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In this article, we construct a fractional multi-wavelet basis based on Legendre polynomials to solve fractional delay linear and nonlinear differential equations. For this we introduce an orthonormal fractional basis for Banach space L 2 [ 0 , 1 ] with suitable inner product which make it effective to decrease computational operations and increase accuracy to find approximate solution of the equations. Also, solving fractional problems by orthogonal basis such as Legendre polynomials has a lower accuracy in comparison with fractional basis. Finally, some examples are solved to show the high accuracy of the presented method, and also to compare with some other works. [ABSTRACT FROM AUTHOR]

In the present article, the p-Laplacian operator is applied for examining hybrid fractional differential equations (HFDEs). Establishing the existence and uniqueness (EU) results and analyzing the Hyers–Ulam (HU) stability for HFDEs incorporating fractional derivatives of different orders with the p-Laplacian operator are the primary objectives of this research. With an understanding of the Green function, we will transform the provided HFDE into a corresponding integral form of hybrid FDEs for EU results. A fixed point theorem is employed to examine the existence of solution (ES), and the Banach contraction mapping principle technique is employed to determine the uniqueness of solution. The result of the HU stability study is examined using dynamical systems and functional analysis approaches. Additionally, an application is presented to demonstrate the findings. [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]

In this manuscript, the numerical approximation of the solution for time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation is described involving the Caputo fractional derivative with the help of two novel techniques namely, Sumudu Residual Power Series (SRPS) method and Homotopy Perturbation Sumudu Transform (HPST) method. CDGSK equation describe the propagation of capillary waves and shallow water waves. The results obtained through numerical approximation shows that both SRPS and HPST under consideration are reliable and efficient. The present framework precisely reflects the behavior of the obtained result for various fractional orders. Additionally, we conduct a comparison between the exact solution and an approximate series solution to affirm the efficacy of the proposed techniques. This analysis highlights the efficiency and precision of both techniques in solving nonlinear fractional differential equations. All the numerical simulations are demonstrated graphically using Maple and Matlab. [ABSTRACT FROM AUTHOR]

In this paper, Lie symmetry analysis method is applied to the generalized time fractional reaction–diffusion equation. We obtain a conditional symmetric group and some conservation laws of the governing equation. The obtained Lie symmetries are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative. Furthermore, we obtained asymptotic stable solutions and convergent power series solutions for the reduced equations. The dynamic behavior of these exact solutions is presented graphically. • Lie symmetry analysis method is applied to fractional reaction-diffusion equation. • A conditional symmetric group and some conservation laws are obtained. • Some asymptotic stable solutions and convergent power series solutions are obtained. [ABSTRACT FROM AUTHOR]

Neural network solvers represent an innovative and promising approach for tackling time-fractional partial differential equations by utilizing deep learning techniques. L1 interpolation approximation serves as the standard method for addressing time-fractional derivatives within neural network solvers. However, we have discovered that neural network solvers based on L1 interpolation approximation are unable to fully exploit the benefits of neural networks, and the accuracy of these models is constrained to interpolation errors. In this paper, we present the high-precision Hermite Neural Solver (HNS) for solving time-fractional partial differential equations. Specifically, we first construct a high-order explicit approximation scheme for fractional derivatives using Hermite interpolation techniques, and rigorously analyze its approximation accuracy. Afterward, taking into account the infinitely differentiable properties of deep neural networks, we integrate the high-order Hermite interpolation explicit approximation scheme with deep neural networks to propose the HNS. The experimental results show that HNS achieves higher accuracy than methods based on the L1 scheme for both forward and inverse problems, as well as in high-dimensional scenarios. This indicates that HNS has significantly improved accuracy and flexibility compared to existing L1-based methods, and has overcome the limitations of explicit finite difference approximation methods that are often constrained to function value interpolation. As a result, the HNS is not a simple combination of numerical computing methods and neural networks, but rather achieves a complementary and mutually reinforcing advantages of both approaches. The data and code can be found at https://github.com/hsbhc/HNS. • A high-order explicit approximation scheme for the Caputo fractional derivative is constructed. • An efficient Hermite Neural Solver (HNS) is designed for solving time fractional partial differential equations. • The impact of Hermite interpolation order on the accuracy of the HNS is analyzed. • The performance of HNS has been thoroughly verified across various problem scenarios. [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]

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]

This paper investigates a stochastic fractional differential variational inequality with Lévy jump (SFDVI with Lévy jump), which comprises a stochastic fractional differential equation with Lévy jump and a stochastic variational inequality. By employing the successive approximation method as well as the projection technique, we establish unique existence of the solution for the SFDVI with Lévy jump under some mild conditions. Moreover, the main results are applied to obtain the unique existence of the solution for the spatial price equilibrium problem in stochastic environments. [ABSTRACT FROM AUTHOR]

This study investigates the existence and Ulam–Hyers stability (UHS) results in the context of mixed Hadamard and Riemann–Liouville Fractional Stochastic Differential Equations (HRFSDEs). The primary focus is on establishing the existence and uniqueness of solutions through the application of the Banach Fixed point Theorem (BFT) coupled with standard stochastic analysis techniques. Subsequently, the UHS results for mixed HRFSDEs are explored utilizing the powerful tool of the Gronwall inequality. The theoretical findings shed light on the stability properties of the considered equations. To validate the obtained results, two numerical examples are presented, demonstrating the practical implications of the stability analysis. • Novel result of existence and uniqueness of a solution to mixed Hadamard and Riemann–Liouville fractional stochastic differential equations. • Novel Ulam–Hyers stability results for mixed Hadamard and Riemann–Liouville fractional stochastic differential equations is proved. • Numerical example is given to confirm the theoretical results obtained. [ABSTRACT FROM AUTHOR]

After uncertainty theory was established, it has become a new branch of mathematics and been applied to describing the indeterministic phenomena as an uncertain dynamic system. As an important part of uncertainty theory, uncertain fractional differential equation is a good tool to model some complex dynamic systems with uncertainties. Due to the lack of corresponding information, models of parametric uncertain fractional differential equation are not always available. Therefore, the requirement for nonparametric estimation of uncertain fractional differential equation is urgent. Utilizing the thought of Legendre polynomial approximation, we propose a method to estimate the nonparametric uncertain fractional differential equations. Then, the error analysis of this method is given. After that, the numerical simulation is applied to illustrating the validity of error analysis and show the errors between the approximating functions and actual functions. The stability of this method has also been verified without highly dense observations. Finally, by means of uncertain hypothesis test and Kolmogorov–Smirnov test, we prove the superiority of uncertain fractional differential equations for some examples. • A nonparametric estimation method is proposed for uncertain fractional differential equations (UFDEs). • Error analysis of the proposed method is given and illustrate with numerical simulation. • Advantages of UFDEs are proved by making comparison with uncertain differential equations and stochastic differential equations. [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]

• Time fractional Tricomi Equation. • Atangana-Baleanu operator of fractional order. • Two step Adams Bashforth method. • Fourier Stability Analysis. • Numerical simulations. Recently, Atangana and Baleanu (AB) introduced a new fractional differentiation concept using non-local and non-singular kernel. Later on, theoretical applications, more practical applications and new numerical methods was established for solving partial differential equations in the meaning of AB derivative. In this study, the new numerical scheme was formulated by Owolabi and Atangana [A. Atangana, K.M.Owolabi, New numerical approach for fractional differential equations. Mathematical Modelling of Natural Phenomena 13.1 (2018) 1–19.] is considered for solving fractional Tricomi equation which involves the Mittag- Leffler kernel. A novel two-step Adams-Bashforth scheme is applied for the approximation of the AB fractional derivative. Stability of the numerical scheme is examined with the help of von Neumann stability analysis and induction principle. To test the applicability and suitability of the proposed method, two notable examples are considered with numerical results presented for some fractional order values. [ABSTRACT FROM AUTHOR]

In this paper, the existence results of the solution and stability are focused for the variable fractional order differential equation. In view of the definitions of three kinds of Caputo variable fractional order operator, the sufficient condition of the solution existence for the variable fractional order differential system is obtained by use of Arzela–Ascoli theorem. Moreover, some criterions of the Mittag–Leffler stability and asymptotical stability are proposed for the variable fractional order differential system according to the Fractional Comparison Principle. [ABSTRACT FROM AUTHOR]

In this manuscript, we aim to refine and characterize nonlinear F -contractions in a more general framework of b -metric spaces. We investigate the existence and uniqueness of such contractions in this setting. We discuss the solutions to differential equations in the setting of fractional derivatives involving Mittag–Leffler kernels (Atangana–Baleanu fractional derivative) by using nonlinear F -contractions that indicate the genuineness of the presented result. [ABSTRACT FROM AUTHOR]

With the rapid development of Internet technology, network users and data are also rapidly accumulating, which leads to significant asymmetry between user needs and massive information. Therefore, how to perceive user intentions and provide users with personalized and accurate information becomes an important research topic; at the same time, the Internet has become another important source of Public Intelligence, widely used in political, economic, military and other fields, people need to explore a new Internet-based public intelligence collection and processing technology. Based on fractional order differential, a new model for predicting user intentions and Public Intelligence mining trends is constructed. The order of the fractional differential equation model is identified by the optimization algorithm. The second and the parameters give the analytical solution of the model, and the resulting prediction equation can be used as a new method for Internet user intention and public intelligence mining prediction. The verification results of real experimental data show that the method can identify the model parameters more accurately, and the obtained fractional differential equation model is more consistent with the actual data. [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]

• Discussing the Hyers-Ulam stability for nonlinear differential equations involving Atangana-Baleanu fractional derivatives. • Fractional differential equations with singularity and nonlinear p-Laplacian operator in Banach's space are studied. • Guo-Krasnoselskii theorem was consider to obtain the results. In this paper we are established the existence of positive solutions (EPS) and the Hyers-Ulam (HU) stability of a general class of nonlinear Atangana-Baleanu-Caputo (ABC) fractional differential equations (FDEs) with singularity and nonlinear p -Laplacian operator in Banach's space. To find the solution for the EPS, we use the Guo-Krasnoselskii theorem. The fractional differential equation is converted into an alternative integral structure using the Atangana-Baleanu fractional integral operator. Also, HU-stability is analyzed. We include an example with specific parameters and assumptions to show the results of the proposal. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and the numerical estimates of solutions for a specific types of fractional differential equations. The nonlinear part of the problem, however, presupposes certain hypotheses. Particularly, for exact localization of the parameter, the existence of a non-zero solution is established, which requires the sublinearity of the nonlinear part at the origin and infinity. We also take into consideration several theoretical and numerical examples. One of the main novelties of this paper is to use the variational method to investigate the properties of solutions of boundary values problems involving Atangana–Baleanu fractional derivative for the first time. [ABSTRACT FROM AUTHOR]

In this article, we derive sufficient conditions to prove the oscillation for solutions of Caputo fractional differential equations with Mittag–Leffler nonsingular kernel of the form { ( a A B C D κ 0 ξ) (t) + ϕ 1 (t , ξ) = θ (t) + ϕ 2 (t , ξ) , t > a ξ k (a) = b k (k = 0 , 1 , ... , n) , where n < κ 0 ≤ n + 1 and a A B C D κ 0 is the left-fractional Caputo derivative with Mittag–Leffler nonsingular kernel or the Atangana–Baleanu fractional derivative in the sense of Caputo. An example is given to validate part of the proven results. [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]

• 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]

• Several chaotic systems with interesting behaviors are presented. • Atangana–Baleanu fractional derivative operator is considered. • Existence and uniqueness of general system as well as local stability analysis are examined. • A range of chaotic patterns examined through time series are obtained for different instances of fractional orders. In this paper, a range of chaotic systems with some interesting behaviors such as multi-scroll attractors, self-excited and hidden attractors, period-doubling to chaos, periodic and chaotic bursting oscillations, and different multiple coexisting attractors have been considered and modelled with the new Atangana–Baleanu fractional derivative operator in time. Existence and uniqueness of general system as well as local stability analysis are examined. In the simulation framework, a range of chaotic patterns examined through time series were obtained for different instances of fractional orders. Comparison between the integer (with p = 1) and noninteger (0 < p < 1) order results are given. [ABSTRACT FROM AUTHOR]

• Several definitions of fractional order derivatives have been proposed, including: Riemann-Liouville, Grunwald-Letnikov, Liouville-Caputo, Caputo-Fabrizio, Atangana- Baleanu and Gomez operators of fractional derivative with different areas of application. • Many powerful methods for finding approximated solutions are presented to study the solutions of linear/ nonlinear fractional equations contains FDE, FPDE and FIDE. • The cases of fractional equations with time delay and variable order fractional equations are describe to study the robust, accurate and efficient numerical schemes for these equations. In this review paper, we are mainly concerned with the numerical methods for solving fractional equations, which are divided into the fractional differential equations (FDEs), time-fractional, space-fractional and space-time-fractional partial differential equations (FPDEs), fractional integro-differential equations (FIDEs) and delay fractional differential and/or fractional partial differential equations (DFDE/DFPDEs). The concept of the variable-order fractional operators will also be reviewed. At the same time, the techniques for improving the accuracy and computation and storage are also introduced. [ABSTRACT FROM AUTHOR]

In this article, we consider integro-differential equations involving the recently explored Atangana–Baleanu fractional derivatives which contain the generalized Mittag-Leffler functions in their kernels. Utilizing fixed point techniques, we examine the existence and uniqueness of solutions to such equations in Banach spaces. Moreover, we consider an example and investigate numerical outcomes for various values of the fractional order. Then, we consider the stability of the tackled integro-differential equation in the frame of Ulam. [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]

• 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 model considering with different fractional derivative definitions give rise to different behaviours. • Hopf bifurcation and chaos exists for the model considering with conformable fractional order derivative. • Source rate of immune cells plays critical role in cancer Dynamics. In this paper, tumor-immune system interaction has been considered by two fractional order models. The first and the second model consist of system of fractional order differential equations with Caputo and conformable fractional derivative respectively. First of all, the stability of the equilibrium points of the first model is studied. Then, a discretization process is applied to obtain a discrete version of the second model where conformable fractional derivative is taken into account. In discrete model, we analyze the stability of the equilibrium points and prove the existence of Neimark-Sacker bifurcation depending on the parameter σ. Moreover, the dynamical behaviours of the models are compared with each other and we observe that the discrete version of conformable fractional order model exhibits chaotic behavior. Finally, numerical simulations are also presented to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

• Logistic model with feedback control of fractional order is studied. • Non-standard finite difference method (NSFDM) is implemented to the proposed model. • Uniform asymptotic stability of this model is established based on an appropriate Lyapunov function. • Numerical examples are performed to show the validity of the established theoretical results. In this paper, we propose and study a Logistic model with feedback control of fractional order that is extended directly from a system of ordinary differential equations. Uniform asymptotic stability of this model is established based on an appropriate Lyapunov function and an important consequence of this result; we present a simple proof for the global stability of the original system of ordinary differential equations. Besides, to numerically solve and simulate the proposed fractional model, unconditionally positive nonstandard finite difference schemes are constructed and analyzed. Finally, the numerical simulations obtained by the constructed nonstandard finite difference schemes are compared with the Grunwald–Letnikov scheme to reveal that the proposed scheme is convenient for solving the proposed model and confirm the validity of the established results. [ABSTRACT FROM AUTHOR]

• The model of polluted lake system is introduced in a short way. • Fractional type of polluted lake system is constructed. • Modified model is analyzed in detail with Atangana–Baleanu derivative. • A numerical example is given to demonstrate the efficiency of the newly proposed operator. In this paper, we use Atangana–Baleanu derivative which is defined with the Mittag–Leffler function and has all the properties of a classical fractional derivative for solving the system of fractional differential equations. The classical model of polluted lakes system is modified by using the concept of fractional differentiation with nonsingular and nonlocal fading memory. The new numerical scheme recommended by Toufik and Atangana is used to analyze the modified model of polluted lakes system. Some numerical illustrations are presented to show the effect of the new fractional differentiation. [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 index concept is generalized for categorizing system of fractional differential algebraic equations and analyzing the solvability of such systems. • A model of a simple pendulum in a Newtonian fluid is obtained and described by using nonlinear fractional differential algebraic equations. • An application in an electrical circuit is introduced and the solvability of the obtained linear system is studied. • A simple and efficient numerical method for solving a system of FDAEs with different fractional derivatives including Liouville-Caputo's definition, Caputo-Fabrizio's definition and a definition with Mittag-Leffler kernel are introduced. Abstract One of the important classes of coupled systems of algebraic, differential and fractional differential equations (CSADFDEs) is fractional differential algebraic equations (FDAEs). The main difference of such systems with other class of CSADFDEs is that their singularity remains constant in an interval. However, complete classifying and analyzing of these systems relay mainly to the concept of the index which we introduce in this paper. For a system of linear differential algebraic equations (DAEs) with constant coefficients, we observe that the solvability depends on the regularity of the corresponding pencils. However, we show that in general, similar properties of DAEs do not hold for FDAEs. In this paper, we introduce some practical applications of systems of FDAEs in physics such as a simple pendulum in a Newtonian fluid and electrical circuit containing a new practical element namely fractors. We obtain the index of introduced systems and discuss the solvability of these systems. We numerically solve the FDAEs of a pendulum in a fluid with three different fractional derivatives (Liouville–Caputo's definition, Caputo–Fabrizio's definition and with a definition with Mittag–Leffler kernel) and compare the effect of different fractional derivatives in this modeling. Finally, we solved some existing examples in research and showed the effectiveness and efficiency of the proposed numerical method. [ABSTRACT FROM AUTHOR]

Highlights • The leaving smoking model is analyzed with newly-defined Atangana–Baleanu derivative. • The existence and uniqueness of smoking model of solutions are obtained. • Numerical results related to smoking model are given for different fractional orders. • The obtained numerical results are compared to real world applications. Abstract This paper ensures meticulous mathematical study for analyzing the dynamics of smoking model and its public health with Atangana–Baleanu derivative. The existence and uniqueness problems to relevant model are examined via fixed point theory. The results are implemented by giving some illustrative graphics including the variation of fractional order. [ABSTRACT FROM AUTHOR]

• To generalize truncated M -fractional derivative as a novel and effective derivative under interval uncertainty. • To investigate the existence and uniqueness conditions of the solution for the interval fractional differential equations (IFDEs) based on this new derivative. • To present the applicability of this novel interval fractional derivative for IFDEs based on the notion of w -increasing. • To develop the significant and applicable classical properties of this novel fractional derivative under interval uncertainty. Abstract In the recent years some efforts were made to propose simple and well-behaved fractional derivatives that inherit the classical properties from the first order derivative. In this regards, the truncated M -fractional derivative for α -differentiable function was recently introduced that is a generalization of four fractional derivatives presented in the literature and has their important features. In this research, we aim to generalize this novel and effective derivative under interval uncertainty. The concept of interval truncated M -fractional derivative is introduced and some of the distinguished properties of this interesting fractional derivative such as Rolle's and mean value theorems, are developed for the interval functions. In addition, the existence and uniqueness conditions of the solution for the interval fractional differential equations (IFDEs) based on this new derivative are also investigated. Finally, we present the applicability of this novel interval fractional derivative for IFDEs based on the notion of w -increasing (w -decreasing) by solving a number of test problems. [ABSTRACT FROM AUTHOR]

Highlights • This article considers the interplay among four financial factors. • We use integer order and fractional order differential equation systems to model a financial system. • Both mathematical analyses and numerical simulations are carried out to illustrate the characteristic of the model. • The model displays a variety of rich dynamic behaviors including chaos over a wide range of system parameters. Abstract In this paper, we use integer order and fractional order differential equation systems to model a financial system. Based on the interaction among several financial factors, a model is constructed. Both mathematical analyses and numerical simulations are carried out to illustrate the characteristic of the model. We find that the system displays a variety of rich dynamic behaviours including chaos over a wide range of system parameters. Our investigation indicates that the interplay among several financial factors lead to chaos under some circumstances. We then design control laws to synchronization two integer order financial systems and two fractional order financial systems. Numerical simulations are presented to verify the effectiveness of the designed control laws. [ABSTRACT FROM AUTHOR]

Highlights • A fractional optimal control problem governed by a cancer-obesity model is considered. • We provide stability analysis. • We investigate three treatment strategies, namely, chemotherapy, immunotherapy and their combination. • We compare contribution of the non-singular kernel to the choice of the treatment strategy. • We examine the choice of the cost functional to eradicate the tumor burden and regenerate the normal cell population. Abstract The aim of this study is to examine a fractional optimal control problem (FOCP) governed by a cancer-obesity model with and without singular kernel, separately. We propose a new model including the population of immune cells, tumor cells, normal cells, fat cells, chemotherapeutic and immunotherapeutic drug concentrations. Existence and stability of the tumor free equilibrium point and coexisting equilibrium point are investigated analytically. We obtain the numerical solution of the fractional cancer-obesity model using L1 formula. The aim behind the FOCP is to find the optimal doses of chemotherapeutic and immunotherapeutic drugs which minimize the difference between the number of tumor cells and normal cells. To do so, we insert some weight constants as the coefficients of tumor and normal cells in the cost functional so that normal cell population is larger compared to tumor burden. On the other hand, we investigate the effect of obesity to the choice and schedules of treatment strategies in case of low and high caloric diets. Moreover, we discuss the choice of the differentiation operator, namely operators with and without singular kernel. Lastly, some illustrative examples are shown to examine the impact of the fractional derivatives of different orders on cancer-obesity model and we observe the contribution of the cost functional to eradicate tumor burden and regenerate normal cell population. Our model predicts the negative effect of obesity on the health of patient and we show that the most efficient treatment choice to eradicate the tumor is to apply combined therapy together with low caloric diet. [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]

In this manuscript, we describe fractional differential equations with neutral, integral boundary conditions and mixed delay using Atangana–Baleanu derivatives, which include the generalized Mittag-Leffler kernel. We determine the existence and uniqueness of results and analysis by fixed point method. Moreover, we explained the stability of the fractional differential equation in the frame of Ulam–Hyers. Then, we investigate an example with various values and illustrate the outcomes graphically. • We consider the nonlinear fractional mixed integro-differential equations with fractional boundary conditions. • The existence result is obtained by using standard fixed point theory. • An example is given to illustrate our analytical and numerical results. By using MATLAB coding graph is drawn. [ABSTRACT FROM AUTHOR]

This research article centers on the formulation and analysis of a novel prey–predator model that integrates the impacts of predation fear, prey refuge, and carry-over effects. The model is formulated and analyzed using fractional differential equations (FDEs). The model incorporates ecological concepts such as anti-predator behaviours and memory effects to maintain a better understanding of prey–predator interactions. The mathematical model is developed based on a basic prey–predator model with a Michelis–Menten functional response and includes fear-induced carry-over effects, prey refuge, and fractional order derivatives. The study investigates the well-posedness, stability, and Hopf bifurcation of the proposed model and conducts detailed numerical investigations. The integration of different anti-predatory mechanisms and the use of FDEs contribute to a comprehensive understanding of the dynamics of prey–predator interactions and provide useful insights into the complexities of ecological systems. • This article focus on the analysis of a prey-predator model in the realm of FDEs. • The stability and Hopf bifurcation analysis are carried out. • The interplay of the parameters is examined in detail with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, some comments are proposed for the paper by Najafi and Taleghani (2022). Some properties for fractional Liu process and fractional calculus are used in the proof of existence and uniqueness results for fractional Liu uncertain differential equation (Theorem 7). We show that incorrect properties of fractional calculus is used, and lead to subsequent incorrect proof, furthermore, we make some corrections and present new results. • Some mistakes were found in Najafi's paper (Najafi and Taleghani, 2022). • In order to avoid similar mistakes in further research on uncertain differential equations, the comments for Najafi's paper should be needed. [ABSTRACT FROM AUTHOR]

In this article, a general nonlinear system of functional differential equations for two types of operators is considered. One of them includes R D β i which are n-operators in the Riemann–Liouville's sense of derivative while the second is based on a series operator L ( R D ϱ i ) where all the R D ϱ i 's are Riemann–Liouville's fractional operators with the assumption of β i , ϱ i ∈ (0 , 1 ]. These two types of operators are combined with the help of a nonlinear Φ p -operator. This novel construction is more useful in scientific problems. The novel system of FDEs is studied for the solution existence, uniqueness analysis, stability analysis and numerical computation is also carried out. For an application of the presumed general coupled system, a fractional order Leukemia therapy model with its numerical simulations and optimization, is given. The Leukemia model describes the propagation of infected cells. A fractional-order finite-time terminal sliding mode control is designed to control Leukemia for the fractional-order dynamics for eliminating Leukemic cells while maintaining an adequate number of normal cells by the application of a chemotherapeutic agent that is considered as safe. The Lyapunov stability theory has been employed to analyze the controllers. The comparative simulations are presented for a better illustration of the work and show the superior tracking and convergence performance of the proposed control method. [ABSTRACT FROM AUTHOR]

The purpose of our research is to identify some sufficient conditions for the stochastic controllability of fractional-order semilinear stochastic systems. The required outcomes are achieved by dividing the system under consideration into two systems: a linear stochastic system and a semilinear deterministic system. The key outcomes are achieved by employing the Schauder fixed point technique, nonlinear monotonicity, and keeping just fractional-order α ∈ (1 / 2 , 1). One example is presented for demonstrating the outcomes. • The purpose of our research is to identify some sufficient conditions for the stochastic controllability of fractional-order semilinear stochastic systems. • The required outcomes are achieved by dividing the system under consideration into two systems: a linear stochastic system and a semilinear deterministic system. • The key outcomes are achieved by employing the Schauder fixed point technique, nonlinear monotonicity, and keeping just fractional-order (1/2,1). [ABSTRACT FROM AUTHOR]

In this article, we deal with a fully fractional thermostat model in the settings of the Riemann–Liouville fractional derivatives. The equivalences between the fractional differential equations and the corresponding Volterra–Fredholm integral equations are rigorously derived. By choosing an appropriate weighted Banach space of continuous functions, we employ two standard fixed-point theorems, Leray–Schauder alternative and Banach contraction principle, to establish the existence of blow-up solutions — the unbounded mathematical solutions in the operational interval. Furthermore, an implicit numerical scheme based on the right product rectangle rule is presented, which provides the numerical approximation of the obtained solution. Some examples are provided to validate our theoretical findings, along with numerical simulations of the solutions. • A fully fractional thermostat model involving the Riemann–Liouville derivatives. • The equivalence between the FDE and the corresponding integral equation is obtained. • Some new existence and uniqueness results are established. • An implicit numerical scheme based on the right product rectangle rule is derived. • Numerical simulations of implicit solutions are presented. [ABSTRACT FROM AUTHOR]

Highlights • Nutrient–phytoplankton–zooplankton model involving variable-order fractional differential operators is studied. • Numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. • Numerical simulation results are provided for illustrating the effectiveness and applicability of the algorithm. Abstract We extended the nutrient–phytoplankton–zooplankton model involving variable-order fractional differential operators of Liouville–Caputo, Caputo–Fabrizio and Atangana–Baleanu. Variable-order fractional operators permits model and describe accurately real world problems, for example, diffusion or spread of nutrients or species in different states. Particularly, we model the interaction of nutrient phytoplankton and its predator zooplankton. The variable-order fractional numerical scheme based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation was consider. Numerical simulation results are provided for illustrating the effectiveness and applicability of the algorithm to solve variable-order fractional differential equations. [ABSTRACT FROM AUTHOR]

Highlights • The fractional differences and sums within delta and nabla are studied summarized and related by some identities so that it becomes helpful to proceed to other types of fractional differences. • The definition of left and right fractional differences with discrete exponential kernels and their corresponding sums have been given and studied on the time scale hZ. (CFC and CFR h-fractional differences and their corresponding CF h-fractional sums) • The definition of left and right fractional differences with discrete MittagLeffler kernels and their corresponding sums have been given and studied on the time scale hZ. (ABC and ABR h-fractional differences and their corresponding AB h-fractional sums) • Working on hZ will guarantee better numerical solutions for complex system models depending on discrete fractional calculus and it will affect the domain of convergence for the fractional differences and sums. Abstract The aim of this article is to recall and study fractional derivatives with singular kernels on h Z and define fractional derivatives with non-singular exponential and Mittag–Leffler kernels on h Z and study some of their properties. We shall follow the nabla time scale analysis and relate the h − nabla classical discrete fractional derivatives to the delta existing ones studied before by some authors. Some dual identities between left and right and delta and nabla, left and right h − fractional difference types will be investigated. The nabla h − discrete versions of the Mittag-Leffler functions will be recalled by means of the nabla h − fatorial functions and nabla h − Taylor polynomials. The discrete Laplace on h Z and its convolution theory are used often to proceed in our investigation. The obtained results will generalize the nabla classical discrete fractional differences and the nabla discrete fractional differences with discrete exponential and M L − kernels studied recently by Abdeljawad and Baleanu by setting h = 1. [ABSTRACT FROM AUTHOR]

Highlights • Local and global stability analysis. • Numerical technique for fractional differentiation. • Modelling with Atangana–Baleanu derivative based on nonlocal and nonsingular kernels. • Numerical simulation results showing chaotic behaviours for different values of fractional index. Abstract The intra-specific relation two predators and a prey dependent food chain system is considered in this paper. To explore the dynamic richness of such system, we replace the classical time-derivative with either the Caputo or the Atangana–Baleanu fractional derivative operators. Two notable numerical schemes for the approximation of such derivatives are formulated. Local and global stability analysis are investigated to ensure the correct choice of the biologically meaning parameters. The condition for occurrence of the Hopf-bifurcation is also observed. We justify the performance of these schemes by reporting their absolute error when applied to nonlinear fractional differential equations. In addition, numerical simulations with different α values and experimented parameter values confirm the analytical results shows that modelling with fractional derivative could give rise to a more richer chaotic dynamics. [ABSTRACT FROM AUTHOR]

Highlights • We investigate Atanagana-Baleanu Derivatives. • We apply reproducing kernel Hilbert space method to fractional differential initial value problems containing the Atanagana-Baleanu derivatives. • We show our results by table and figures. Abstract Atangana and Baleanu gave a derivative with fractional order to search the valuable inquiries. We present a novel technique for investigating fractional differential equations including the Atangana-Baleanu fractional derivative in this paper. We give some examples for this method for equations of fractional initial value problems. [ABSTRACT FROM AUTHOR]

Highlights • Semigroup principle failures to capture natural phenomena. • The future of modeling real world problem relies on fractional differential operators with non-index law property. • Atangana–Baleanu fractional differential operators are convolution of Riemann–Liouville–Caputo derivative with the Mittag–Leffler function. • Index law is not valid in fractional differentiation. Abstract Recently fractional differential operators with non-index law properties have being recognized to have brought new weapons to accurately model real world problems particularly those with non-Markovian processes. This present paper has two double aims, the first was to prove the inadequacy and failure of index law fractional calculus and secondly to show the application of fractional differential operators with no index law properties to statistic and dynamical systems. To achieve this, we presented the historical construction of the concept of fractional differential operators from Leibniz to date. Using a matrix based on the fractional differential operators, we proved that, fractional operators obeying index law cannot model real world problems taking place in two states, more precisely they cannot describe phenomena taking place beyond their boundaries, as they are scaling invariant, more precisely our results show that, mathematical models based on these differential operators are not able to describe the inverse memory, meaning the full history of a physical problem cannot be described accurately using these derivatives with index law properties. On the other hand, we proved that, differential operators with no index-law properties are scaling variant, thus can describe situations taking place in different states and are able to localize the frontiers between two states. We present the renewal process properties included in differential equation build out of the Atangana–Baleanu fractional derivative and counting process, which is connected to its inter-arrival time distribution Mittag–Leffler distribution which is the kernel of these derivatives. We presented the connection of each derivative to a statistical family, for instance Riemann–Liouville–Caputo derivatives are connected to the Pareto statistic, which has no well-defined average when alpha is less than 1 corresponding to the interval where fractional operators mostly defined. We established new properties and theorem for the Atangana–Baleanu derivative of an analytic function, in particular we proved that, they are convolution of the Mittag–Leffler function with the Riemann–Liouville–Caputo derivatives. To see the accuracy of the non-index law derivative to in modeling real chaotic problems, 4 examples were considered, including the nine-term 3-D novel chaotic system, King Cobra chaotic system, the Ikeda delay system and chaotic chameleon system. The numerical simulations show very interesting and novel attractors. The king cobra system with the Atangana–Baleanu presented a very novel attractor where at the earlier time we observed a random walk and latter time we observed the real sharp of the cobra. The Ikeda model with Atangana–Baleanu presented different attractors for each value of fractional order, in particular we obtain a square and circular explosions. The results obtained in this paper show that, the future of modeling real world problem relies on fractional differential operators with non-index law property. Our numerical results showed that, to not model physical problems with fractional differential operators with non-singular kernel and imposing index law in fractional calculus is rightfully living with closed eyes without ever taking a risk to open them. [ABSTRACT FROM AUTHOR]

Highlights • We propose a new generalize numerical schemes for simulating variable-order fractional differential operators. • Numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. • Numerical simulations for the chaotic financial system and memcapacitor-based circuit are obtained. Abstract Variable-order differential operators can be employed as a powerful tool to modeling nonlinear fractional differential equations and chaotical systems. In this paper, we propose a new generalize numerical schemes for simulating variable-order fractional differential operators with power-law, exponential-law and Mittag-Leffler kernel. The numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes were applied to simulate the chaotic financial system and memcapacitor-based circuit chaotic oscillator. Numerical examples are presented to show the applicability and efficiency of this novel method. [ABSTRACT FROM AUTHOR]