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The purpose of this paper is to derive the analytical expression of fractional order reducing generation operator (or inverse accumulating generating operation) and study its properties.

This disaggregation method includes three main steps. First, by utilizing Gamma function expanded for integer factorial, this paper expands one order reducing generation operator into integer order reducing generation operator and fractional order reducing generation operator, and gives the analytical expression of fractional order reducing generation operator. Then, studies the commutative law and exponential law of fractional order reducing generation operator. Lastly, gives several examples of fractional order reducing generation operator and verifies the commutative law and exponential law of fractional order reducing generation operator.

The authors pull the analytical expression of fractional order reducing generation operator and verify that fractional order reducing generation operator satisfies commutative law and exponential law.

Expanding the reducing generation operator would help develop grey prediction model with fractional order operators and widen the application fields of grey prediction models.

The analytical expression of fractional order reducing generation operator, properties of commutative law and exponential law for fractional order reducing generation operator are first studied.

The purpose of this paper is to unify the expression of fractional grey accumulating generation operator and the reducing generation operator, and build the FDGM(1,1) model with the unified fractional grey generation operator.

By systematically studying the properties of the fractional accumulating operator and the reducing operator, and analyzing the sensitivity of the order value, a unified expression of the fractional operators is given. The FDGM(1,1) model with the unified fractional grey generation operator is established. The relationship between the order value and the modeling error distribution is studied.

The expression of the fractional accumulating generation operator and the reducing generation operator can be unified to a simple expression. For −1<r < 1, the fractional grey generation operator satisfies the principle of new information priority. The DGM(1,1) model is a special case of the FDGM(1,1) model with r = 1.

The sensitivity of the unified operator is verified through random numerical simulation method, and the theoretical proof was not yet possible.

The FDGM(1,1) model has a higher modeling accuracy and modeling adaptability than the DGM(1,1) by optimizing the order.

The expression of the fractional accumulating generation operator and the reducing generation operator is firstly unified. The FDGM(1,1) model with the unified fractional grey generation operator is firstly established. The unification of the fractional accumulating operator and the reducing operator improved the theoretical basis of grey generation operator.

Understanding the mechanical and thermal behavior of materials is the goal of the branch of study known as fractional thermoelasticity, which blends fractional calculus with thermoelasticity. It accounts for the fact that heat transfer and deformation are non-local processes that depend on long-term memory. The sphere is free of external stresses and rotates around one of its radial axes at a constant rate. The coupled system equations are solved using the Laplace transform. The outcomes showed that the viscoelastic deformation and thermal stresses increased with the value of the fractional order coefficients.

The results obtained are considered good because they indicate that the approach or model under examination shows robust performance and produces accurate or reliable results that are consistent with the corresponding literature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

This study introduces a proposed viscoelastic photoelastic heat transfer model based on the Moore-Gibson-Thompson framework, accompanied by the incorporation of a new fractional derivative operator. In deriving this model, the recently proposed Caputo proportional fractional derivative was considered. This work also sheds light on how thermoelastic materials transfer light energy and how plasmas interact with viscoelasticity. The derived model was used to consider the behavior of a solid semiconductor sphere immersed in a magnetic field and subjected to a sudden change in temperature.

This paper aims to study the complex behavior of a dynamical system using fractional and fractal-fractional (FF) derivative operators. The non-classical derivatives are extremely useful for investigating the hidden behavior of the systems. The Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) derivatives are considered for the fractional structure of the model. Further, to add more complexity, the authors have taken the system with a CF fractal-fractional derivative having an exponential kernel. The active control technique is also considered for chaos control.

The systems under consideration are solved numerically. The authors show the Adams-type predictor-corrector scheme for the AB model and the Adams–Bashforth scheme for the CF model. The convergence and stability results are given for the numerical scheme. A numerical scheme for the FF model is also presented. Further, an active control scheme is used for chaos control and synchronization of the systems.

Simulations of the obtained solutions are displayed via graphics. The proposed system exhibits a very complex phenomenon known as chaos. The importance of the fractional and fractal order can be seen in the presented graphics. Furthermore, chaos control and synchronization between two identical fractional-order systems are achieved.

This paper mentioned the complex behavior of a dynamical system with fractional and fractal-fractional operators. Chaos control and synchronization using active control are also described.

This paper aims to investigate the existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions. With regard to this nonlinear boundary value problem, three popular fixed point theorems, namely, Krasnoselskii’s theorem, Leray–Schauder’s theorem and Banach contraction principle, are employed to theoretically prove and guarantee three novel theorems. The main outcomes of this work are verified and confirmed via several numerical examples.

In order to accomplish our purpose, three fixed point theorems are applied to the problem under consideration according to some conditions that have been established to this end. These theorems are Krasnoselskii's theorem, Leray Schauder's theorem and Banach contraction principle.

In accordance to the applied fixed point theorems on our main problem, three corresponding theoretical results are stated, proved, and then verified via several numerical examples.

The existence and uniqueness of solution for generalized Sturm–Liouville and Langevin equations formulated using Caputo–Hadamard fractional derivative operator in accordance with three nonlocal Hadamard fractional integral boundary conditions are studied. To the best of the authors’ knowledge, this work is original and has not been published elsewhere.

Creep behavior of concrete at high temperature has become a major concern in building structures, such as factories, bridges, tunnels, airports and nuclear buildings. Therefore, a simple and accurate prediction model for the high-temperature creep behavior of concrete is crucial in engineering applications.

In this paper, the variable-order fractional operator is introduced to capture the high-temperature creep behavior of concrete. By assuming that the variable-order function is a linear function with time, the proposed model benefits from the advantages of both formal simplicity and the physical significance for macroscopic intermediate materials. The effectiveness of the model is demonstrated by data fitting with existing experimental results of high-temperature creep of two representative concretes.

The results show that the proposed model fits well with the experimental data, and the value of order is increasing with the increase of the applied stress levels, which meets the fact that higher stress can accelerate the rate of creep. Furthermore, the relationship between the model parameters and loading conditions is deeply analyzed. It is found that the material coefficients are constant at a constant temperature, while the order function parameters are determined by the applied stress levels. Finally, the variable-order fractional model can be further written into a general equation of time and applied stress.

This paper provides a simple and practical variable-order fractional model for predicting the creep behavior of concrete at high temperature.

The purpose of this paper is to find the solution for special cases of regular-long wave equations with fractional order using q-homotopy analysis transform method (q-HATM).

The proposed technique (q-HATM) is the graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme and fractional derivative defined with Atangana-Baleanu (AB) operator.

The fixed point hypothesis considered to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional-order model. To illustrate and validate the efficiency of the future technique, the authors analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order.

To illustrate and validate the efficiency of the future technique, we analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order. The obtained results elucidate that, the proposed algorithm is easy to implement, highly methodical, as well as accurate and very effective to analyse the behaviour of nonlinear differential equations of fractional order arisen in the connected areas of science and engineering.

Traditional grey models are integer order whitening differential models; these models are relatively effective for the prediction of regular raw data, but the prediction error of irregular series or shock series is large, and the prediction effect is not ideal.

The new model realizes the dynamic expansion and optimization of the grey Bernoulli model. Meanwhile, it also enhances the variability and self-adaptability of the model structure. And nonlinear parameters are computed by the particle swarm optimization (PSO) algorithm.

Establishing a prediction model based on the raw data from the last six years, it is verified that the prediction performance of the new model is far superior to other mainstream grey prediction models, especially for irregular sequences and oscillating sequences. Ultimately, forecasting models are constructed to calculate various energy consumption aspects in Chongqing. The findings of this study offer a valuable reference for the government in shaping energy consumption policies and optimizing the energy structure.

It is imperative to recognize its inherent limitations. Firstly, the fractional differential order of the model is restricted to 0 < a < 2, encompassing only a three-parameter model. Future investigations could delve into the development of a multi-parameter model applicable when a = 2. Secondly, this paper exclusively focuses on the model itself, neglecting the consideration of raw data preprocessing, such as smoothing operators, buffer operators and background values. Incorporating these factors could significantly enhance the model’s effectiveness, particularly in the context of medium-term or long-term predictions.

This contribution plays a constructive role in expanding the model repertoire of the grey prediction model. The utilization of the developed model for predicting total energy consumption, coal consumption, natural gas consumption, oil consumption and other energy sources from 2021 to 2022 validates the efficacy and feasibility of the innovative model.

These findings, in turn, provide valuable guidance and decision-making support for both the Chinese Government and the Chongqing Government in optimizing energy structure and formulating effective energy policies.

This research holds significant importance in enriching the theoretical framework of the grey prediction model.

The highlights of the paper are as follows:

1. A novel grey Bernoulli prediction model is proposed to improve the model’s structure.

2. Fractional derivative, fractional accumulating generation operator and Bernoulli equation are added to the new model.

3. The proposed model can achieve full compatibility with the traditional mainstream grey prediction models.

4. Energy consumption in Chongqing verifies that the performance of the new model is much better than that of the traditional grey models.

5. The research provides a reference basis for the government to formulate energy consumption policies and optimize energy structure.

A novel grey Bernoulli prediction model is proposed to improve the model’s structure.

Fractional derivative, fractional accumulating generation operator and Bernoulli equation are added to the new model.

The proposed model can achieve full compatibility with the traditional mainstream grey prediction models.

Energy consumption in Chongqing verifies that the performance of the new model is much better than that of the traditional grey models.

The research provides a reference basis for the government to formulate energy consumption policies and optimize energy structure.

With the increasing demands of industrial applications, it is imperative for robots to accomplish good contact-interaction with dynamic environments. Hence, the purpose of this research is to propose an adaptive fractional-order admittance control scheme to realize a robot–environment contact with high accuracy, small overshoot and fast response.

Fractional calculus is introduced to reconstruct the classical admittance model in this control scheme, which can more accurately describe the complex physical relationship between position and force in the interaction process of the robot–environment. In this control scheme, the pre-PID controller and fuzzy controller are adopted to improve the system force tracking performance in highly dynamic unknown environments, and the fuzzy controller is used to improve the trajectory, transient and steady-state response by adjusting the pre-PID integration gain online. Furthermore, the stability and robustness of this control algorithm are theoretically and experimentally demonstrated.

The excellent force tracking performance of the proposed control algorithm is verified by constructing highly dynamic unstructured environments through simulations and experiments. In simulations and experiments, the proposed control algorithm shows satisfactory force tracking performance with the advantages of fast response speed, little overshoot and strong robustness.

The control scheme is practical and simple in the actual industrial and medical scenarios, which requires accurate force control by the robot.

A new fractional-order admittance controller is proposed and verified by experiments in this research, which achieves excellent force tracking performance in dynamic unknown environments.

Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type.

The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations.

Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique.

Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.