Influence Of Fractional Order Research Articles

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

An advanced approach for the electrical responses of discrete fractional-order biophysical neural network models and their dynamical responses

The multiple activities of neurons frequently generate several spiking-bursting variations observed within the neurological mechanism. We show that a discrete fractional-order activated nerve cell framework incorporating a Caputo-type fractional difference operator can be used to investigate the impacts of complex interactions on the surge-empowering capabilities noticed within our findings. The relevance of this expansion is based on the model’s structure as well as the commensurate and incommensurate fractional-orders, which take kernel and inherited characteristics into account. We begin by providing data regarding the fluctuations in electronic operations using the fractional exponent. We investigate two-dimensional Morris–Lecar neuronal cell frameworks via spiked and saturated attributes, as well as mixed-mode oscillations and mixed-mode bursting oscillations of a decoupled fractional-order neuronal cell. The investigation proceeds by using a three-dimensional slow-fast Morris–Lecar simulation within the fractional context. The proposed method determines a method for describing multiple parallels within fractional and integer-order behaviour. We examine distinctive attribute environments where inactive status develops in detached neural networks using stability and bifurcation assessment. We demonstrate features that are in accordance with the analysis’s findings. The Erdös–Rényi connection of asynchronization transformed neural networks (periodic and actionable) is subsequently assembled and paired via membranes that are under pressure. It is capable of generating multifaceted launching processes in which dormant neural networks begin to come under scrutiny. Additionally, we demonstrated that boosting connections can cause classification synchronization, allowing network devices to activate in conjunction in the future. We construct a reduced-order simulation constructed around clustering synchronisation that may represent the operations that comprise the whole system. Our findings indicate the influence of fractional-order is dependent on connections between neurons and the system’s stored evidence. Moreover, the processes capture the consequences of fractional derivatives on surge regularity modification and enhance delays that happen across numerous time frames in neural processing.

Simultaneously primary and super-harmonic resonance of a van der Pol oscillator with fractional-order derivative

The dynamic characteristics and stability of a fractional-order van der Pol oscillator under simultaneously primary and super-harmonic resonance are analyzed by the multiscale method. At first, the first-order approximate analytical solution of the system is obtained, and the analytical results are verified by numerical simulation. In addition, the concepts of equivalent linear damping and equivalent linear stiffness of simultaneously primary and super-harmonic resonance are proposed to enhance the comprehension of fractional parameters roles. Based on Lyapunov's stability method, qualitative analysis is performed on the stability region and the type of equilibrium points of the steady-state solution. The influences of fractional order and coefficient on system stability, equivalent damping, and equivalent stiffness are discussed. It is found that the change of fractional order and coefficient will cause a deviation of the vibration curve and affect the multiple solutions, resonant frequency, and stability range of the system, which is useful to design or control the similar dynamic system.

Nonlocal theory of propagation and reflection of plane waves in higher order thermo diffusive semiconducting medium

The present article is related to propagation and reflection of elastic wave through a nonlocal generalized thermo-diffusive semiconducting elastic solid. The non-local theory is employed to study the wave behavior. Three phase lag model with higher order fractional order derivative is incorporated to discuss heat propagation through the medium, in addition with two phase lags diffusion equation. The Helmholz vector rule is applied to decompose the system into longitudinal and transverse components. The frequency dispersion relation indicates the presence of four coupled longitudinal and one un-coupled shear vertical wave. The speed of the waves is plotted against angular frequency for local and nonlocal medium. The cutoff frequency of the waves is also depicted graphically. The longitudinal P-wave is taken to be an incident wave at the free surface of the solid to compute the reflection coefficients. The influences of fractional order and nonlocal parameters on amplitude ratios are also studied. The effect of these parameters is found to be significant. The results are proved in the context of energy conservation. The results obtained from the current investigation are very useful for scientists working on problems of geophysics and various fields of mechanics.

Modeling and Harmonic Analysis of a Fractional-Order Zeta Converter

The Zeta converter is an essential and widely used high-order converter. The current modeling studies on Zeta converters are based on the model that devices, such as capacitors and inductors, are of integer order. For this reason, this paper takes the Zeta converter as the research object and conducts an in-depth study on its fractional-order modeling. However, the existing modeling and analysis methods have high computational complexity, the analytical solutions of system variables are tedious, and it is difficult to describe the ripple changes of state variables. This paper combines the principle of harmonic balance with the equivalent small parameter method (ESPM); the approximate analytic steady-state solution of the state variable can be obtained in only three iterative steps in the whole solving process. The DC components and ripples of the state variables obtained by the proposed method were compared with those obtained by the Oustaloup’s filter-based approximation method; the symbolic period results obtained by ESPM had sufficient precision because they included more combinations of higher harmonics. Finally, the influence of fractional order on harmonics were analyzed. The obtained results show that the proposed method has the advantage of being less computational and easily describing changes in the ripple of the state variables. The simulation results are provided for validity of the theoretical analysis.

Fractional Dual-Phase-Lag Non-Fourier Heat Transfer in a Bimaterial with a Circular Interface Insulator

The transient temperature response of a bimaterial with a circular insulated interface region is studied under sudden heating or cooling. The time-fractional dual-phase-lag heat conduction model is adopted to simulate the non-Fourier effect. The problem is reduced to an initial-boundary value problem. The Laplace transform is applied to convert the problem to a mixed boundary value problem, and then the Hankel transform reduces it to a Fredholm integral equation. Special situations for asymptotic thermal behavior near the insulated circular edge and for the steady-state cases are discussed, respectively. The dynamic intensity factors of heat flux and temperature gradient near the insulated circular edge are computed numerically through Stehfest’s Laplace inversion transform technique. The influences of fractional order and relaxation times on the instantaneous temperature change are analyzed. The exact solution of temperature fields for the steady-state case is derived and displayed graphically. The wave-like diffusion behavior of the fractional dual-phase-lag model is interpreted.

Exploration of bifurcation and stability in a class of fractional-order super-double-ring neural network with two shared neurons and multiple delays

Complex multi-ring coupling structures are common in real-world networks. However, the current work is mainly limited to unidirectional multi-ring or those sharing a neuron. This paper is devoted to the stability and Hopf bifurcation of a class of double-ring neural network with two shared neurons and multiple time delays, where n and m neurons are distributed on the double-ring, respectively. First, we obtain the characteristic equation of the network at the trivial equilibrium point by using the Coates flow graph method. Then, based on this, some sufficient conditions for the stability and Hopf bifurcation of the double-ring network under two connection modes are given. Finally, we provide some numerical examples to illustrate the validness of the theoretical results, and the influences of fractional order, network size and the distance between two shared neurons on Hopf bifurcation are also discussed.

Resonance and bifurcation of fractional quintic Mathieu-Duffing system.

In this paper, the main subharmonic resonance of the Mathieu-Duffing system with a quintic oscillator under simple harmonic excitation, the route to chaos, and the bifurcation of the system under the influence of different parameters is studied. The amplitude-frequency and phase-frequency response equations of the main resonance of the system are determined by the harmonic balance method. The amplitude-frequency and phase-frequency response equations of the steady solution to the system under the combined action of parametric excitation and forced excitation are obtained by using the average method, and the stability conditions of the steady solution are obtained based on Lyapunov's first method. The necessary conditions for heteroclinic orbit cross section intersection and chaos of the system are given by the Melnikov method. Based on the separation of fast and slow variables, the bifurcation phenomena of the system under different conditions are obtained. The amplitude-frequency characteristics of the total response of the system under different excitation frequencies are investigated by analytical and numerical methods, respectively, which shows that the two methods achieve consistency in the trend. The influence of fractional order and fractional derivative term coefficient on the amplitude-frequency response of the main resonance of the system is analyzed. The effects of nonlinear stiffness coefficient, parametric excitation term coefficient, and fractional order on the amplitude-frequency response of subharmonic resonance are discussed. Through analysis, it is found that the existence of parametric excitation will cause the subharmonic resonance of the Mathieu-Duffing oscillator to jump. Finally, the subcritical and supercritical fork bifurcations of the system caused by different parameter changes are studied. Through analysis, it is known that the parametric excitation coefficient causes subcritical fork bifurcations and fractional order causes supercritical fork bifurcations.

Effect of fractional order on unsteady magnetohydrodynamics pulsatile flowof blood inside an artery

This manuscript aims to investigate the velocity profile for the blood flow through an artery subject to magnetic field. It has been investigated how periodic acceleration of the body and slip conditions affect the irregular pulsatile blood flow across a porous media inside an artery if a magnetic field is present, under the assumption that blood is an incompressible electrically conducting fluid. A mathematical formulation involving Caputo fractional derivative serves as the basis of study. An analytical solution for fluid velocity is developed with the help of finite Hankel and Laplace transforms. The influence of fractional order on the fluid velocity is illustrated with the help of graphical simulations. The obtained results will be helpful in future research for the treatment of stenosis and other cardiovascular diseases.

MODELING AND CHARACTERISTIC ANALYSIS OF FRACTIONAL-ORDER BOOST CONVERTER BASED ON THE CAPUTO–FABRIZIO FRACTIONAL DERIVATIVES

This paper presents a novel approach for modeling Boost converters using the Caputo–Fabrizio (C-F) definition-based fractional-order model to address singular characteristics in fractional-order definitions and enhance model accuracy. A small signal modeling method is proposed to improve the accuracy of circuit parameter design and to derive state-averaged models, state-space equations, and transfer functions. The influence of capacitor and inductor orders on steady-state characteristics is analyzed and the influence of fractional-order on ripple characteristics is investigated through simulation. When the fractional-order approaches 1, the output voltage increases and the inductance current decreases, with waveform jitter mitigation. Moreover, boundary conditions for continuous conduction mode operation are established based on ripple characteristics. The numerical and circuit-oriented simulations verify the correctness of the proposed model. Finally, the orders and accurate parameters of capacitors and inductors based on the C-F definition are determined and the experiments are conducted. The comparison between the experimental and simulation results demonstrates that the proposed model can accurately describe the steady-state characteristics of the practical circuit systems, which further validates the accuracy of the proposed method.

Dynamical analysis of fractional oscillator system with cosine excitation utilizing the average method

In this paper, an analytical and numerical algorithm for solving displacement response of fractional oscillator system with cosine excitation is established. The analytical method is to solve steady‐state response and transient response solutions of the system by means of average method and then the total displacement response solution is the sum of steady‐state solution and transient solution. The numerical method is to use the Grünwald–Letnikov definition of fractional derivatives to discretize the fractional differential term in the system and then to reduce the order of the original system. Then, the approximate response solution of the system under the general periodic excitation is obtained by using the Fourier series expansion method and the superposition principle of linear system. Finally, the effectiveness and feasibility of the analytical technique are verified by numerical simulation, and the influence of fractional order, linear damping coefficient and fractional derivative coefficient on the amplitude of the steady‐state response and the total displacement response of the system is analyzed.

An analytical integration Legendre polynomial series approach for Lamb waves in fractional order thermoelastic multilayered plates

The Legendre polynomial series approach (LPSA) has been widely used to solve guided wave propagation in various structures since 1999. The LPSA directly introduces the boundary conditions into the control equations through the rectangular window function. However, in the solving process, the Legendre polynomial series, the rectangular window function, and their derivatives are introduced into the integral kernel functions, which results in a lot of CPU time on the abundant numerical integration calculations. To overcome this defect, an analytical integration Legendre polynomial series approach (AILPSA) is presented and is used to solve the Lamb waves in fractional order thermoelastic multilayered plates. Coupled wave equations and heat conduction equation are solved by the AILPSA and the LPSA, respectively. Comparison between two approaches indicates the computational efficiency of the AILPSA is improved by more than 90%. In addition, a backward error estimation is given in the convergency analysis to make up for the deficiency of the numerical experiments. Finally, the influence of fractional order on the thermoelastic Lamb wave is discussed.

Firing activities in a fractional-order Hindmarsh–Rose neuron with multistable memristor as autapse

Considering the fact that memristors have the characteristics similar to biological synapses, a fractional-order multistable memristor is proposed in this paper. It is verified that the fractional-order memristor has multiple local active regions and multiple stable hysteresis loops, and the influence of fractional-order on its nonvolatility is also revealed. Then by considering the fractional-order memristor as an autapse of Hindmarsh–Rose (HR) neuron model, a fractional-order memristive neuron model is developed. The effects of the initial value, external excitation current, coupling strength and fractional-order on the firing behavior are discussed by time series, phase diagram, Lyapunov exponent and inter spike interval (ISI) bifurcation diagram. Three coexisting firing patterns, including irregular asymptotically periodic (A-periodic) bursting, A-periodic bursting and chaotic bursting, dependent on the memristor initial values, are observed. It is also revealed that the fractional-order can not only induce the transition of firing patterns, but also change the firing frequency of the neuron. Finally, a neuron circuit with variable fractional-order is designed to verify the numerical simulations.

Analysis of fractional hybrid differential equations with impulses in partially ordered Banach algebras

In this paper, we investigate a class of fractional hybrid differential equations with impulses, which can be seen as nonlinear differential equations with a quadratic perturbation of second type and a linear perturbation in partially ordered Banach algebras. We deduce the existence and approximation of a mild solution for the initial value problems of this system by applying Dhage iteration principles and related hybrid fixed point theorems. Compared with previous works, we generalize the results to fractional order and extend some existing conclusions for the first time. Meantime, we take into consideration the effect of impulses. Our results indicate the influence of fractional order for nonlinear hybrid differential equations and improve some known results, which have wider applications as well. A numerical example is included to illustrate the effectiveness of the proposed results.

Bridge-Ship Collision Avoidance Control Based on AFSMC with a FRNN Estimator

For collision avoidance and maneuvering control in bridge areas, an adaptive fractional sliding mode control with fractional recurrent neural network (FRNN-AFSMC) is proposed. The uncertainties are estimated by FRNN, and the fractional gradient is adopted to improve the recurrent neural network (RNN). Its convergence has been proven. The influence of fractional order on algorithm performance is analyzed, and the simulation platform of ship collision avoidance control is built. Dynamic collision avoidance of multiple ships is simulated and verified. The results show the feasibility and effectiveness of dynamic autonomous collision avoidance motion control in a dynamic ocean environment.

ON DYNAMIC BEHAVIOR OF A DISCRETE FRACTIONAL-ORDER NONLINEAR PREY–PREDATOR MODEL

This work is devoted to explore the dynamics of the proposed discrete fractional-order prey–predator model. The model is the generalization of the conventional discrete prey–predator model to its corresponding fractional-order counterpart. The fixed points of the proposed model are first found and their stability analyses are carried out. Then, the nonlinear dynamical behaviors of the model, including quasi-periodicity and chaotic behaviors, are investigated. The influences of fractional order and different parameters in the model are examined using several techniques such as Lyapunov exponents, bifurcation diagrams, phase portraits and [Formula: see text] complexity. The feedback control method is suggested to suppress the chaotic dynamics of the model and stabilize any selected unstable fixed point of the system.

A quadrature-free Legendre polynomial approach for the fast modelling guided circumferential wave in anisotropic fractional order viscoelastic hollow cylinders

Compared to the traditional integer order viscoelastic model, a fractional order derivative viscoelastic model is shown to be advantageous. The characteristics of guided circumferential waves in an anisotropic fractional order Kelvin–Voigt viscoelastic hollow cylinder are investigated by a quadrature-free Legendre polynomial approach combining the Weyl definition of fractional order derivatives. The presented approach can obtain dispersion solutions in a stable manner from an eigenvalue/eigenvector problem for the calculation of wavenumbers and displacement profiles of viscoelastic guided wave, which avoids a lot of numerical integration calculation in a traditional polynomial method and greatly improves the computational efficiency. Comparisons with the related studies are conducted to validate the correctness of the presented approach. The full three dimensional spectrum of an anisotropic fractional Kelvin–Voigt hollow cylinder is plotted. The influence of fractional order and material parameters on the phase velocity dispersion and attenuation curves of guided circumferential wave is discussed in detail. Moreover, the difference of the phase velocity dispersion and attenuation characteristics between the Kelvin–Voigt and hysteretic viscoelastic models is also illustrated. The presented approach along with the observed wave features should be particularly useful in non-destructive evaluations using waves in viscoelastic waveguides.

DYNAMICS OF A FRACTIONAL-ORDER BAM NEURAL NETWORK WITH LEAKAGE DELAY AND COMMUNICATION DELAY

This paper lucubrates the bifurcations of a fractional-order bidirectional associative memory neural network (FOBAMNN) with leakage and communication delays. Firstly, leakage delay is taken as a bifurcation parameter to detect the bifurcations of the proposed neural network (NN). Leakage delay-induced bifurcation results are commendably established. Then communication delay is regarded as a bifurcation parameter to investigate the bifurcations of the developed FOBAMNN, and the stability interval and bifurcation point are derived. It reveals that the devised FOBAMNN possesses outstanding stability performance if selecting a lesser value of them, while the bifurcation emerges of FOBAMNN and eventually leads to performance deterioration with an outsize one. Besides, the influence of fractional order on the bifurcation points is carefully studied. It perceives that a proper fractional order can heighten the stability performance of the developed FOBAMNN. Ultimately, numerical simulations are exploited to underpin the developed theory.

Nonlinear Vibration Analysis of Viscoelastic Smart Sandwich Plates Through the use of Fractional Derivative Zener Model

In this paper, the nonlinear vibrational behavior of a sandwich plate with embedded viscoelastic material is studied through the use of constitutive equations with fractional derivatives. The studied sandwich structure is consisted of a viscoelastic core that is located between the faces of functionally graded magneto-electro-elastic (FG-MEE). In order to determine the frequency-dependent feature of the viscoelastic layer, four-parameter fractional derivative model is utilized. The material properties of FG-MEE face sheets have been distributed considering the power law scheme along the thickness. In addition, for derivation of the governing equations on the sandwich plate, first-order shear deformation plate theory along with von Karman-type of kinematic nonlinearity are implemented. The derived partial differential equations (PDEs) have been transformed to the ordinary differential equations (ODEs) through the Galerkin method. After that, the nonlinear vibration equations for the sandwich plate have been solved by multiple time scale perturbation technique. Moreover, for evaluating the effect of different parameters such as electric and magnetic fields, fractional order, the ratio of the core-to-face thickness and the power low index on the nonlinear vibration characteristics of sandwich plates with FG-MEE face sheets, the parametric analysis has been performed. The obtained results revealed the enhanced nonlinear natural frequency through an increment in the fractional order. Furthermore, the prominent influence of fractional order on the nonlinear frequency of sandwich plate was declared at the negative electric potential and positive magnetic potential.

Circumferential thermoelastic Lamb wave in fractional order cylindrical plates

AbstractAn improved Legendre polynomial series approach (AILPSA) is presented to investigate the circumferential thermoelastic Lamb wave in a fractional order orthotropic cylindrical plate. In the AILPSA, the analytical integration is developed based on the orthogonality and recursive properties of the Legendre polynomial to simplify the integral computation involved in the solving progress. As a consequence, the computational efficiency is improved significantly. Results are compared with those from the previous article to confirm the validity of the introduced method. Using the AILPSA, guided wave characteristics in various fractional order orthotropic cylindrical plates are investigated by solving the eigenvalues and eigenvectors of the system of algebraic equations. The influences of fractional order and radius‐thickness ratio on dispersion curve and displacement, temperature amplitudes are illustrated. The results show that the fractional order almost has no effect on the phase velocities of quasi‐elastic wave modes, but has notable effect on their attenuations. A smaller fractional order means a smaller temperature amplitude for the quasi‐elastic wave mode, but means a larger temperature amplitude for the thermal wave mode. A smaller radius thickness ratio indicates a smaller attenuation and a larger temperature amplitude.