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Integer orders differential system is a special case of fractional-order differential system. Integer orders chaotic system that we usually study is ideally approximate to realistic chaotic system. Fractional-order chaotic system has broader and changeable values of order and more complex dynamical behavior than integer order chaotic system. Thus, fractional-order differential equation can describe the nonlinear characteristics of actual chaotic system more exactly, which has more prominent research meanings and application value. This paper designs a new four-wing four-dimensional heterogeneous fractional-order chaotic system, when the values of fractional-order qi (i = 1,2,3,4) are not identical (q1 = q2 = 0.9, q3 = q4 = 0.8, in step size of 0.1), the attractors of this chaotic system will all show four-wing shapes in any direction. After analysis this chaotic system's stability and existence, this paper also introduces a nonlinear state feedback controller, and adopts the chain shape circuit to conduct experiment simulation through Multisim software 10.0. The results of circuit simulation and Matlab numerical operation have the same chaotic attractor phase diagram. This demonstrates the effectiveness of this four-wing four-dimensional heterogeneous fractional-order chaotic system's design and the feasibility of the feedback controller in the circuit; meanwhile, it provides referable bases for the application in actual circuits. [ABSTRACT FROM AUTHOR]

Linear systems described by fractional differential equations (FDEs) with variable coefficients involving Riemann--Liouville and Caputo derivatives are examined in the paper. For these systems, a solution of the initial-value problem is derived in terms of the generalized Peano--Baker series and a time-optimal control problem is formulated. The optimal control problem is treated from the convex-analytical viewpoint. Necessary and sufficient conditions for time-optimal control similar to that of Pontryagin's maximum principle are obtained. Theoretical results are supported by examples. [ABSTRACT FROM AUTHOR]

This paper uses the principle of compressed mapping to discuss the existence and uniqueness of the explicit finite difference method for the fractional diffusion equation with time delay. The Laplace transform method obtains the necessary expression form of the solution. At the same time, the existence theorem and the existence and uniqueness theorem of the solution to the boundary value problem is established. Finally, an example is given to verify the correctness of the conclusion. The experimental results show that the parallel algorithm proposed in this paper agrees with the exact solution.

This paper’s main motivation is to study the notion of weighted pseudo almost automorphic (WPAA) functions and establish the existence results of piecewise continuous mild solution of fractional order integro-differential equation with instantaneous impulses. The usual WPAA functions may not work since the solution of impulsive differential equations may not be continuous. Thus in order to give a broader spectrum, we introduce this concept. We establish main results by using the Banach contraction mapping principle and Sadovskii’s fixed point theorem. An example is shown to exhibit our analytic findings. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study, in mean square sense, a class of random fractional linear differential equation where the initial condition and the forcing term are assumed to be second-order random variables. The solution stochastic process of its associated Cauchy problem is constructed combining the application of a mean square chain rule for differentiating second-order stochastic processes and the random Fröbenius method. To conduct our study, first the classical Caputo derivative is extended to the random framework, in mean square sense. Furthermore, a sufficient condition to guarantee the existence of this operator is provided. Afterwards, the solution of a random fractional initial value problem is built under mild conditions. The main statistical functions of the solution stochastic process are also computed. Finally, several examples illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, is defined the operator Dλ,ν,nα,β : A → A, given by Dλ,ν,nα,β f(z) = (1 - α - β)RνDnf(z) + αRνΩλzf(z) + βDnΩλzf(z), for z ∈ U, where Rν is the Ruscheweyh derivative, Dn is the Sălăgean operator, Ωλz is a fractional differintegral operator introduced by S. Owa and H. M. Srivastava, A = {f ∈ H(U) : f(z) = z + a2z² + a3z³ + ..., z ∈ U}, α, β ≥ 0, ν > -1, n ∈ N0 = {0, 1, 2, 3, ...}, -∞ < λ < 2. A certain subclass of analytic functions in the open unit disk, Rλ,ν,nα,β (δ), where 0 ≤ δ ≤ 1, is introduced using the new operator. Are obtained some properties of the class Rλ,ν,nα,β (δ) and some differential subordinations using the operator Dλ,ν,nα,β. [ABSTRACT FROM AUTHOR]

This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder's fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory. [ABSTRACT FROM AUTHOR]

In this paper, we study the generalized fractional operators pertaining to the generalized Mittag-Leffler function and multi-index Mittag-Leffler function. Some applications of the established results associated with generalized Wright function are also deduced as corollaries. The results are useful in solving the problems of science, engineering and technology where the Mittag-Leffler function occurs naturally. [ABSTRACT FROM AUTHOR]

In this paper, we discus the existence of solutions for Riemann-Liouville fractional differential inclusions supplemented with Erdélyi-Kober fractional integral conditions. We apply endpoint theory, Krasnoselskii's multi-valued fixed point theorem and Wegrzyk's fixed point theorem for generalized contractions. For the illustration of our results, we include examples. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of boundary value problems of nonlinear fractional differential equations with integral boundary conditions. Some new existence and uniqueness results are obtained by using Banach fixed point theorem. Other existence results are also presented by using Krasnoselskii theorem. [ABSTRACT FROM AUTHOR]

In this work, the well known invariant subspace method has been modified and extended to solve some partial differential equations involving Caputo-Fabrizio (CF) or Atangana-Baleanu (AB) fractional derivatives. The exact solutions are obtained by solving the reduced systems of constructed fractional differential equations. The results show that this method is very simple and effective for constructing explicit exact solutions for partial differential equations involving new fractional derivatives with nonlocal and non-singular kernels, such solutions are very useful to validate new numerical methods constructed for solving partial differential equations with CF and AB fractional derivatives. [ABSTRACT FROM AUTHOR]

Reachability and minimum energy control of descriptor fractional discrete-time linear systems with different fractional orders are addressed. Using the Weierstrass-Kronecker decomposition theorem of the regular pencil, a solution to the state equation of descriptor fractional discrete-time linear systems with different fractional orders is given. The reachability condition of this class of systems is presented and used for solving the minimum energy control problem. The discussion is illustrated with numerical examples. [ABSTRACT FROM AUTHOR]