






New Aspects of CaputoFabrizio Fractional Derivative


Mohammed AlRefai,
Kamal Pal,


Abstract
: 

In this paper, we consider classes of linear and nonlinear fractional differential equations involving the CaputoFabrizio
fractional derivative of nonsingular 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 wellknown 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
dierential equations involving Hadamard fractional derivative with nonlocal mixed
boundary conditions with multiple orders. An example is given to demonstrate the application of
our results. 






LIOUVILLEWEYL FRACTIONAL HAMILTONIAN SYSTEMS: EXISTENCE RESULT


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 $(L(t)u,u)\geq Mu^{2}$ 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 AmbrosettiRabinowitz 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 TwoDimensional SpaceTime 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 CaputoFabrizio fractional derivative, to solve for the numerical solution of twodimensional spacetime 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 CaputoFabrizio Derivative


Sabrina Roscani,


Abstract
: 

This paper deals with the fractional CaputoFabrizio derivative and some basic properties related. A computation of this fractional derivative to power functions is given in terms of MittagLefler functions. The inverse operator named the fractional Integral of CaputoFabrizio 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):8792, 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 GinzburgLandauSchrödinger Equation in Modelling Superconductivity


ASIM PATRA,


Abstract
: 

In the present paper, the Complex GinzburgLandauSchrödinger (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. timesplitting 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 wellknown BernsteinKantorovich
Choquet and BernsteinDurrweyerChoquet polynomial Choquetintegral
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 Volterratype 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 realworld 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 Volterratype equations. For this aim, one supervised backpropagation type learning algorithm which is planned on a threelayered feedforward 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 MittagLeffler 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 MittagLeffler 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 RungeKutta 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 LevyFeller advectiondispersion equation using Jacobi polynomials


Nasser H. Sweilam,


Abstract
: 

In this paper, a practical formula expressing explicitly the fractionalorder derivatives, in the sense of RieszFeller operator, of Jacobi polynomials is presented. Jacobi spectral collocation method together with the trapezoidal rule is used to reduce the fractional Levy
Feller advectiondispersion equation (LFADE) to a system of algebraic equations which greatly simplies solving like this fractional dierential equation. Numerical simulations with some comparisons are introduced to conrm the effectiveness and reliability of the
proposed technique for the LevyFeller fractional partial differential equations. 






Numerical method for space and timefractional telegraph equation with generalized Lagrange multipliers


Manoj Kumar,


Abstract
: 

In the present article, we applied a novel methodology to solve spacefractional and timefractional 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 underactuated system. The system model is derived using EulerLagrange formulation. The response of FMPC and a traditional model predictive control (MPC) for the considered underactuated 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 integrodifferential type of singularly perturbed problems


Muhammed Syam,


Abstract
: 

In this paper, we study a class of first order fractional nonlinear Volterra integrodifferential 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 integrodifferntial 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 7DOF 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 αfractional differentiable functions and the integration by parts rule for αfractional differentiable functions. 






Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with RiemannLiouville derivative


Owolabi Kolade Matthew,


Abstract
: 

Stability analysis and numerical treatment of chaotic fractional differential system in RiemannLiouville 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 Subdiffusion Equation


M.Adel,


Abstract
: 

Fractional reaction subdiffusion equations are widely used in recent years to simulate physical phenomena. In this paper, we consider the variable order linear and nonlinear reaction subdiffusion equation. A numerical study for the variable order linear and nonlinear
reaction subdiffusion equations is introduced by using a class of finite difference methods.
These methods are extension of the weighted average methods for ordinary (nonfractional) reaction subdiffusion 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 RiemannLiouville fractional differential equations with
fractional boundary conditions. 




