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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.

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 propose a fractional order optimization method based on perturbation bound and gamma function of a DGM(r,1).

By analyzing and minimizing perturbation bound, the sub-optimal solution on fractional order interval is obtained through offline solving without iterative calculation. By this method, an optimized fractional order non-equidistant ROGM (OFONEROGM) is applied in fitting and prediction water quality parameters for a surface water pollution monitoring system.

This method can narrow fractional order interval in this work. In a surface water pollution monitoring system, the fitting and prediction performances of OFONEROGM are demonstrated comparing with integer order non-equidistant ROGM (IONEROGM).

A method of offline solving the sub-optimal solution on fractional order interval is proposed. It can narrow the optimized fractional order range of NEROGM without iterative calculation. A large number of calculations are eliminated. Besides that, optimized fractional order interval is only related to the number of original data, and convenient for practical application. In this work, an OFONEROGM is modeled for predicting water quality trend for preventing water pollution or stealing sewage discharge. It will provide guiding significance in water quality parameter fitting and predicting for water environment management.

This paper aims to introduce a novel 4D hyperchaotic fractional-order system which can produce one-to-four-wing hyperchaotic attractors. In the study of chaotic systems with variable-wing attractors, although some chaotic systems can generate one-to-four-wing attractors, none of them are hyperchaotic attractors, which is incomplete for the dynamic characteristics of chaotic systems.

A novel 4D fractional-order hyperchaotic system is proposed based on the classical three-dimensional Lü system. The complex and abundant dynamic behaviors of the fractional-order system are analyzed by phase diagrams, bifurcation diagrams and the corresponding Lyapunov exponents. In addition, SE and C₀ algorithms are used to analyze the complexity of the fractional-order system. Then, the influence of order q on the system is also investigated. Finally, the circuit is implemented using physical components.

The most particular interest is that the system can generate one-to-four-wing hyperchaotic attractors with only one parameter variation. Then, the hardware circuit experimental results tally with the numerical simulations, which proves the validity and feasibility of the fractional-order hyperchaotic system. Besides, under different initial conditions, coexisting attractors can be obtained by changing the parameter d or the order q. Then, the complexity analysis of the system shows that the fractional-order chaotic system has higher complexity than the corresponding integer-order chaotic system.

The circuit structure of the fractional-order hyperchaotic system is simple and easy to implement, and one-to-four-wing hyperchaotic attractors can be observed in the circuit. To the best of the knowledge, this unique phenomenon has not been reported in any literature. It is of great reference value to analysis and circuit realization of fractional-order chaotic systems.

In this paper, the formulation and analytic solutions for fractional continuously variable order dynamic models, namely, fractional continuously mass-spring damper (continuously variable fractional order) systems, have been presented. The authors will demonstrate via two cases where the frictional damping given by fractional derivative, the order of which varies continuously – while the mass moves in a guide. Here, the continuously changing nature of the fractional-order derivative for dynamic systems has been studied for the first time. The solutions of the fractional continuously variable order mass-spring damper systems have been presented here by using a successive recursive method, and the closed form of the solutions has been obtained. By using graphical plots, the nature of the solutions has been discussed for the different cases of continuously variable fractional order of damping force for oscillator. The purpose of the paper is to formulate the continuously variable order mass-spring damper systems and find their analytical solutions by successive recursion method.

The authors have used the viscoelastic and viscous – viscoelastic dampers for describing the damping nature of the oscillating systems, where the order of the fractional derivative varies continuously.

By using the successive recursive method, here, the authors find the solution of the fractional continuously variable order mass-spring damper systems, and then obtain close-form solutions. The authors then present and discuss the solutions obtained in the cases with the continuously variable order of damping for an oscillator through graphical plots.

Formulation of fractional continuously variable order dynamic models has been described. Fractional continuous variable order mass-spring damper systems have been analysed. A new approach to find solutions of the aforementioned dynamic models has been established. Viscoelastic and viscous – viscoelastic dampers are described. The discussed damping nature of the oscillating systems has not been studied yet.

Investigation of the smoking model is important as it has a direct effect on human health. This paper focuses on the numerical analysis of the fractional order giving up smoking model. Nonetheless, due to observational or experimental errors, or any other circumstance, it may contain some incomplete information. Fuzzy sets can be used to deal with uncertainty. Yet, there may be some inconsistency in the membership as well. As a result, the primary goal of this proposed work is to numerically solve the model in a type-2 fuzzy environment.

Triangular perfect quasi type-2 fuzzy numbers (TPQT2FNs) are used to deal with the uncertainty in the model. In this work, concepts of r2-cut at r1-plane are used to model the problem's uncertain parameter. The Legendre wavelet method (LWM) is then utilised to solve the giving up smoking model in a type-2 fuzzy environment.

LWM has been effectively employed in conjunction with the r2-cut at r1-plane notion of type-2 fuzzy sets to solve the model. The LWM has the advantage of converting the non-linear fractional order model into a set of non-linear algebraic equations. LWM scheme solutions are found to be well agreed with RK4 scheme solutions. The existence and uniqueness of the model's solution have also been demonstrated.

To deal with the uncertainty, type-2 fuzzy numbers are used. The use of LWM in a type-2 fuzzy uncertain environment to achieve the model's required solutions is quite fascinating, and this is the key focus of this work.

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.

The purpose of this paper is to detail the design of a fractional controller which was developed for the suppression of the flexural vibrations of the first mode of a smart beam.

During the design of the fractional controller, in addition to the classical control parameters such as the controller gain and the bandwidth; the order of the derivative effect was also included as another design parameter. The controller was then designed by considering the closed loop frequency responses of different fractional orders of Continued Fraction Expansion (CFE) method.

The first, second, third and fourth order approximations of CFE method were studied for the performance analysis of the controller. It was determined that the increase in the order resulted in better vibration level suppression at the resonance. The robustness analysis of the developed controllers was also conducted.

The experimentally obtained free and forced vibration results indicated that the increase in the order of the approximations yielded better performance around the first flexural resonance region of the smart beam and proved to yield better performance than the classical integer order controllers.

Evaluation of the performance of a developed fractional controller was realized by using different approach orders of the CFE method for the suppression of the flexural vibrations of a smart beam.

This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at high frequencies exhibits a fractional effect. In addition, for the fractional-order formulation, this paper aims to provide a more accurate numerical algorithm for solving the fractional differential equations.

This paper analyzes the frequency-dependence of soil electrical properties based on fractional calculus theory. A collocation method based on the Puiseux series is proposed to solve fractional differential equations.

The algorithm proposed in this paper can be used to solve fractional differential equations of arbitrary order, especially for 0.5th-order equations, obtaining accurate numerical solutions. Calculating the impact response of the grounding electrode based on the fractional calculus theory can obtain a more accurate result.

This paper proposes an algorithm for solving fractional differential equations of arbitrary order, especially for 0.5th-order equations. Using fractional calculus theory to analyze the frequency-dependent effect of soil electrical properties, provides a new idea for ground-related transient calculation.

In this contribution, the solution of Mass-Spring-Damper Systems in the fractional context by using Caputo derivative and Orthogonal Collocation Method is investigated. For this purpose, different case studies considering constant and periodic sources are evaluated. The dimensional consistency of the model is guaranteed by introducing an auxiliary parameter. The obtained results are compared with those found by using both the analytical solution and the predictor-corrector method of Adams–Bashforth–Moulton type. The influence of the fractional order on the mechanical system is evaluated.

In the present contribution, an extension of the Orthogonal Collocation Method to solve fractional differential equations is proposed.

In general, the proposed methodology was able to solve a classical mechanical engineering problem with different characteristics.

The development of a new numerical method to solve fractional differential equations is the major contribution.