Theory Of Fractional Differential Equations Research Articles

Stability and bifurcation analysis of a fractional-order single-gene regulatory model with delays under a novel PDα control law

In this paper, we propose a novel fractional-order proportional-derivative (PD) strategy to achieve the control of bifurcation of a fractional-order gene regulatory model with delays. The stability theory of fractional differential equations proved that with delays, some explicit conditions for the local asymptotical stability and Hopf bifurcation are given for the controlled fractional-order genetic model. It is demonstrated that the fractional-order gene regulatory model becomes controllable by adjusting the control gain parameters. In addition, the effect of fractional-order parameter on the dynamical behaviors is shown. Finally, numerical simulations are carried out to testify the validity of the main results and the availability of the fractional-order PD controller.

ЭКОНОМИЧЕСКИЙ ТРЕК МОДЕЛИ ОНТОГЕНЕЗА РЕГИОНАЛЬНЫХ ВУЗОВ

The article is devoted to the issues of modeling the economic processes of ontogenesis that arise in the universities of a new formation. The aim of the study is to substantiate the subject area and build a mathematical model of the economic ontogenesis of a higher educational institution of a new level, the specificity of which is the fact that an educational institution has the opportunity to become an integrator, a kind of platform for ensuring economic superiority at the regional and global levels. The subject area of the model is based on the concepts and methods of the theory of innovation. The activities of the university are considered in three spaces: the space of knowledge, the space of consent and the space of innovation. Using the methods of the theory of fractional differential equations, a continuous economic and mathematical model has been built, which, at certain values of the input parameters, has a unique solution and correctly correlates with the experimental data. The proposed model is the main element of the multi-criteria model of interaction between the university economy and the economy of the region.

Fractional inequalities associated with a generalized Mittag-Leffer function and applications

Mittag-Leffer function is very useful in the theory of fractional differential equations. Ostrowski?s inequality is very important in numerical computations and error analysis of numerical quadrature rules. In this paper, Ostrowski?s inequality via generalized Mittag-Leffer function is established. In application point of view, bounds of fractional Hadamard?s inequalities are straightforward consequences of these inequalities. The presented results are also contained in particular several fractional inequalities and have connection with some known and already published results.

Global $$O(t^{-\alpha })$$ Synchronization of Fractional-Order Non-autonomous Neural Network Model with Time Delays Through Centralized Data-Sampling Approach

This paper aims to investigate the global $$O(t^{-\alpha })$$ synchronization of a class of fractional-order non-autonomous neural networks with time delay. Using centralized data-sampling principle and the theory of fractional differential equations, sufficient criteria for the $$O(t^{-\alpha })$$ synchronization is derived. Centralized data-sampling control is applied in the drive-response-based coupled neural networks to achieve global $$O(t^{-\alpha })$$ synchronization. It is a more effective strategy as it gives better control performance. Numerical examples are also given to illustrate the validity of the theoretical result.

Consensus-based decision making in non-linearly multi-coupled IoT networked SLAM operations

This study presents a modeling method for control system of multi-coupled non-linear cyber physical system with the help of second-order dynamics of communicating membrane system of neuronal population based on a population coding algorithm. Here, the communicating membrane system is modeled based on several IoT devices connected in peer to peer form and the theory of fractional differential equations is as governing function over them. Here, the difficulty is in developing a collaborative learning algorithm for finding collaborative key points within high-dimensional decision space. In order to overcome such a problem, a gradient-based method is required to enable distributed control and collaborative decision making. Here, at every interval of the sampling time, all the subsystems are optimally synchronized in P population system by multiset-rewriting rules including the effect of symport/antiport systems. This is an interesting technical problem, related to the borderline between universality and non-universality of continuity of distributed coupling of multi-coupled control system. The presented study discusses the experimental proof of the algorithmic framework and model for SLAM operation. The consensus architecture in this domain will enable the development of evolving architecture of a non-linear cyber physical system. The close loop stability and the recursive feasibility of the evolved architecture are also studied.

Global Mittag-Leffler stability and synchronization analysis of fractional-order quaternion-valued neural networks with linear threshold neurons

This paper talks about the stability and synchronization problems of fractional-order quaternion-valued neural networks (FQVNNs) with linear threshold neurons. On account of the non-commutativity of quaternion multiplication resulting from Hamilton rules, the FQVNN models are separated into four real-valued neural network (RVNN) models. Consequently, the dynamic analysis of FQVNNs can be realized by investigating the real-valued ones. Based on the method of M-matrix, the existence and uniqueness of the equilibrium point of the FQVNNs are obtained without detailed proof. Afterwards, several sufficient criteria ensuring the global Mittag-Leffler stability for the unique equilibrium point of the FQVNNs are derived by applying the Lyapunov direct method, the theory of fractional differential equation, the theory of matrix eigenvalue, and some inequality techniques. In the meanwhile, global Mittag-Leffler synchronization for the drive–response models of the addressed FQVNNs are investigated explicitly. Finally, simulation examples are designed to verify the feasibility and availability of the theoretical results.

基于激励滑模控制的分数阶神经网络的修正投影同步研究

The modified projective synchronization of a class of fractionalorder neural networks was studied. An appropriate active controller was firstly selected to facilitate the design of the sliding mode controller. Afterwards, a suitable switching plane and 2 effective reaching laws were defined and several criteria were established to ensure the synchronization of the driveresponse systems based on the sliding mode control theory and the theory of fractional differential equations. Numerical examples verify the validity and feasibility of the theoretical results.

Global Mittag-Leffler Synchronization of Fractional-Order Neural Networks Via Impulsive Control

This paper aims at analyzing the impulsive synchronization of fractional-order neural works. Firstly, in view of control theory, by constructing a suitable impulsive response system with the designed controller, the synchronization error system between the drive system and the corresponding response system is given. Afterwards, based on the theory of impulsive differential equation, the theory of fractional differential equation, Lyapunov direct method, and inequality techniques, some effective sufficient criteria are established to guarantee the global Mittag-Leffler stability for the synchronization error system. Finally, several simulation examples are designed to demonstrate the effectiveness and feasibility of the obtained results.

A note on asymptotic behaviour of Mittag–Leffler functions

ABSTRACTIn this paper, we give a short note on the asymptotic behaviour of the two parameter Mittag–Leffler function. Useful results are collected for the reader and also explicit estimation formulas for this function are obtained which will play a role in existence and stability theory of fractional differential equations.

Quasi-uniform synchronization of fractional-order memristor-based neural networks with delay

Quasi-uniform synchronization of delayed fractional-order memristor-based neural networks (FMNNs) is discussed in this paper. On the basis of the theory of fractional differential equations and the theory of differential inclusion, the synchronization error system between the concerned drive system and the associated response system is formulated, and then, by employing Hölder inequality, Cp inequality and Gronwall-Bellman inequality, several sufficient criteria are proposed to ensure the quasi-uniform synchronization for the considered delayed FMNNs. Three simulation examples are also presented to illustrate the availability and correctness of the theoretical results.

基于分数阶微分方程的手机病毒传播模型研究

随着手机普及率的提高，手机病毒的传播也愈发严重。因此，对于病毒传播规律的研究必不可少。本文建立并研究了一类基于分数阶微分方程的手机病毒传播模型，利用分数阶微分方程的相关理论，详细分析了平衡点的存在性及其局部稳定性，并通过数值试验验证了理论结果的正确性。通过研究，我们得到在基本再生数小于1的情况下，未感染平衡点是局部渐进稳定的，病毒会消亡；在基本再生数大于1时，感染平衡点局部渐近稳定，病毒将扩散。根据所得到的理论结果给出了控制手机病毒传播的有效措施，为手机病毒的预测、控制和防治提供了重要的参考依据。 With the increasing popularity of mobile phones, the spread of mobile virus has become increas-ingly serious. Therefore, the study of the spread of the virus is necessary. In this paper, we first establish an epidemic model of mobile phone virus based on the fractional differential equation. Then by means of the theory of fractional differential equations, we analyse the existence and sta-bility condition of the equilibrium of the model. It is showed that if the basic reproduction number is less than 1, the infection free equilibrium is locally asymptotically stable and virus will die out, and if the basic reproductive number is greater than 1, the infection equilibrium is stable and the virus will spread. Some numerical experiments are carried out to confirm the obtained results. In addition, some effective measures are given to control the spread of the mobile virus, which pro-vides referential support for virus prediction, control and prevention.

Centralized and Decentralized Data-Sampling Principles for Outer-Synchronization of Fractional-Order Neural Networks

This paper aims to investigate the outer-synchronization of fractional-order neural networks. Using centralized and decentralized data-sampling principles and the theory of fractional differential equations, sufficient criteria about outer-synchronization of the controlled fractional-order neural networks are derived for structure-dependent centralized data-sampling, state-dependent centralized data-sampling, and state-dependent decentralized data-sampling, respectively. A numerical example is also given to illustrate the superiority of theoretical results.

On Gronwall’s type integral inequalities with singular kernels

In this paper, the generalizations of Gronwall?s type integral inequalities with singular kernels are established. In applications, theorems on stability estimates for the solutions of the nonliner integral equation and the integral-differential equation of the parabolic type are presented. Moreover, these inequalities can be used in the theory of fractional differential equations.

Analytical solutions of fractional foam drainage equation by residual power series method

The current work highlights the following issues: a brief survey of the development in the theory of fractional differential equations has been raised. A very recent technique based on the generalized Taylor series called—residual power series (RPS)—is introduced in detailed manner. The time-fractional foam drainage equation is considered as a target model to test the validity of the RPS method. Analysis of the obtained approximate solution of the fractional foam model reveals that RPS is an alternative method to be added for the fractional theory and computations and considered to be a significant method for exploring nonlinear fractional models.

Adaptive pinning cluster synchronization of fractional-order complex dynamical networks

The problem about cluster synchronization of fractional-order CDNs is studied via a pinning adaptive approach in this paper. Based on the stability theory of fractional differential equations, some sufficient criteria for local and global cluster synchronization of fractional-order CDNs are derived. In this paper, the coupling configuration matrix can be asymmetric as well as reducible and the inner coupling matrix can also be asymmetric. Moreover, the number of pinning nodes in each cluster can be evaluated. Especially, when the coupling strength is large enough and the coupling configuration matrix is symmetric, cluster synchronization can be achieved via pinning a single node in each cluster. Finally, some typical examples are given to illustrate the correctness and effectiveness of our results, a surprising finding is that the synchronization performance will become better as the fractional order decreases in this simulation.

On the global existence of solutions to a class of fractional differential equations

We present two global existence results for an initial value problem associated to a large class of fractional differential equations. Our approach differs substantially from the techniques employed in the recent literature. By introducing an easily verifiable hypothesis, we allow for immediate applications of a general comparison type result from [V. Lakshmikantham, A.S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal. TMA 69 (2008), 2677–2682].

A fractional differential equation for a MEMS viscometer used in the oil industry

A mathematical model is developed for a micro-electro-mechanical system (MEMS) instrument that has been designed primarily to measure the viscosity of fluids that are encountered during oil well exploration. It is shown that, in one mode of operation, the displacement of the device satisfies a fractional differential equation (FDE). The theory of FDEs is used to solve the governing equation in closed form and numerical solutions are also determined using a simple but efficient central difference scheme. It is shown how knowledge of the exact and numerical solutions enables the design of the device to be optimised. It is also shown that the numerical scheme may be extended to encompass the case of a nonlinear spring, where the resulting FDE is nonlinear.

Basic theory of fractional differential equations

In this paper, the basic theory for the initial value problem of fractional differential equations involving Riemann–Liouville differential operators is discussed employing the classical approach. The theory of inequalities, local existence, extremal solutions, comparison result and global existence of solutions are considered.

Computation of the generalized Mittag-Leffler function and its inverse in the complex plane

The generalized Mittag-Leffler function Eα, β(z) has been studied for arbitrary complex argument z∈ℂ and parameters α∈ℝ+ and β∈ℝ. This function plays a fundamental role in the theory of fractional differential equations and numerous applications in physics. The Mittag-Leffler function interpolates smoothly between exponential and algebraic functional behaviour. A numerical algorithm for its evaluation has been developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations for Eα, β(z) in the complex z-plane are reported here. We find that all complex zeros emerge from the point z=1 for small α. They diverge towards −∞+(2k−1)πi for α→1− and towards −∞+2kπi for α→1+ (k∈ℤ). All the complex zeros collapse pairwise onto the negative real axis for α→2. We introduce and study also the inverse generalized Mittag-Leffler function Lα, β(z) defined as the solution of the equation Lα, β(Eα, β(z))=z. We determine its principal branch numerically.

粘弾性振動子の強制調和振動

The equation of motion for the forced harmonic vibration of a viscoelastic oscillator is formulat-ed in terms of a fractional derivative in addition to ordinary derivatives. This equation of motion is exactly solved using the theory of fractional differential equations. An equivalent model of the viscoelastic oscillator is derived and used to estimate the responses of the oscillator. The validity of the equivalent model is proved by comparing the solutions of the exact model and the equivalent model for harmonic excitation. Damping property of the vibration of the viscoelastic oscillator is elucidated by the obtained responses.