Existence Results for Systems of Nonlinear Caputo Fractional Differential Equations

We aim, in this work, to demonstrate the existence of minimal and maximal coupled quasi-solutions for nonlinear Caputo fractional differential systems with order q ∈ (1,2). Our approach is based on mixed monotone iterative techniques developed under the concept of lower and upper quasi-solutions. Our results extend those obtained for ordinary differential equations and fractional ones.

Harmonic balance method with alternating frequency/time domain technique for nonlinear dynamical system with fractional exponential

Comparisons of the common methods for obtaining the periodic responses show that the harmonic balance method with alternating frequency/time （HB-AFT） do- main technique has some advantages in dealing with nonlinear problems of fractional exponential models. By the HB-AFT method, a rigid rotor supported by ball bearings with nonlinearity of Hertz contact and ball passage vibrations is considered. With the aid of the Floquet theory, the movement characteristics of interval stability are deeply studied. Besides, a simple strategy to determine the monodromy matrix is proposed for the stability analysis.

Fractional nonlinear energy sinks

The cubic or third-power(TP)nonlinear energy sink(NES)has been proven to be an effective method for vibration suppression,owing to the occurrence of targeted energy transfer(TET).However,TET is unable to be triggered by the low initial energy input,and thus the TP NES would get failed under low-amplitude vibration.To resolve this issue,a new type of NES with fractional nonlinearity,e.g.,one-third-power(OTP)nonlinearity,is proposed.The dynamic behaviors of a linear oscillator(LO)with an OTP NES are investigated numerically,and then both the TET feature and the vibration attenuation performance are evaluated.Moreover,an analogy circuit is established,and the circuit simulations are carried out to verify the design concept of the OTP NES.It is found that the threshold for TET of the OTP NES is two orders of magnitude smaller than that of the TP NES.The parametric analysis shows that a heavier mass or a lower stiffness coefficient of the NES is beneficial to the occurrence of TET in the OTP NES system.Additionally,significant energy transfer is usually accompanied with efficient energy dissipation.Consequently,the OTP NES can realize TET under low initial input energy,which should be a promising approach for micro-vibration suppression.

An Exploration on Adaptive Iterative Learning Control for a Class of Commensurate High-order Uncertain Nonlinear Fractional Order Systems

This paper explores the adaptive iterative learning control method in the control of fractional order systems for the first time. An adaptive iterative learning control(AILC) scheme is presented for a class of commensurate high-order uncertain nonlinear fractional order systems in the presence of disturbance.To facilitate the controller design, a sliding mode surface of tracking errors is designed by using sufficient conditions of linear fractional order systems. To relax the assumption of the identical initial condition in iterative learning control(ILC), a new boundary layer function is proposed by employing MittagLeffler function. The uncertainty in the system is compensated for by utilizing radial basis function neural network. Fractional order differential type updating laws and difference type learning law are designed to estimate unknown constant parameters and time-varying parameter, respectively. The hyperbolic tangent function and a convergent series sequence are used to design robust control term for neural network approximation error and bounded disturbance, simultaneously guaranteeing the learning convergence along iteration. The system output is proved to converge to a small neighborhood of the desired trajectory by constructing Lyapnov-like composite energy function(CEF)containing new integral type Lyapunov function, while keeping all the closed-loop signals bounded. Finally, a simulation example is presented to verify the effectiveness of the proposed approach.

A FRACTIONAL NONLINEAR EVOLUTIONARY DELAY SYSTEM DRIVEN BY A HEMI-VARIATIONAL INEQUALITY IN BANACH SPACES

This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space.Utilizing the KKM theorem,a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established.By using a fixed point theorem for a condensing setvalued map,the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions.Finally,an example is presented to illustrate our main results.

Stabilizing controller design for nonlinear fractional order systems with time varying delays

To deal with stabilizing of nonlinear affine fractional order systems subject to time varying delays,two methods for finding an appropriate pseudo state feedback controller are discussed.In the first method,using the Mittag-Lefler function,Laplace transform and Gronwall inequality,a linear stabilizing controller is derived,which uses the fractional order of the delayed system and the upper bound of system nonlinear functions.In the second method,at first a sufficient stability condition for the delayed system is given in the form of a simple linear matrix inequality(LMI)which can easily be solved.Then,on the basis of this result,a stabilizing pseudo-state feedback controller is designed in which the controller gain matrix is easily computed by solving an LMI in terms of delay bounds.Simulation results show the effectiveness of the proposed methods.

Characteristic fractional step finite difference method for nonlinear section coupled system

For the section coupled system of multilayer dynamics of fluids in porous media, a parallel scheme modified by the characteristic finite difference fractional steps is proposed for a complete point set consisting of coarse and fine partitions. Some tech- niques, such as calculus of variations, energy method, twofold-quadratic interpolation of product type, multiplicative commutation law of difference operators, decomposition of high order difference operators, and prior estimates, are used in theoretical analysis. Optimal order estimates in 12 norm are derived to show accuracy of the second order approximation solutions. These methods have been used to simulate the problems of migration-accumulation of oil resources.

FIXED POINTS OF α-TYPE F-CONTRACTIVE MAPPINGS WITH AN APPLICATION TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

In this paper, we introduce new concepts of a-type F-contractive mappings which are essentially weaker than the class of F-contractive mappings given in [21, 22] and different from a-GF-contractions given in [8]. Then, sufficient conditions for the existence and uniqueness of fixed point are established for these new types of contractive mappings, in the setting of complete metric space. Consequently, the obtained results encompass various generalizations of the Banach contraction principle. Moreover, some examples and an application to nonlinear fractional differential equation are given to illustrate the usability of the new theory.

A Numerical Algorithm Based on Quadratic Finite Element for Two-Dimensional Nonlinear Time Fractional Thermal Diffusion Model

In this article,a high-order scheme,which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme,is developed for looking for the numerical solution of a two-dimensional nonlinear time fractional thermal diffusion model.The time Caputo fractional derivative is approximated by using the L2-1formula,the first-order derivative and nonlinear term are discretized by some second-order approximation formulas,and the quadratic finite element is used to approximate the spatial direction.The error accuracy O(h3+
t2)is obtained,which is verified by the numerical results.

Finite-Time Stability for Nonlinear Fractional Differential Equations with Time Delay

The finite-time stability and the finite-time contractive stability of solutions for nonlinear fractional differential equations with bounded delay are investigated. The derivative of Lyapunov function along solutions of the considered system is defined in terms of the Caputo fractional Dini derivative. Based on the Lyapunov-Razumikhin method, several sufficient criteria are established to guarantee the finite-time stability and the finite-time contractive stability of solutions for the related systems. An example is provided to illustrate the effectiveness of the obtained results.

High-Order Local Discontinuous Galerkin Algorithm with Time Second-Order Schemes for the Two-Dimensional Nonlinear Fractional Diffusion Equation

In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.

Closed form soliton solutions of three nonlinear fractional models through proposed improved Kudryashov method

We introduce a new integral scheme namely improved Kudryashov method for solving any nonlinear fractional differential model.Specifically,we apply the approach to the nonlinear space-time fractional model leading the wave to spread in electrical transmission lines(s-tfETL),the time fractional complex Schrödinger(tfcS),and the space-time M-fractional Schrödinger-Hirota(s-tM-fSH)models to verify the effectiveness of the proposed approach.The implementing of the introduced new technique based on the models provides us with periodic envelope,exponentially changeable soliton envelope,rational rogue wave,periodic rogue wave,combo periodic-soliton,and combo rational-soliton solutions,which are much interesting phenomena in nonlinear sciences.Thus the results disclose that the proposed technique is very effective and straight-forward,and such solutions of the models are much more fruitful than those from the generalized Kudryashov and the modified Kudryashov methods.

Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation

The fractional derivatives in the sense of Caputo and the homotopy analysis method are used to construct an approximate solution for the nonlinear space-time fractional derivatives Klein-Gordon equation. The numerical results show that the approaches are easy to implement and accurate when applied to the nonlinear space-time fractional derivatives Klein- Gordon equation. This method introduces a promising tool for solving many space-time fractional partial differential equations. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.

The Solution to Impulse Boundary Value Problem for a Class of Nonlinear Fractional Functional Differential Equations

In this paper, we investigate the existence of solution for a class of impulse boundary value problem of nonlinear fractional functional differential equation of mixed type. We obtain the existence results of solution by applying some well-known fixed point theorems. An example is given to illustrate the effectiveness of our result.

Solving Fractional Differential Equations via Fixed Points of Chatterjea Maps

In this paper,we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces.Furthermore,we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated withSλand consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential equations.We also establish certain interesting examples to illustrate the usability of our results.

ADI Finite Element Method for 2D Nonlinear Time Fractional Reaction-Subdiffusion Equation

In this paper, an alternating direction Galerkin finite element method is presented for solving 2D time fractional reaction sub-diffusion equation with nonlinear source term. Firstly, one order implicit-explicit method is used for time discretization, then Galerkin finite element method is adopted for spatial discretization and obtain a fully discrete linear system. Secondly, Galerkin alternating direction procedure for the system is derived by adding an extra term. Finally, the stability and convergence of the method are analyzed rigorously. Numerical results confirm the accuracy and efficiency of the proposed method.

Two-dimensional localized modes in nonlinear systems with linear nonlocality and moiré lattices

Periodic structures structured as photonic crystals and optical lattices are fascinating for nonlinear waves engineering in the optics and ultracold atoms communities.Moiréphotonic and optical lattices—two-dimensional twisted patterns lie somewhere in between perfect periodic structures and aperiodic ones—are a new emerging investigative tool for studying nonlinear localized waves of diverse types.Herein,a theory of two-dimensional spatial localization in nonlinear periodic systems with fractional-order diffraction(linear nonlocality)and moiréoptical lattices is investigated.Specifically,the flat-band feature is well preserved in shallow moiréoptical lattices which,interact with the defocusing nonlinearity of the media,can support fundamental gap solitons,bound states composed of several fundamental solitons,and topological states(gap vortices)with vortex charge s=1 and 2,all populated inside the finite gaps of the linear Bloch-wave spectrum.Employing the linear-stability analysis and direct perturbed simulations,the stability and instability properties of all the localized gap modes are surveyed,highlighting a wide stability region within the first gap and a limited one(to the central part)for the third gap.The findings enable insightful studies of highly localized gap modes in linear nonlocality(fractional)physical systems with shallow moirépatterns that exhibit extremely flat bands.

LMI Based Synchronization Control of Nonlinear Affine Fractional Order Chaotic Systems Considering Input Constraint

The current paper deals with synchronization problem among chaotic nonlinear fractional order systems considering input saturation constraint.To develop the idea,at first a generalized sector condition and a memory-less nonlinear function are employed to deal with the saturation problem.Then a new state feedback controller is designed to achieve synchronization in master-slave chaotic fractional order nonlinear systems with input saturation.Using the state feedback controller,the asymptotic stability of whole dynamic error model between master and slave is achieved.The stability of the closed-loop system is guaranteed using Lyapunov theory and sufficient stability conditions are formulated in terms of caputo fractional derivative of a quadratic Lyapunov function and Linear Matrix Inequalities(LMI).Finally,to verify the effectiveness of the proposed control scheme,some simulation results are employed to show the effectiveness of the proposed methodology.

A LINEARLY-IMPLICIT ENERGY-PRESERVING ALGORITHM FOR THE TWO-DIMENSIONAL SPACE-FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION BASED ON THE SAV APPROACH

The main objective of this paper is to present an efficient structure-preserving scheme,which is based on the idea of the scalar auxiliary variable approach,for solving the twodimensional space-fractional nonlinear Schrodinger equation.First,we reformulate the equation as an canonical Hamiltonian system,and obtain a new equivalent system via introducing a scalar variable.Then,we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space direction.After that,applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete version.As expected,the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step.Finally,numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.

Event-Triggered Adaptive Fuzzy Finite Time Control of Fractional-Order Non-Strict Feedback Nonlinear Systems

In this article,the problem of event-triggered adaptive fuzzy finite time control of nonstrict feedback fractional order nonlinear systems is investigated.By using the property of fuzzy basis function,the obstacle caused by algebraic loop problems is successfully circumvented.Moreover,a new adaptive event-triggered scheme is designed under the unified framework of backstepping control method,which can largely reduce the amount of communications.The stability of the closed-loop system is ensured through fractional Lyapunov stability analysis.Finally,the effectiveness of the proposed scheme is verified by simulation examples.