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In the last decade, hundreds of papers have been published on fractional modeling. The approach is often similar, with the motivation of incorporating a "memory effect": take the integer-order differential-equation model, and replace the ordinary derivative to the left-hand side by a fractional operator/derivative. Does this make sense? Which is the memory there? What is the physical and biological interpretation of fractional models? This paper aims at investigating these important issues. With fluxes, transitions, and concepts from survival analysis (hazard function), we give a meaning for Caputo compartmental models and their non-Markovian property. Essentially, a fractional index is connected with the instantaneous risk of leaving a compartment, conditioned on the past stay in there. We distinguish between Caputo models that are purely fractional (i.e., the usual ones) and partially fractional (i.e., a mix of ordinary and Riemann–Liouville components), and we explore initialization as well. We build new Euler-type numerical schemes, rooted in probability, and assess them for different models: a simple death process, logistic growth, SIR equations, the dynamics of an HIV infection, and the evolution of the smoking habit. A detailed discussion on the pros and cons of the fractional methodology is made along the article. • Which is the role of a fractional operator in an equation? What is memory? • The paper investigates these issues, for Caputo fractional compartmental models. • We analyze the hazard function (risk of transition), initialization, non-Markovianity. • Some Caputo models in the literature are unphysical. Critical evaluation is included. • Numerical experiments are conducted with probabilistic Euler-type discrete schemes. [ABSTRACT FROM AUTHOR]

In this research, we introduce and investigate a variable-order fractional memristor neural network, focusing on its engineering applications in synchronization and image encryption. This study stands out as a pioneering effort in proposing such an architecture for image encryption purposes. Distinct from conventional fractional-order systems, our model incorporates a time-varying fractional derivative, leading to more complex behaviors. Through numerical simulations, we vividly demonstrate the chaotic dynamics of the system. Our results further reveal the system's outstanding performance in image encryption applications. To augment the system's efficiency, we introduce a robust control strategy that guarantees smooth stabilization and synchronization of the variable-order fractional system. Considering the unique variable-order fractional nature of the system, we provide theoretical validations and empirical evidence supporting its stability and convergence properties. Additionally, we present synchronization outcomes between pairs of such neural networks employing our robust control approach. Our numerical analyses firmly substantiate the superiority of our control strategy, particularly highlighting its precision, robustness, and ability to maintain chattering-free performance under external disturbances. • We introduce a novel neural network for advanced image encryption. • We showcased the emerging chaotic dynamics for enhanced encryption. • The results indicate superior encryption performance with robust control. • We exhibit a smooth synchronization and stabilization strategy. • Our empirical results prove evidence of stability and convergence. [ABSTRACT FROM AUTHOR]

This paper investigates the Sundaresan technique for modeling fractional order systems. Sundaresan, Prasad, and Krishnaswamy published this method in 1978 for modeling oscillatory and non-oscillatory systems based on the second-order integer transfer function. This technique is based on the transient response parameters. A problem of convergence of the derivative of the response in the frequency domain makes it impossible to follow Sundaresan's solution in his original paper with integer order when it is a fractional order case. The paper proposes an equation that outlines this problem. Due to the limited knowledge of the inverse Mittag-Leffler function, a reduced form of this equation is explicit to avoid the inverse problem. Results with simulated and real curve shapes show that the method works well, with a good approximation to the curve, both with simulation and real system curves. • Develop an equation for fine-tuning the parameters of the pseudo-second-order system. • Novel fractional-order identification based on Sundaresan technique. • The method proposed is a generalized method of the classic Sundaresan technique. [ABSTRACT FROM AUTHOR]

Various real-time processes are effectively represented with a fractional nonlinear autoregressive exogenous (F-NARX) model where estimation of model parameters is considered as an essential task. In this study, a population-based gazelle optimization algorithm (GOA) inspired by the evolutionary characteristics of gazelle's is exploited for the parameter estimation of the key term-separated F-NARX system. The Grünwald-Letnikov derivative, a fractional order calculus operator is integrated to develop the F-NARX system from a conventional non-linear autoregressive system. The mean square error-based merit function is developed and the effectiveness of the GOA for the F-NARX system is analyzed in terms of speedy convergence, estimation accuracy, complexity and robustness for different noise scenarios. The extendibility of the GOA is assessed through the estimation of stiff parameters of the electrically stimulated muscle model (ESMM) required for rehabilitation of paralyzed muscles. The efficacy of the GOA is endorsed through Wilcoxon signed rank statistical test in comparison with recent counterparts of Runge Kutta optimization algorithm, Whale optimization algorithm, and Harris Hawks optimization algorithm. • Gazelle optimization algorithm, GOA, is presented for parameter estimation of the fractional nonlinear, F-NARX systems. • The mean square error-based merit function is developed to evaluate the effectiveness of the GOA. • The robustness, convergence, and accuracy of the GOA are endorsed for electrically stimulated muscle model identification. • The efficacy of the GOA is established by statistical testing in comparison with the state-of-the-art counterparts. [ABSTRACT FROM AUTHOR]

Fractional calculus of complex orders in the complex domain is a rapidly growing field of interest among many mathematicians. While fractional differential equations in real variables have received much attention recently, attempts to solve such equations in complex variables have been rather scant. This research work deals with the complex order fractional differential equation with boundary conditions. The existence of solutions is established by using Dhage's fixed point theorem with some conditions, whereas the application of the Banach contraction principle obtains the uniqueness of the solution. Moreover, Ulam–Hyers stability of the considered problem is also discussed in this work. Examples and application are presented to verify the obtained results. [ABSTRACT FROM AUTHOR]

The object of this study is to define existence, controllability and exponential stability results within a Sobolev-type fractional stochastic neutral equations of order 1 < ω < 2 with sectorial operator and optimal control. To do this, first, this study relies on the confluence of stochastic analysis, fractional calculus, sectorial operator, and the applicability of Sadovskii's fixed point method. First, we highlight the existence of mild solutions to the fractional stochastic control equation and then introduce the concept of approximate controllability. Next, employing an impulsive Poisson system yields sufficient conditions for guaranteeing the exponential stability of the mild solution in the mean square moment. Further, our inquiry expands into the existence of optimal control. Finally, an example is provided to illustrate the obtained theory. • This paper explores the stability of the fractional delay system of order ω ∈ (1 , 2). • This paper is the first study of a fractional jumps system with sectorial operators. • We establish conditions for the fractional system and extend to the controllability. • We determine mean square stability of fractional system ω ∈ (1 , 2) with Poisson jumps. • Finally, an optimal control pair is provided to show the solvability of the problem. [ABSTRACT FROM AUTHOR]

In this paper, the controllability for Hilfer fractional neutral stochastic differential equations with infinite delay and nonlocal conditions has been investigated. Using concepts from fractional calculus, semigroup of operators, fixed-point theory, measures of noncompactness, and stochastic theory the main controllability conclusion is attained. The applications of the key findings are finally illustrated with two examples. [ABSTRACT FROM AUTHOR]

In this paper, our main purpose is to search and obtain the controllability for Sobolev-type fuzzy Hilfer fractional integro-differential inclusions with Clarke subdifferential. Some sufficient conditions for the controllability results of this inclusion problem are proposed by using related techniques of fuzzy set theory, Sobolev-type, fractional calculus and Clarke subdifferential. The theorem of the controllability results is proved by Bohnenblust–Karlin fixed point theorem. In addition, we show an example to explain the controllability results of this inclusion problem. • We study a form of Sobolev-type equations combined with fuzzy differential inclusions. • We list some properties and hypotheses used for the main controllability theorem. • We give an example and a numerical example. [ABSTRACT FROM AUTHOR]

This research focuses on utilizing the Mittag-Leffler kernel within stochastic differential systems to estimate the controllability of nonlocal Atangana–Baleanu fractional derivatives. By assuming the automatic control of the corresponding linear system, a novel set of necessary and sufficient conditions for the approximate controllability of the fractional stochastic differential inclusions of Atangana–Baleanu is established. Furthermore, the study explores the approximate controllability of the proposed system with infinite delay. The investigation relies on the fixed-point theorem for multivalued operators and fractional calculus to derive these outcomes. Lastly, an illustrative example is provided to highlight the practical implications of the research findings. [ABSTRACT FROM AUTHOR]

Population-based metaheuristic algorithms have gained significant attention in research community due to its effectiveness in solving complex optimization problems in diverse fields. In this study, knacks of population-based Runge-Kutta optimizer (RUN) are exploited for the identification of fractional input non-linear exogenous auto-regressive (FINARX) system with key term separation. The fractional order calculus operator of the Grünwald-Letnikov derivative is exploited to develop FINARX from a conventional non-linear auto-regressive exogenous system. The identification scheme for FINARX model is implemented through a mean-square-error-based fitness function. RUN utilizes the slope variations calculated by the well-known Runge-Kutta method for an effective search mechanism in the exploration and exploitation phases. Moreover, an enhanced solution quality mechanism is employed for speedy convergence and keeping the movement toward the best solution by escaping the local optima. The robustness of the algorithm is analyzed by multiple variations of non-linearity as well as different noise scenarios. The performance of the RUN to identify the FINARX system is validated in terms of convergence rate, fitness value, robustness, and accuracy in weight estimation. The effectiveness of the RUN is further assessed through exhaustive simulations with their statistics as well as comparison with the standard recent counterparts, including the Whale optimization algorithm, Reptile Search algorithm, and Aquila optimizer on different performance indices for the FINARX system. [ABSTRACT FROM AUTHOR]

The main objective of the present article is to employ the ψ -Hilfer pseudo-fractional derivative (HPFD) to examine the reachability criteria for fractional dynamical systems with mixed delays in control of order ϑ ∈ (0 , 1) and type ϱ ∈ [ 0 , 1 ]. we derived the sufficient and necessary conditions for the reachability criterion of fractional linear dynamical systems by utilizing the positiveness of Grammian matrices, which are defined by the Mittag-Leffler functions. The sufficient conditions for the reachability criteria of fractional nonlinear dynamical systems are obtained by using Banach's fixed point theorem. To help grasp the theoretical results, only a limited number of numerical examples are provided. • To examine the reachability of dynamical systems with mixed delays in control. • To describe the dynamics of systems with mixed delays in control using ψ -HPFD. • To get necessary and sufficient conditions for the reachability of linear systems. • To derive sufficient conditions for the reachability of nonlinear systems. [ABSTRACT FROM AUTHOR]

This paper provides a novel approach to studying the chaos control and synchronization of fractional-order hybrid Darcy–Brinkman systems (FOHDBs). We use the truncated Galerkin method to transform the system of partial differential equations (PDEs) into ordinary differential equations (ODEs) based on Fourier modes in a nonlinear low-dimensional Brinkman model. In the nonlinear complex system, hybrid nanoparticles and fluid flow physical characteristics are considered to be external disturbances. As a result, the implications of model uncertainty and external disruptions are fully absorbed into the problem, further complicating it and resulting in highly complex and unpredictable behaviors. In addition, To assure the occurrence of the sliding motion in a finite amount of time, an appropriate robust fractional sliding mode technique is proposed. Sliding motion is shown to occur in finite time using the fractional Lyapunov stability. Following that, we discussed control techniques for achieving infinitesimally close to an equilibrium of chaotic states and tracking errors. It is important to note that FOHDBs and external stochastic disturbances can be controlled using the proposed fractional nonsingular terminal sliding mode control technique. Finally, the effectiveness and applicability of the suggested finite-time control technique are graphically illustrated by the analytical and numerical simulations. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for solving fractional delay differential equations of variable order using the spectral element method. The proposed method overcomes the limitations of traditional spectral methods, such as poor approximation in long intervals and inefficiency in high degrees. By introducing a variable order differentiation matrix and using basic Lagrangian functions to approximate the solution in each element, the method achieves high accuracy and efficiency. A penalty method is also applied to minimize the jump of fluxes at interface points, and the effectiveness of this approach is analyzed. Finally, three benchmark problems are solved, and the convergence analysis demonstrates the effectiveness and efficiency of the proposed method. In essence, this paper offers a significant contribution to the literature on fractional differential equations and their numerical solution methodologies. • New approach for solving fractional delay differential equations using spectral element method. Overcomes limitations of traditional methods. • Introduces variable order differentiation matrix and Lagrangian functions for high accuracy. Penalty method minimizes flux jumps at interface points. • Rigorous convergence analysis of proposed method. • Effectiveness demonstrated with practical examples and discretization. [ABSTRACT FROM AUTHOR]

This paper aims to develop definitions and properties about the generalized Laplace transform, fractional integral and derivative on time scales. On the basis of the α − derivative and generalized exponential function, the Laplace transform is extended to the generalized case with respect to another function on time scales. Then, the generalized fractional integral and derivative on time scales are defined by employing the inverse generalized Laplace transform to unify a variety of definitions about continuous and discrete fractional calculus. Moreover, some significant theorems are derived to further increase the availability of these proposed operators. Finally, two examples with different kernel functions are given to verify the feasibility of the theoretical results. • Some new definitions of calculus and generalized exponential function are proposed. • The generalized Laplace transform with respect to another function on time scales is established. • The generalized fractional integral and derivative on time scales are defined firstly. • These results provide crucial tools for fractional calculus theory applied to various practical problems. [ABSTRACT FROM AUTHOR]

In this study, a novel investigation in developing intelligent adaptive Bayesian networks (IABN) is carried out to solve the fractional order Parkinson's disease model (FOPDM) represented with a system of three fractional differential equations reflecting the brain's electrical activity rhythms at different cerebral cortex positions. The IABN are developed by using the competency of the multi-layer architecture of neural networks with backpropagation through Bayesian regularization optimization scheme. The reference dataset for IABN is created through Grunwald-Letnikov fractional derivative-based numerical solver for FOPDM in case of varying sensor locations on the cerebral cortex, as well as, considering different fractional orders in FOPDM. The proposed IABN is applied to the created datasets arbitrarily partitioned into training and testing sets by optimizing the fitness criterion based on mean square error (MSE) metric. The detailed analyses of the proposed IABN through extensive simulations and comparison with the reference numerical solutions of FOPDM in terms of MSE training/testing plots, absolute error, training/testing regression plots, and error histogram results, endorse the worth of the proposed intelligent scheme for different fractional orders. [Display omitted] • The fractional order model of Parkinson's disease is presented representing the brain electrical activity rhythms. • Intelligent adaptive Bayesian networks, IABN, are proposed to investigate the dynamics of fractional model. • Grunwald-Letnikov fractional derivative based numerical solver is used to generate the reference dataset. • The accuracy of the IABN is verified by comparison with reference numerical solutions for varying fractional orders. [ABSTRACT FROM AUTHOR]

The new idea of Atangana and Baleanu introduced recently for fractional derivatives has received tremendous attention from researchers of various fields. However, this idea has not been applied for heat dissipation in transmission line of electrical circuit. Although the phenomenon of heat dissipation potentially damages the electrical devices which leads to catastrophic failure and shortens the life expectancy of costly electrical components. This manuscript signifies an electro-analog model of fractional diffusion process which enables to simulate heat dissipation in circuit board. The modeling of the problem has been established through modern operators of fractional calculus via the governing differential equation. By invoking Laplace transform on fractional governing equation of heat dissipation in transmission line of circuit, the analytic solutions are investigated through the exponential and Mittage Leffler kernel of non-integer order differentiations. The duality of analytic solutions has been compared with Caputo–Fabrizio and Atangana–Baleanu fractional differentiations graphically using embedded pertinent parameters. The comparison of both approaches of fractional differentiations has disclosed the heat transfer of circuit board and diffusion process with dissipation vividly. [ABSTRACT FROM AUTHOR]

Human being live in three-dimensional space; they can accurately visualize processes taking place in one, two and three dimensions. Although the set of bi-complex numbers and quaternion have attracted attention of many researchers in physics and related branches, they do not really represent processes taking place in the space where human being are located. We suggested a new set of complex number called "the Trinition". The new set is comprised between complex number with one imaginary part and complex number with three imaginary parts called quaternion/bi-complex numbers. We established a bijection between the new set and the three-dimensional space. We presented some important properties of the new set. We showed that all chaotic attractors in three dimension are simply three-dimensional mapping in the new set. Fewer examples of mapping in such set were presented. A new methodology that can be used to obtain more strange attractors are equally suggested. The methodology combines fractional chaotic models and some fractal mapping within the new set. Some illustrative figures are presented. [ABSTRACT FROM AUTHOR]

Economics and finance are extremely complex nonlinear systems involving human subjects with many subjective factors. There are numerous attribute properties that cannot be described by the theory of integer-order calculus; so it is necessary to theoretically study the internal complexity of the economic and financial system using the method of bifurcation and chaos of fractional nonlinear dynamics. Fractional calculus can more accurately describe the existence characteristics of complex physical, financial or medical systems, and can truly reflect the actual state properties of these systems; therefore the application of fractional order in chaotic systems has great significance to study the mathematical analysis of nonlinear dynamic systems, and the use of fractional calculus theory to model the complexity evolution of fractional chaotic financial systems has attracted more and more scholars' attention. On the basis of summarizing and analyzing previous studies, this paper qualitatively analyzes the stability of equilibrium solution of fractional-order chaotic financial system, and explores the complexity evolution law of the financial system near the equilibrium point and the occurring conditions of asymptotic chaotic state near this equilibrium point, and simulate the complexity evolution of chaotic financial systems using the Admas-Bashforth-Moulton finite difference method for mapping, phase diagram and time series graph. The research results of this paper provide a reference for government to formulate relevant economic policies, decision-making or further research on the complexity evolution of fractional-order chaotic financial systems. [ABSTRACT FROM AUTHOR]

• Adaptive fractional control is design to synchronize chimera states. • Atangana–Baleanu fractional derivative with Mittag-Leffler (M-L) kernel is considered. • Numerical simulations are presented for specific parameters. • The adaptive controller is based on the uncertainty of the coupling parameters. This work presents the Hindmarsh–Rose fractional model of three-state using the Atangana–Baleanu–Caputo fractional derivative with strong memory. The model allows simulating the chimera states in a neural network. To achieve the synchronization was developed a fractional adaptive controller which is based on the uncertainty of the coupling parameters. The synchronization was studied using different fractional-orders and for 15, 40, 65 and 90 neurons. We consider fractional derivatives with nonlocal and non-singular Mittag-Leffler law. The simulations results show that the neurons synchronization is reached using the proposed method. We believe that the application of fractional operators to synchronization of chimera states open a new direction of research in the near future. [ABSTRACT FROM AUTHOR]

Meteorological data is the basis for climate prediction and various scientific research. It is very important to study and prediction meteorological data. At present, the prediction methods for meteorological data are mainly a single intelligent method or a single point time series method based on time series data, which ignoring the interaction between meteorological sites and multiple meteorological factors. In this paper, Meteorological space-time data is decomposed into high frequency component and low frequency component by Hilbert Huang Transform. The high frequency component is modeling and predicting by fractional calculus grey model. The low frequency component is modeling and predicting by multivariate space-time auto regression model. The model verification results show that compared with the existing time series prediction methods, this method can more fully explain the non-stationary and nonlinear dynamic process of multivariate meteorological space-time sequences. [ABSTRACT FROM AUTHOR]

In this paper, the author proposes a new stock financial market stochastic volatility and stock pricing model based on the Taylor formula, by embedding the strictly increasing harmonic steady-state process as a time variable into the Brownian motion with drift terms. The use of variance gamma and normal inverse high-lower distribution are special forms of NTS distribution, combined with principal component analysis (PCA) and artificial neural network (ANN) methods, for nonlinear and multi-scale complex financial time series in financial markets. The model is constructed and predicted, and the forecast and calculation of stock market index and foreign exchange rate are realized. Through calculation and research, the model can fill the blank of complex time series model research in financial market. The stock signal analysis based on fractional calculus equation proposed in the thesis is based on the idea of decomposition-reconstruction-integration, which can improve the prediction accuracy of the model for the time series combined financial model. The Shanghai and Shenzhen 300 Index and foreign exchange rates selected by the paper are taken from the market real data, and the skeleton prediction model is established. It can predict the short-term trend after the stock market closes, confirming the nonlinear, multi-scale and non-stationary the prediction accuracy of the sequential decomposition prediction method of financial time series is effectively improved, and the principal component and artificial neural network method are used to compress redundant data and shorten the prediction time. [ABSTRACT FROM AUTHOR]

The occurrence of economic management crisis has seriously affected the production and operation of enterprises, the stability of capital markets and even the economic security of the entire country and the world. The use of higher mathematics in economic management is very beneficial to the economic restructuring. For example, in the Lagrangian method for solving the constraint optimization problem, the correlation function can be listed in the Lagrangian fractional calculus equation for the economic management early warning problem with many independent variables. Then take one of the factors as the dependent variable and other factors as fixed constants, and bring them into the Lagrangian fractional calculus equation, you can find the variable solution and get the extreme value of the economic management early warning algorithm. Therefore, this paper combines normative research and empirical research to study the algorithm design, theoretical analysis and numerical experiments of Lagrangian-based methods for solving constrained optimization problems. The Lagrangian fractional calculus method is used to evaluate the early warning algorithm of economic management, improve the prediction accuracy and practicability of the model, and conduct empirical research. It is expected to find a way to effectively determine whether a listed company is caught in an economic management crisis and provide early warning for the listed company's own management. [ABSTRACT FROM AUTHOR]

• Establishing a relationship between Atangana-Baleanu-Caputo derivative and semilinear systems. • Extend the principle of application of the control function on different type of fractional systems. • Prove the existence of mild solutions by adopting Darbo's fixed point techniques. • Show the effect of impulses and delay on fractional differential equations with an Atangana-Baleanu-Caputo derivative. • Build a study on delay systems by establishing some spaces. We discuss the controllability of semilinear differential equations of fractional order with impulses and delay. We make use of the Atangana-Baleanu derivative. Our main tools are semigroup theory, the fixed point theorem due to Darbo and their combination with the properties of measures of noncompactness. Our abstract results are well supported by an illustrative example. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of Hopfield neural networks involving fractal–fractional derivatives. The incorporation of fractal–fractional derivatives in the neural network framework brings forth novel modeling capabilities, capturing nonlocal dependencies, complex scaling behaviors, and memory effects. The aim of this study is to provide a comprehensive analysis of the dynamics of Hopfield neural networks with fractal–fractional derivatives, including the existence and uniqueness of solutions, stability properties, and numerical analysis techniques. Numerical analysis techniques, including the Adams–Bashforth method, are employed to accurately simulate the fractal–fractional Hopfield neural network system. Moreover, the obtained numerical data serves as validation for developing predictions using Multilayer Perceptron (MLP) and Long Short-Term Memory (LSTM) neural network methods. The findings contribute to the advancement of both fractional calculus and neural network theory, providing valuable insights for theoretical investigations and practical applications in complex systems analysis. • Examined 3D Hopfield network stability under fractal–fractional derivatives. • Computed solutions for diverse fractal and fractional dimensions. • Utilized LSTM and MLP to study numerical data. • Visualized Deep Learning vs. numerical results. • Employed Deep Learning to predict system solutions. [ABSTRACT FROM AUTHOR]

A discrete fractional-order complex-valued neural network is taken into consideration in the present study. For the existence of the solution of the considered model to be stable in finite time, certain requirements are specified. Our strategy focuses on the use of the recently formulated discrete fractional calculus, mathematical inequalities, Krasnoselskii's fixed point theorem, and the Arzelà–Ascoli theorem. We present afew numerical examples that demonstrate the theoretical results' implementation. [ABSTRACT FROM AUTHOR]

In order to describe the mechanical behaviors of viscoelastic materials that couple memory effects and time-varying viscosity properties toggled between thixotropy and rheopexy, this paper explores and establishes the constitutive equation of a unified Maxwell model with a variable kernel in terms of the ψ -Caputo fractional derivative. By virtue of a Volterra integral equation of the second kind and the ψ -Laplace transform technique, the closed-form expressions of creep and relaxation responses for the proposed model are derived and compared in detail. Furthermore, to enhance practicality, the exponential and power-law time-varying viscosity candidates are embedded into the proposed model, along with exhibiting the corresponding creep and relaxation behaviors in light of illustration and reasoning, respectively. The results may provide a fresh perspective for detecting the relationship between the viscoelastic system with nonlinear time-varying viscosity and generalized fractional calculus. [ABSTRACT FROM AUTHOR]

The biggest wish of the e-commerce industry is to forecast and estimate the taste and repugnance of users by utilizing user ratings for various products. Chaotic users' feedback drives the e-commerce industry to develop efficient, robust, and smart recommendation systems with the capability to deal with the inconsistencies in users' rating patterns and provide extremely related, appropriate, and timely recommendations. A few fractional order-based solutions for an effective matrix factorization procedure are suggested to boost recommender systems' performance regarding recommendations speed and precision. We further investigate a newly proposed enhanced fractional stochastic gradient descent (EFSGD) technique for accelerating recommendations speed through efficient matrix factorization. The Faa di Bruno fractional derivative used in EFSGD exploits the users' historical feedback for providing accurate predictions. As a result of the growing computational complexity of recommendation models, it will a challenge for EFSGD to extract group of prominent features and ignore the cluster of redundant users' and items' latent attributes for increasing the recommendation speed and reducing the computational complexity respectively. Therefore, (a) to resolve chaotic users' ratings patterns problem and (b) reduce the complexity by identifying and selecting groups of highly correlated latent features, an innovative elastic net regularized enhanced fractional adaptive model is developed for efficient matrix factorization. The proposed model for recommender systems outperforms baseline and state-of-the-art in terms of convergence speed, computational cost, and prediction accuracy. The substantial performance of the suggested strategy is verified through various latent factors, ratings prediction-based valuation measures, learning-rates, and fractional order values. Whereas the authenticity is verified through Movie-Lens and FilmTrust datasets. • Elastic net regularized enhanced fractional gradient strategy, EN-FSGD, is presented for chaotic users' rating patterns. • The fractional derivative of the objective function is calculated using Faa di Bruno formula. • The EN-EFSGD achieves a faster convergence rate as compared to the standard counterpart with lower fractional order values. • The superior performance of the EN-EFSGD is verified through two standard benchmark datasets. [ABSTRACT FROM AUTHOR]

In this article, fractional Dirac systems are considered under impulsive conditions and their basic properties are investigated. In this context, the symmetry, eigenvalues, and eigenfunctions of the system were examined and then the existence and uniqueness of the solutions of the system were proved. Finally, the characteristic function of the system is shown. • Dirac systems have an important place in the history of physics. • It is well-known that the basic physics of relativistic quantum mechanics was formulated by the Dirac system. • They provide a natural description of the electron spin and predict the existence of antimatter. • In this article, the fractional form of the Dirac system under impulsive boundary conditions will be investigated. [ABSTRACT FROM AUTHOR]

We present the Liouville–Caputo fractional difference method for the numerical evolution of the sequential differential equation of fractional order. Meanwhile, some binomial coefficients are considered in discrete fractional calculus to find and determine the corresponding sequence of continuous fractional order equations. Finally, a standard numerical test is offered in detail to demonstrate the validity of the main theorem. [ABSTRACT FROM AUTHOR]

• A mathematical model of two-dimensional generalized mythical bird, butterfly wings and paradise bird maps are considered. • Fractional conformable derivative of Khalil's and Atangana's type, the Liouville–Caputo and Atangana–Baleanu derivatives with constant and variable-order derivatives are considered. • Numerical simulations are presented for speci c parameters. • Numerical simulations of the model considering these derivatives were obtained using the Atangana–Tou k numerical method. • Mixed integration methods are considered to obtain very strange and new behaviors. In this paper, we propose a fractional form of two-dimensional generalized mythical bird, butterfly wings and paradise bird maps involving the fractional conformable derivative of Khalil's and Atangana's type, the Liouville–Caputo and Atangana–Baleanu derivatives with constant and variable-order. We obtain new chaotical behaviors considering numerical schemes based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. Also, the dynamics of the proposed maps are investigated numerically through phase plots considering combinations of these derivatives and mixed integration methods for each map. The numerical simulations show very strange and new behaviors for the first time in this manuscript. [ABSTRACT FROM AUTHOR]

• Adaptive fractional control is design to synchronize chimera states. • Atangana–Baleanu fractional derivative with Mittag-Leffler (M-L) kernel is considered. • Numerical simulations are presented for specific parameters. • The adaptive controller is based on the uncertainty of the coupling parameters. In this paper, a mathematical model which considers population dynamics among cancer stem cells, tumor cells, healthy cells, the effects of excess estrogen and the body's natural immune response on the cell populations was considered. Fractional derivatives with power law and exponential decay law in Liouville–Caputo sense were considered. Special solutions using an iterative scheme via Laplace transform were obtained. Furthermore, numerical simulations of the model considering both derivatives were obtained using the Atangana–Toufik numerical method. Also, random model described by a system of random differential equations was presented. The use of fractional derivatives provides more useful information about the complexity of the dynamics of the breast cancer competition model. [ABSTRACT FROM AUTHOR]

Researchers around the world are still wondering about the real origin and causes of hidden oscillating regimes and hidden attractors exhibited by some non-linear complex models. Such models are characterized by a dynamic with a basin of attraction that does not contain neighborhoods of equilibrium points. In this paper, we show that hidden oscillating regimes and hidden attractors can also exist in systems resulting from a combination with fractional differentiation. We apply a fractional derivative with Mittag–Leffler Kernel to a dynamical system with an exponential non-linear term and analyzed the resulting model both analytically and numerically. The combined model, which has no equilibrium points is however shown to display complex oscillating trajectories that culminate in chaos. Numerical simulations show some bifurcation dynamics with respect to the derivative order β and prove that the observed chaotic behavior persists as β varies. These observations made here allow us to say that the fractional model under study belongs to the category of systems with hidden oscillations. [ABSTRACT FROM AUTHOR]

• A comparison of Bank data in different fractional operators are proposed. • Parameter estimations using the least square method is presented. • Different fractional operators such as Caputo, Caputo–Fabrizio and Atangana–Baleanu versus real data is shown. • A novel numerical approaches for the solution of fractional and parameter fractional model are given. • Comparison of all the operators with real data is presented. The present paper investigate the dynamics of the bank data through a competition model with real field data for the year 2004–2014. Initially, we formulate a competition model for the bank data and then use different fractional approaches to simulate the model with the real data for many fractional order parameters α. Then, we present a novel approach for each fractional model and provide a graphical illustration with real data. We show that all these fractional approaches have good resemblance to each other and can be used to model such real data case. We prove in general that the results of the fractional approaches utilized here are good for modeling purpose but also we prove the results of the fractional Atangana–Baleanu operator is more accurate and flexible and can be used confidently to modeled such real case problem. [ABSTRACT FROM AUTHOR]

It is believed that Mittag-Leffler function and its variants govern the solutions of most fractional evolution equations. Is it always true? We establish some unknown and useful properties that characterize some aspects of fractional calculus above power-law. These include, the correspondence relations, the equality of mixed partials, the boundedness, the continuity and the eigen-property for the derivative with non singular kernel. Some related evolution equations are studied using those properties and they unexpectedly show that solutions are not governed by Mittag-Leffler function nor its variants. Rather, solutions are governed by unusual functions of type exponential (37), easier to handle and more friendly than power series like Mittag-Leffler functions. Numerical approximations of a second order nonhomogeneous fractional Cauchy problem are performed and show regularity in the dynamics. An application to a geoscience model, the Rössler system of equations is performed where existence and uniqueness results are provided. The system appears to be chaotic and the properties of the derivative with no singular kernel do not interrupt this chaos. We conclude by a similar application to heat model where we solve the model numerically and graphically reveal solutions comparable with expected exact ones. [ABSTRACT FROM AUTHOR]

In this work, we present a novel numerical method based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation to solve numerically fractional delay differential equations. We focus on the fractional derivative with power-law, exponential decay and Mittag-Leffler kernel of Liouville-Caputo type with constant and variable-order. The numerical methods were applied to simulate the Duffing attractor, El-Niño/Southern-Oscillation, and Ikeda systems. [ABSTRACT FROM AUTHOR]

• We provide an epidemic SIR model with long-range temporal memory governed by delay differential equations with fractional-order. • The existence of steady states and the asymptotic stability of the steady states are discussed. • The occurrence of Hopf bifurcation is captured when the time-delay passes through a critical value. • Theoretical results are validated numerically. In this paper, we provide an epidemic SIR model with long-range temporal memory. The model is governed by delay differential equations with fractional-order. We assume that the susceptible is obeying the logistic form in which the incidence term is of saturated form with the susceptible. Several theoretical results related to the existence of steady states and the asymptotic stability of the steady states are discussed. We use a suitable Lyapunov functional to formulate a new set of sufficient conditions that guarantee the global stability of the steady states. The occurrence of Hopf bifurcation is captured when the time-delay τ passes through a critical value τ *. Theoretical results are validated numerically by solving the governing system, using the modified Adams-Bashforth-Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. In some cases, the phase portrait gets stretched as the order of the derivative is reduced. [ABSTRACT FROM AUTHOR]

• The weighted Caputo–Fabrizio fractional derivative and integral are defined, and related properties are discussed. • Solutions of linear and nonlinear fractional equations are obtained in closed forms, and by implementing Banach fixed point theorem. • Applications to linear and nonlinear integro-differential equations are presented. In this paper, we define the weighted Caputo–Fabrizio fractional derivative of Caputo sense, and study related linear and nonlinear fractional differential equations. The solution of the linear fractional differential equation is obtained in a closed form, and has been used to define the weighted Caputo–Fabrizio fractional integral. We study main properties of the weighted Caputo–Fabrizio fractional derivative and integral. We also, apply the Banach fixed point theorem to establish the existence of a unique solution to the nonlinear fractional differential equation. Two examples are presented to illustrate the efficiency of the obtained results. [ABSTRACT FROM AUTHOR]

• Developing the reproducing kernel Hilbert space method within the Atangana-Baleanu fractional approach for solving the Riccati and Bernoulli equations with respect to initial conditions of necessity. • Constructing two Hilbert spaces for both range and domain space which are basis for the solution methodology of the presented method. • Framing numerical algorithm and procedure of solution with the optimal formulation of the problem. • Introducing computational simulations to delineate suitability and relevance of the calculations created. • Utilizing some useful properties of the Atangana-Baleanu fractional derivative and integral. This article is concerned with design and comprehensive study of a numerical approach for solving Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense. The proposed technique is using the reproducing kernel Hilbert space to approximate the solution to those class of fractional differential equations in the form of uniformly convergent series with respect to space variables. An accurate computational algorithm is presented to confirm the gained analysis. The relevant theorems and characterizations related to Riccati and Bernoulli equations are included among the reproducing kernel theory. Supplementary problems at the end of the article serve as a complete review of the utilized method and theories. The numerical consequences of the proposed approach are practically effective and impressive, whilst the utilized theories lay the foundation stone for addressing such issues. In light of the summary, conclusions and future recommendations are also provided. [ABSTRACT FROM AUTHOR]

The literature on fluid dynamics shows that there still exist number of unusual irregularities observed in wave motions described by the Korteweg–de Vries equation, Burgers equation or the combination of both, called Korteweg–de Vries–Burgers (KdVB) equation. In order to widen the studies in the topic and bring more clearness in the wave dynamics, we extend and analyze the KdVB-equation with two levels of perturbation. We combine the model with one of the fractional derivatives with Mittag–Leffler Kernel, namely the Caputo sense derivative with non-singular and non-local kernel (known as ABC-derivative (Atangana–Beleanu–Caputo)). After a brief look at the dynamics of standard integer KdVB-equation, we analyze the combined fractional KdVB-equation by showing its existence and uniqueness results. Numerical simulations using the fundamental theorem of fractional calculus show that the dynamics for the combined model is similar to the integer order dynamics, but highly parameterized and controlled by the order of the fractional derivative with Mittag–Leffler Kernel. [ABSTRACT FROM AUTHOR]

• Fractional-order models are more accurate than integer-order models in reflecting some properties of objects when studying practical problems. • This paper analyses the relationship between enterprise management and financial performance, mathematically models its relationship. • We constructs fractional differential equations, and tests it through empirical research. • The influence of the characteristics of management age, international experience, team size etc. on the financial performance of the company. With the development of science and technology, people also find that fractional-order models are more accurate than integer-order models in describing some phenomena and reflecting some properties of objects when studying practical problems. Since the fractional derivative is a quasi-differential operator, its memory-preserving non-locality, while beautifully portraying the real problem, also brings considerable difficulties to theoretical analysis and numerical calculation. This paper analyses the relationship between enterprise management and financial performance, analyses the mean and heterogeneity of the characteristics of enterprise management team, mathematically models its relationship, constructs fractional differential equations, and tests it through empirical research. The influence of the characteristics of enterprise management age, international experience, education level, team size and government background on the financial performance of the company. [ABSTRACT FROM AUTHOR]

This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken nowadays. The discussion is started by a brief historical report that starts with the studies of thermal machines and combines in theories such as the statistical mechanics of Boltzmann–Gibbs and the Brownian Movement. In this scenario, in the twentieth century, a series of experiments were reported that were not described by the usual model of diffusion. Such experiments paved the way for deeper investigation into anomalous diffusion, i.e. 〈 (x − 〈 x 〉) 2 〉 ∝ t α. These processes are very abundant in physics, and the mechanisms for them to occur are diverse. For this reason, there are many possible ways of modelling the diffusive processes. This article discusses three analytic approaches to investigate anomalous diffusion: fractional diffusion equation, nonlinear diffusion equation and Langevin equation in the presence of fractional, coloured or multiplicative noises. All these formalisms presented different degrees of complexity and for this reason, they have succeeded in describing anomalous diffusion phenomena. [ABSTRACT FROM AUTHOR]

• Fractional differential equation can more accurately describe the variation of natural phenomena. • Dirichlet function, as an abstract mathematical model, has many unique properties in calculus. • It can also be used to construct counterexamples in calculus and to clarify many fuzzy concepts to deepen the understanding of mathematical concepts. • Necessary and sufficient conditions for the controllability of fractional linear differential systems in abstract space are stated. • The finite difference decomposition method of fractional calculus in the abstract space for the Dirichlet function equation is also studied. • The existence of solutions for boundary value problems of fractional differential equations with Dirichlet boundary value condition in the abstract space is discussed by using the critical point theory. With the development of nonlinear science, it is found that the fractional differential equation can more accurately describe the variation of natural phenomena.Therefore, the study of fractional differential equations and their boundary value problems is of great significance for solving nonlinear problems.The Dirichlet function, as an abstract mathematical model, has many unique properties in calculus.It points out special circumstances when describing many mathematical concepts, It can also be used to construct counterexamples in calculus and to clarify many fuzzy concepts to deepen the understanding of mathematical concepts.Therefore, this paper mainly studies the necessary and sufficient conditions for the controllability of fractional linear differential systems in abstract space.The finite difference decomposition method of fractional calculus in the abstract space for the Dirichlet function equation is also studied.The existence of solutions for boundary value problems of fractional differential equations with Dirichlet boundary value condition in the abstract space is discussed by using the critical point theory. [ABSTRACT FROM AUTHOR]

The Optimal Control Problem (OCP) for variable-order fractional differential systems with time delay is considered. The fractional time derivative is Atangana–Baleanu derivatives in a Caputo sense. The existence and the uniqueness results are derived. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as an integral function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) with variable-order and time delay. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Atangana–Baleanu derivative, we obtain an optimality system for the optimal control. To obtain the optimality conditions for the given problem, the generalization of the Dubovitskii–Milyutin Theorem was applied. To illustrate the results we introduce some examples. [ABSTRACT FROM AUTHOR]

Highlights • The innovative Fractional-order PID (FOPID) controller is applied to control stochastic bifurcation for a generalized VDP system driven by a color noise. • An equivalent expression in a form of slow-varying variables for FOPID controller is derived, which can clearly display the evolution process of FOPID controller with the time. • A new numerical algorithm is proposed to simulate the system with the coexistence of fractional integral and fractional derivative. • Bifurcation control by the orders of fractional calculus and the coefficient in FOPID controller is discussed in detail and demonstrated by figures. Abstract Fractional-order PID (FOPID) controller, as the results of recent development of fractional calculus, is becoming wide-used in many deterministic dynamical systems, but not in stochastic dynamical systems. This paper explores stochastic bifurcation of a generalized Van del Pol (VDP) system under the control of FOPID controller. Firstly, introducing the transformation between fast-varying and slow-varying variables of the system response, and utilizing the properties of fractional calculus, we obtain a new expression in the form of slow-varying variables for FOPID controller. Based on this work, the stochastic averaging method is applied to obtain the Fokker–Planck–Kolmogorov (FPK) equation and the stationary probability density function (PDF) of the amplitude response. Then a new numerical algorithm is proposed to testify the analytical results in the case of the coexistence of fractional integral and fractional derivative. After that, stochastic bifurcations induced by the order of the fractional integral, the order of the fractional derivative and the coefficient in FOPID controller are investigated in detail. The agreement between analytical and numerical results verifies the correctness and effectiveness of our proposed methods. [ABSTRACT FROM AUTHOR]

Highlights • Taking the fractional-order simplified Lorenz system as an example, we firstly systematically analyze the effect of different numerical method, whether or not it is commensurate order, the change of system parameter and solution step on the lowest order of the system to generate chaos. • By selecting the optimal results from a previous analysis, we obtain the lowest order of the fractional-order simplified Lorenz system is 1.136, which is lower than all results reported previously. • We proposed a corollary: the lowest order of fractional-order chaotic system under the exact solution condition is lower than the approximate solution from numerical simulation. Abstract Considering the fractional-order simplified Lorenz system as an example, we analyze the lowest order under the effects of variations in the numerical methods employed to solve the system. We varied the commensurability of the system equations, the system parameter and iteration step to reveal how they affect the appearance of chaos. The results show that the lowest order is obtained using the Adomian decomposition method (ADM), and it is smaller than that obtained through the Adams-Bashforth-Moulton (ABM) algorithm. The fractional-order system in the case of incommensurate order always has a smaller lowest order for chaos than does the system with commensurate order. We found that the lowest order of the fractional-order system decreases as the system parameter increases or as the iteration step decreases. In addition, by selecting the optimal results from a previous analysis, we obtain the lowest order of the fractional-order simplified Lorenz system is 1.136, which is lower than all results previously reported. Finally, we propose a corollary: the lowest order of a fractional-order chaotic system returned by an exact solution condition is lower than the approximate solution derived by numerical simulations. This work facilitates us to find a lower lowest order of fractional-order nonlinear systems, which are relevant to the study of fractional-order nonlinear system for engineering application. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the fractional order dynamical system theory is used to describe the complex behaviour of partially penetrating wells (PPWs) in a typical reservoir whose geometry is governed by fractal tools. The Green's function approach, as a generalised impulse response function, is adopted to model the fluid flow in any type of reservoir with a partially penetrating (vertical) well producing from it. Having obtained the initial description of a typical PPW, using the Laplace transform a new dimensionless constant-flow-rate solution is introduced, when wellbore storage and skin effects are significant. The pressure-transient behaviour of a PPW is discussed following two synthetic examples which illustratively depict the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]

Highlights • The fractional Herglotz variational principle (FHVP) in phase space is proposed. • Hamilton canonical equations for FHVP in phase space are derived. • Noether's theorem for FHVP in phase space is proved. • Some applications of the results we obtained are discussed. Abstract The aim of this paper is to bring together two approaches, Heglotz variational principle and fractional calculus, to deal with non-conservative systems in phase space. Namely, we study the functional of Herglotz type whose extremum is sought, by the differential equation that involves Caputo fractional derivatives in phase space. Firstly, Herglotz variational principle under fractional Hamilton action in phase space is presented, and its Hamilton canonical equations are derived. Secondly, two basic formulae for the variation of the fractional Hamilton–Herglotz action in phase space are obtained. Furthermore, the definition and the criterion of Noether symmetry for fractional Herglotz variational principle are given, and the corresponding Noether's theorem is established. Under appropriate conditions, the Noether's theorem can reduce to the classical one of Herglotz type in phase space. Finally, two examples are given to illustrate the application of the results. [ABSTRACT FROM AUTHOR]

Highlights • Boundary layer flow of viscous fluid is developed with the help Caputo–Fabrizio (CF) and (ABC) fractional derivatives. • Solutions for temperature, concentration and velocity fields are obtained by Laplace transform method. • As a result, we have observed that both the fractional models (CF) and (AB) are better in describing the memory effect of the physical problem. • (ABC) model is comparatively good in telling the history of the temperature, concentration and velocity fields. Abstract We have analyzed the magnetohydrodynaimcs (MHD) unsteady free convection flow of incompressible Newtonian fluid passing over an inclined plate through porous medium with variable temperature and concentration at the boundary. Additionally, we have also seen the effects of heat sink and chemical reaction. We have solved dimensionless equations governing the physical problem by Laplace transform method. Firstly, we have found the analytical results for concentration, temperature and velocity fields of classical model. After that we have extended the classical model to some fractional models specifically Caputo–Fabrizio (CF) and Atangana–Baleanu (ABC). Semi analytical results are attained for concentration, temperature and velocity fields for both models and then compared with solutions of classical one. Influence of Fembedded parameters on concentration, temperature and velocity domains can be perceived through MathCad software. As a result, we have observed that both the fractional models (CF) and (ABC) are better in describing the history of the physical problem. Further it is noted that, (ABC) model is well-suited in stimulating the history functions of temperature, concentration and velocity fields. [ABSTRACT FROM AUTHOR]

Highlights • A novel framework to analyze the evolution of adjacent trajectories of a nonlinear fractional-order system within a given region is given. • Sufficient conditions guarantee the mutual asymptotic convergence are presented. • Conditions for contraction, partial contraction, contraction to a manifold and stability of an equilibrium point are given. • Applications for a variety of current problems (adaptive control, observer design and synchronizing schemes) are given. • Reviewers' comments have been addressed and theory and examples have been updated accordingly. Abstract In this work we present a novel platform for analysis of fractional-order nonlinear systems that, from a differential analysis as well as contraction analysis point of view, gives the sufficient conditions for the mutual convergence of nearby trajectories whose distance decrease asymptotically bounded by a Mittag-Leffler vanishing function. Particular cases, such as partial contraction and contraction to a linear manifold are studied. Applications to stability analysis, adaptive control, observer design and synchronization of chaotic fractional-order systems are derived in order to demonstrate the effectiveness of the proposed paradigm. [ABSTRACT FROM AUTHOR]