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Fractional-order systems generalize classical differential systems and have empirically shown to achieve fine-grain modeling of the temporal dynamics and frequency responses of certain real-world phenomena. Although the study of integer-order memory element (mem-element) emulators has persisted for several years, the study of fractional-order mem-elements has received little attention. To promote the study of the characteristics and applications of mem-element systems in fractional calculus and memory systems, a novel universal fractional-order mem-elements interface for constructing three types of fractional-order mem-element emulators is proposed in this paper. With the same circuit topology, the floating fractional-order memristor, the fractional-order memcapacitor, and fractional-order meminductor emulators can be implemented by simply combining the impedances of different passive elements. PSPICE circuit simulation and printed circuit board hardware experiments validate the dynamical behaviors and effectiveness of our proposed emulators. In addition, the dynamic relationship between fractional-order parameters and values of fractional-order impedance is explored in MATLAB simulation. The proposed fractional-order mem-element emulators built based on the universal interface are constructed with a small number of active and passive elements, which not only reduces the cost but also promotes the development of fractional-order mem-element emulators and application research for the future. [ABSTRACT FROM AUTHOR]

The previous urban growth model by L. M. A. Bettencourt was developed under the framework of a constant β scaling law in an ordinary differential equation based model assuming instantaneous dynamic growth. In this paper, we improve the model by considering the memory effects based on fractional calculus. By testing this new fractional model to different urban attributes related to sustainable growth, such as congestion delay, water supply, and electricity consumption for selected countries (the USA, China, Singapore, Canada, Switzerland, New Zealand), this new model may provide better agreement to the annual population growth by numerically finding the optimal fractional parameter for different attributes. Based on the theoretical time-independent scaling of β = 5 / 6 (sub-linear) and β = 7 / 6 (super-linear), we also analyze the population growth of 42 countries from 1960 to 2018. Furthermore, time-dependent scaling law extracted from empirical data is shown to provide further improvements. With better agreement between this proposed fractional model and the collected empirical population growth data, useful parameters can be estimated. For example, the maintenance cost and additional cost related to the sustainable growth (for a given city's attribute) can be quantitatively determined for the informed decision and urban planning for the sustainable growth of cities. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the reachability of linear and non-linear systems in the sense of the ψ-Hilfer pseudo-fractional derivative in g-calculus by means of the Mittag–Leffler functions (one and two parameters). In this sense, two numerical examples are discussed in order to elucidate the investigated results. [ABSTRACT FROM AUTHOR]

The localized Hartree-Fock potential has proven to be a computationally efficient alternative to the optimized effective potential, preserving the numerical accuracy of the latter and respecting the exact properties of being self-interaction free and having the correct -1/r asymptotics. In this paper we extend the localized Hartree-Fock potential to fractional particle numbers and observe that it yields derivative discontinuities in the energy as required by the exact theory. The discontinuities are numerically close to those of the computationally more demanding Hartree-Fock method. Our potential enjoys a "direct-energy" property, whereby the energy of the system is given by the sum of the single-particle eigenvalues multiplied by the corresponding occupation numbers. The discontinuities c↑ and c↓ of the spin-components of the potential at integer particle numbers N↑ and N↓ satisfy the condition c↑N↑ + c↓N↓ = 0. Thus, joining the family of effective potentials which support a derivative discontinuity, but being considerably easier to implement, the localized Hartree-Fock potential becomes a powerful tool in the broad area of applications in which the fundamental gap is an issue. [ABSTRACT FROM AUTHOR]

To the best of authors' knowledge, little work was referred to the study of a noisy vibro-impact oscillator with a fractional derivative. Stochastic bifurcations of a vibro-impact oscillator with two kinds of fractional derivative elements driven by Gaussian white noise excitation are explored in this paper. We can obtain the analytical approximate solutions with the help of non-smooth transformation and stochastic averaging method. The numerical results from Monte Carlo simulation of the original system are regarded as the benchmark to verify the accuracy of the developed method. The results demonstrate that the proposed method has a satisfactory level of accuracy. We also discuss the stochastic bifurcation phenomena induced by the fractional coefficients and fractional derivative orders. The important and interesting result we can conclude in this paper is that the effect of the first fractional derivative order on the system is totally contrary to that of the second fractional derivative order. [ABSTRACT FROM AUTHOR]

Cumulative residual entropy (CRE) is a measure of uncertainty and departs from other entropy in that it is established on cumulative residual distribution function instead of density function. In this paper, we prove some important properties of discrete CRE and propose fractional multiscale cumulative residual entropy (FMCRE) as a function of fractional order α , which combines CRE with fractional calculus, probability of permutation ordinal patterns, and multiscale to overcome the limitation of CRE. After adding amplitude information through weighted permutation ordinal patterns, we get fractional weighted multiscale cumulative residual entropy (FWMCRE). FMCRE and FWMCRE extend CRE into a continuous family and can be used in more situations with a suitable parameter. Moreover, they can capture long-range phenomena more clearly and have higher sensitivity to the signal evolution. Results from simulated data verify that FMCRE and FWMCRE can identify time series accurately and have immunity to noise. We confirm that the length of time series has little effect on the accuracy of distinguishing data, and even short series can get results exactly. Finally, we apply FMCRE and FWMCRE on stock data and confirm that they can be used as metrics to measure uncertainty of the system as well as distinguishing signals. FWMCRE can also track changes in stock markets and whether adding amplitude information must be decided by the characteristics of data. [ABSTRACT FROM AUTHOR]

Tempered fractional processes offer a useful extension for turbulence to include low frequencies. In this paper, we investigate the stochastic phenomenological bifurcation, or stochastic P-bifurcation, of the Langevin equation perturbed by tempered fractional Brownian motion. However, most standard tools from the well-studied framework of random dynamical systems cannot be applied to systems driven by non-Markovian noise, so it is desirable to construct possible approaches in a non-Markovian framework. We first derive the spectral density function of the considered system based on the generalized Parseval's formula and the Wiener-Khinchin theorem. Then we show that it enjoys interesting and diverse bifurcation phenomena exchanging between or among explosivelike, unimodal, and bimodal kurtosis. Therefore, our procedures in this paper are not merely comparable in scope to the existing theory of Markovian systems but also provide a possible approach to discern P-bifurcation dynamics in the non-Markovian settings. [ABSTRACT FROM AUTHOR]

This paper addresses a novel multiscale fractional order distribution entropy based on a similarity matrix (MFS-DistEn) approach to quantify the information of time series on multiple time scales. It improves the metric method of distance matrix in the original DistEn algorithm and further defines the similarity degree between each vector so that we could measure the probability density distribution more accurately. Besides, the multiscale distribution entropy based on similarity matrix combines the advantages of both the multiscale analysis and DistEn and is able to identify dynamical and scale-dependent information. Inspired by the properties of Fractional Calculus, we select the MFS-DistEn notation as the main indicator to present the relevant properties. The characteristics of the generalized MFS-DistEn are tested in both simulated nonlinear signals generated by the autoregressive fractionally integrated moving-average process, logistic map, and real world data series. The results demonstrate the superior performance of the new algorithm and reveal that tuning the fractional order allows a high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The improved similarity DistEn still has relatively lower sensitivity to the predetermined parameters and decreases with an increase of scale. [ABSTRACT FROM AUTHOR]

In this paper, we examine a modified auxiliary equation method. We applied this novel method on Wu-Zhang system. This model used to describe (1 + 1)-dimensional dispersive long wave in two horizontal directions on shallow waters. This model is one of the fractional nonlinear partial differential equations. We used conformable derivatives properties to convert nonlinear fractional partial differential equation into the ordinary differential equation with integer order. We obtained many different kinds of solutions such as kink and anti-kink, dark, bright, shock, singular, periodic solitary wave. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher's equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. This is the first time that this method is used to solve nonlinear equations in Atangana-Baleanu sense. We present the effectiveness and accuracy of the proposed method by computing the absolute error and the residual error functions. The results show that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the problem of controlling the source function for an optimal control problem involving the fractional wave equation. We show that an optimal solution exists and it is unique for the considered fractional optimal control problem. We calculate the Frechet derivative of the cost functional by means of an adjoint problem and derive necessary optimality conditions. Also, we introduce an efficient numerical approximation for the fractional wave equation with the Atangana-Baleanu derivative. [ABSTRACT FROM AUTHOR]

Mathematical analysis with the numerical simulation of the newly formulated fractional version of the Adams-Bashforth method using the Atangana-Baleanu operator which has both nonlocal and nonsingular properties is considered in this paper. We adopt the fixed point theory and approximation method to prove the existence and uniqueness of the solution via general two-component time fractional differential equations. The method is tested with three nonlinear chaotic dynamical systems in which the integer-order derivative is modeled with the proposed fractional-order case. The simulation result for different α values in (0 , 1 ] is presented. [ABSTRACT FROM AUTHOR]

Recent years have witnessed special attention on complex network based time series analysis. To extract evolutionary behaviors of a complex system, an interesting strategy is to separate the time series into successive segments, map them further to graphlets as representatives of states, and extract from the state (graphlet) chain transition properties, called graphlet based time series analysis. Generally speaking, properties of time series depend on the time scale. In reality, a time series consists of records that are sampled usually with a specific frequency. A natural question is how the evolutionary behaviors obtained with the graphlet approach depend on the sampling frequency? In the present paper, a new concept called the sampling frequency dependent visibility graphlet is proposed to answer this problem. The key idea is to extract a new set of series in which the successive elements have a specified delay and obtain the state transition network with the graphlet based approach. The dependence of the state transition network on the sampling period (delay) can show us the characteristics of the time series at different time scales. Detailed calculations are conducted with time series produced by the fractional Brownian motion, logistic map and Rössler system, and the empirical sentence length series for the famous Chinese novel entitled A Story of the Stone. It is found that the transition networks for fractional Brownian motions with different Hurst exponents all share a backbone pattern. The linkage strengths in the backbones for the motions with different Hurst exponents have small but distinguishable differences in quantity. The pattern also occurs in the sentence length series; however, the linkage strengths in the pattern have significant differences with that for the fractional Brownian motions. For the period-eight trajectory generated with the logistic map, there appear three different patterns corresponding to the conditions of the sampling period being odd/even-fold of eight or not both. For the chaotic trajectory of the logistic map, the backbone pattern of the transition network for sampling 1 saturates rapidly to a new structure when the sampling period is larger than 2. For the chaotic trajectory of the Rössler system, the backbone structure of the transition network is initially formed with two self-loops, the linkage strengths of which decrease monotonically with the increase of the sampling period. When the sampling period reaches 9 , a new large loop appears. The pattern saturates to a complex structure when the sampling period is larger than 11. Hence, the new concept can tell us new information on the trajectories. It can be extended to analyze other series produced by brains, stock markets, and so on. [ABSTRACT FROM AUTHOR]

The fractional parameter in a generalized fractional model is set to control the degree of absorption. However, it does not have an explicit physical meaning, even though it may be estimated from seismic data. Therefore, it is necessary to establish a common reference, which is physically significant for the fractional parameter in model applications. In this paper, a reference is presented according to the constant Q model. The proposed reference can be used to analyze the fractional parameter in different value interval ranges. When the fractional parameter is small, the related absorptive mechanism is equivalent to a constant Q model. When the fractional parameter is large, it reveals an attenuation mechanism corresponding to a frequency-dependent Q. This analysis makes the fractional parameter more practical in other applications. The study also investigates how to derive the generalized linearized reflection coefficient with the fractional parameter for amplitude variation with offset/frequency. The linearized formulas are used to directly analyze the effects of the parameter contrast. They can also be used to directly estimate the related parameters in detail. According to the study, a known fractional parameter could be analyzed in practice. The synthetic results confirmed that the theory could extend the application of the generalized fractional wave equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of infinitely many solutions for a fractional Kirchhoff–Schrödinger–Poisson system. Based on variational methods, especially the fountain theorem for the subcritical case and the symmetric mountain pass theorem established by Kajikiya for the critical case, we obtain infinitely many solutions for the system under certain assumptions. The novelties of this article lie in the appearance of the possibly degenerate Kirchhoff function and weak assumptions on the nonlinear term which are quite mild. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to highlight the main comment raised on the published manuscript [A. Atangana and K. M. Owolabi, Math. Model. Nat. Phenom. 13, 3 (2018)] by Garrappa [Commun. Nonlinear Sci. Numer. Simul. 70, 302–306 (2019)]. It was shown that the scheme proposed by Garrappa [Commun. Nonlinear Sci. Numer. Simul. 70, 302–306 (2019)] did not capture the memory and nonlocality and led to unreliable results. Therefore, we decided to highlight and validate this issue by means of a scheme where misprinting or typos were observed. Further, we propose some examples where some of them were reported by Garrappa [Commun. Nonlinear Sci. Numer. Simul. 70, 302–306 (2019)]. It is shown further by considering different examples of the nature of linear and nonlinear problems, and we show that the scheme presented by Atangana and Owolabi [Math. Model. Nat. Phenom. 13, 3 (2018)] is correct and gives 100% agreement for the case of linear problems with the other methods in the literature, while for the case of nonlinear problems, it gives a reasonable agreement and thus the claim by Garrappa [Commun. Nonlinear Sci. Numer. Simul. 70, 302–306 (2019)] is baseless. [ABSTRACT FROM AUTHOR]

In this paper, we study linear and nonlinear fractional eigenvalue problems involving the Atangana-Baleanu fractional derivative of the order 1 < δ < 2. We first estimate the fractional derivative of a function at its extreme points and apply it to obtain a maximum principle for the linear fractional boundary value problem. We then estimate the eigenvalues of the nonlinear eigenvalue problem and obtain necessary conditions to guarantee the existence of eigenfunctions. We also obtain a uniqueness result and a norm estimate of solutions of the linear problem. The obtained maximum principle and results are based on a condition that connects the boundary conditions, the order of the fractional derivative, and the Mittag-Leffler kernel. This condition is different from the ones obtained in previous results with different types of fractional derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the stability of the equilibrium point and Hopf bifurcation point in the three-component time-fractional differential equation, which describes the predator-prey interaction between different species. In the dynamics, the classical first-order derivative in time is modelled by either the Caputo or the Atangana-Baleanu fractional derivative of order α , 0 < α < 1. We utilized a fractional version of the Adams-Bashforth formula to discretize these fractional derivatives in time. The results of the linear stability analysis presented are confirmed by computer simulation results. [ABSTRACT FROM AUTHOR]

In order to control the nonlinear high-speed train with high robustness, the fractional order control of nonlinear switching systems is studied. The fractional order controller is designed for a class of nonlinear switching systems by the fractional order backstepping method. In this paper, a simple and effective online updating scheme of model coefficients is proposed by using the flexibility of the model predictive control algorithm and its wide range of model accommodation. A stochastic discrete nonlinear state space model describing the mechanical behavior of a single particle in a high-speed train is constructed, and the maximum likelihood estimation of the parameters of a high-speed train is transformed into an optimization problem with great expectations. Finally, numerical comparison experiments of motion characters of two high-speed trains are given. The results show the effectiveness of the proposed identification method. [ABSTRACT FROM AUTHOR]

In this paper, we apply the q -homotopy analysis transform method to the mathematical model of the cancer chemotherapy effect in the sense of Caputo fractional. We find some new approximate numerical results for different values of parameters of alpha. Then, we present novel simulations for all cases of results conducted by considering the values of parameters of alpha in terms of two- and three-dimensional figures along with tables including critical numerical values. [ABSTRACT FROM AUTHOR]

The use of mathematical methods has become an indispensable research tool and method in the establishment and improvement of many disciplines. Therefore, mathematical methods have also been included in the intelligence analysis system of public security information science. Intelligence is a summary of information that exists in all aspects of our lives. This information is distributed according to time based on certain rules. The application of mathematical analysis methods can more accurately extract effective information and predict future trends. As we all know, classical calculus is a powerful tool for dealing with many dynamic processes in the field of applied science. However, there are many complex systems in nature that cannot be characterized by classical integer-order calculus models, especially in information processing analysis. The fractional-order system model can better describe its system performance. This paper introduces the time series analysis method into the public security intelligence analysis system, combines the fractional differential operator to construct the mathematical model, analyzes the network intelligence, predicts the future occurrence of the case, and compares the predicted data with the actual data to verify. The method is predictive of the true credibility of the results. [ABSTRACT FROM AUTHOR]

In this paper, we study finite-time stability of fractional differential systems with variable coefficients, which includes the homogeneous and nonhomogeneous delayed cases. Based on the theories of fractional differential equations, we obtain three theorems on the finite-time stability, which give some sufficient conditions on finite-time stability, respectively, for homogeneous systems without and with time delay and for the nonhomogeneous system with time delay. [ABSTRACT FROM AUTHOR]

In this paper, we consider the mth order Cauchy-Euler's differential equation involving a para-Grassmann variable of order p. In the Grassmann case (i.e., p = 1), we determine the solution for arbitrary order m. In the case of arbitrary order p, we give a solution for the cases m = 1, 2. [ABSTRACT FROM AUTHOR]

In this paper, we use fractional hydrogenic wave functions as trial functions in a variational calculation of the ground state energies of the H 2 + ion and the He atom. We find improved estimates over those obtained using the standard integer angular momentum atomic trial states. We will see that a benefit of using these fractional functions is the generation of a directional charge distribution, which may serve to orient the atoms in chemical bonds. [ABSTRACT FROM AUTHOR]