Value Of Fractional Order Research Articles

Chaos in fractional order financial model with fractal–fractional derivatives

Recently, a new differential operator which combines fractal differentiation and fractional differentiation with different kernels such as power law, exponential decay, and the Mittag–Leffler function has been introduced. We apply fractal–fractional derivative operators to study chaos in financial chaotic model and implement a numerical procedure to obtain their graphical results. In order to determine the existence of chaos for the chosen value of fractional order, we examine the effects of the saving rate, the per-investment cost, the elasticity of demand, and the Lyapunov exponent. In the case of the classical derivative with power law the obtained attractors presented no similarities. While the attractors obtained via fractal–fractional derivative show some crossover effects. The outcomes of this study are novel and extremely important in dealing with financial issues.

Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

This study focuses on the dynamics of drug concentration in the blood. In general, the concentration level of a drug in the blood is evaluated by the mean of an ordinary and first-order differential equation. More precisely, it is solved through an initial value problem. We proposed a new modeling technique for studying drug concentration in blood dynamics. This technique is based on two fractional derivatives, namely, Caputo and Caputo-Fabrizio derivatives. We first provided comprehensive and detailed proof of the existence of at least one solution to the problem; we later proved the uniqueness of the existing solution. The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques. Stability via the Ulam-Hyers (UH) technique was also investigated. The application of the proposed model on two real data sets revealed that the Caputo derivative was more suitable in this study. Indeed, for the first data set, the model based on the Caputo derivative yielded a Mean Squared Error (MSE) of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360. Caputo-Fabrizio-basedderivative appeared to be the second-best method for the problem, with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532. For the second experiment, Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066, whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299. Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach.

Dynamical effects of hypergraph links in a network of fractional-order complex systems.

In recent times, the fractional-order dynamical networks have gained lots of interest across various scientific communities because it admits some important properties like infinite memory, genetic characteristics, and more degrees of freedom than an integer-order system. Because of these potential applications, the study of the collective behaviors of fractional-order complex networks has been investigated in the literature. In this work, we investigate the influence of higher-order interactions in fractional-order complex systems. We consider both two-body and three-body diffusive interactions. To elucidate the role of higher-order interaction, we show how the network of oscillators is synchronized for different values of fractional-order. The stability of synchronization is studied with a master stability function analysis. Our results show that higher-order interactions among complex networks help the earlier synchronization of networks with a lesser value of first-order coupling strengths in fractional-order complex simplices. Besides that, the fractional-order also shows a notable impact on synchronization of complex simplices. For the lower value of fractional-order, the systems get synchronized earlier, with lesser coupling strengths in both two-body and three-body interactions. To show the generality in the outcome, two neuron models, namely, Hindmarsh-Rose and Morris-Leccar, and a nonlinear Rössler oscillator are considered for our analysis.

Analysis of photoplethysmogram signal to estimate heart rate during physical activity using fractional fourier transform – A sampling frequency independent and reference signal-less method

Background and objectiveAcquiring accurate and reliable health information using a PPG signal in wearable devices requires suppressing motion artifacts. This paper presents a method based on the Fractional Fourier transform (FrFT) to effectively suppress the motion artifacts in a Photoplethysmogram (PPG) signal for an accurate estimation of heart rate (HR). MethodsBy analyzing various PPG signals recorded under various physiological conditions and sampling frequencies, the proposed work determines an optimal value of the fractional order of the proposed FrFT. The proposed FrFT-based algorithm separates the motion artifacts component from the acquired PPG signal. Finally, the HR estimation accuracy during the strong motion artifact-affected windows is improved using a post-processing technique. The efficacy of the proposed method is evaluated by computing the root mean square error (RMSE). ResultsThe performance of the proposed algorithm is compared with methods in recent studies using test and training datasets from the IEEE Signal Processing Cup (SPC). The proposed method provides the mean absolute error of 1.88 beats per minute (BPM) on all twenty-three recordings. ConclusionsThe proposed method uses the Fourier method in the fractional domain. A noisy signal is rotated into an intermediate plane between the time and frequency domains to separate the signal from the noise. The algorithm incorporates FrFT analysis to suppress motion artifacts from PPG signals to estimate HR accurately. Further, a post-processing step is used to track the HR for accurate and reliable HR estimation. The proposed FrFT-based algorithm doesn't require additional reference accelerometers or hardware to estimate HR in real-time. The noise and signal separation is optimum for a fractional order (a) value in the vicinity of 0.6. The optimized value of fractional order is constant irrespective of the physical activity and sampling frequency.

Chaotic dynamics in tritrophic interaction with volatile compounds in plants with power law kernel

In this article, we study a fractional-order mathematical model representing tritrophic interaction amongst plants, herbivores, and carnivores with Caputo derivative. The existence and uniqueness of the system are investigated by fixed point theory, while the stability is studied by Hyers-Ulam and generalized Hyers-Ulam stability analysis. The Adams-Bashforth-Moulton scheme is used for numerical calculations. From numerical simulations, it is observed that when the fractional order decreases the system converges to a stable state. It is observed that for a small value of fractional order, the system approaches a stable state rapidly as compared to the integer order. The chaotic behaviour of the system is studied using the Lyapunov spectrum. It is noted that two positive exponents of the proposed model show that the system is hyper-chaotic. It is also observed that a small value of attraction constant disrupts the system due to volatile organic compounds.

Numerical simulation of SIR childhood diseases model with fractional Adams–Bashforth method

This work investigates a fractional Susceptible Infected Recovered (SIR) model to study childhood disease. We analyzed the proposed model by applying Caputo, Caputo–Fabrizio, and Atangana–Baleanu fractional derivatives. Here, we use the fractional Adams–Bashforth method to solve the childhood disease model with nonlocal operator. The proposed numerical technique is developed by combining the fundamental theorem of integral calculus with Lagrange's interpolation. This numerical approach is more efficient than the other existing numerical techniques and is easily computed using Matlab as a programming language. The existence and uniqueness of this fractional model are studied with the fixed‐point theorem. These fractional derivatives show different asymptotic behaviors for the distinct values of the fractional order. The numerical results are presented graphically as well as in tabulated form for some value of fractional order.

Closed-Loop Tuning of Cascade Controller for Load Frequency Control of Multi-Area Distributed Generation Resources Optimized by ASOS Algorithm

This paper provides closed loop tuning of cascaded-tilted integral derivative controller (CC-TID) for load frequency control (LFC) of micro grid system. A micro grid system is the arrangement of distributed generation resources such as wind turbine generator (WTG), fuel cell (FC), aqua electrolyser (AE), diesel engine generator (DEG) and battery energy storage system (BESS). Different controllers such as proportional integral derivative (PID), two degree of freedom (2DOFPID), three degree of freedom (3DOFPID) and tilted integral derivative (TID) are used not only to sustain the disparity between real power generation and load demand but also accomplish zero steady state error to enrich the frequency and tie power regulations. The anticipated controller encompasses both the value of cascade (CC) and fractional order (FO) controls for better elimination of system instabilities. In the proposed CC-3DOFPID-TID controller, TID controller is castoff as a slave controller and 3DOFPID controller aided the role of dominant controller. The controlled parameters are optimized by adaptive symbiotic organism search (ASOS) algorithm for keen results of difficulties in LFC. To persist in ecosystem, symbiotic relations are predictable by organism through imitators. Further the dynamic behaviours of controller optimized by ASOS, teaching learning based optimization (TLBO) and differential evolution particle swarm optimization (DEPSO) are compared by extensive simulations in MATLAB/SIMULINK. Moreover the supremacy of proposed controller is performed through system dynamics comparison among PID, 2DOFPID, 3DOF-PID and CC-3DOFPID-TID controllers. Finally sensitivity of proposed controller has proven though random load perturbation.

Viscoelastoplastic Displacement Solution for Deep Buried Circular Tunnel Based on a Fractional Derivative Creep Model

Time-dependent deformation of surrounding rock is a common phenomenon for tunnels situated in soft rock stratum or hard rock stratum with high geo-stress. To describe this phenomenon, a creep model combining the Abel dashpot and a non-Newton viscous element was adopted, and the analytical solution about the viscoelastoplastic deformation for circular tunnel was obtained based on this creep model. Then, the auxiliary tunnel of Jinping II hydropower station was taken as an example to reveal the influence of creep parameters on the creep deformation. The research shows that (1) the creep model can well describe the whole creep stage of rocks, that is, the decay, constant, and accelerated creep stages, (2) the creep deformation has a positive relation with the value of fractional order of Abel dashpot and the order of the non-Newton viscous element, and (3) the creep curves between test results and analytical solutions are well consistent with each other, which demonstrate the validity of the analytical solution.

Dynamical behaviors in the FitzHugh–Nagumo system with a memory trace

In this paper, the dynamical behaviors of the FitzHugh–Nagumo (FHN) system with a memory trace, which has time-fractional derivatives, are investigated. For the case of a classical order, the constant input current can change the stability of the equilibrium point in a single FHN unit, and the equilibrium is unstable in a certain range of the current. A decrease of the order of the time-fractional derivative may lead to a linear reduction of the range and the appearance of a solution of mixed-mode oscillations, which consist of subthreshold small-amplitude oscillation and suprathreshold large-amplitude oscillation. In the parameter space of the input current and the fractional order, the region of existing the mixed-mode oscillation is linearly widened when the fractional order moves toward its small value. If a suprathreshold perturbation is periodically applied, there exist some obvious bands, on which the excited period is locked to the perturbation period according to some rational ratios. As a result, the bands can be narrowed by decreasing the value of fractional order and their location has a slight drift toward the small value of the perturbation period. In addition, the properties of solitary traveling waves and wave train solutions are also studied in the one-dimensional space. It is illustrated that the traveling pulse is wider for a smaller value of fractional order, and its velocity is larger. Further, some relations of wave trains have a great change when the value of the fractional order is changed.

Etkili yöntem kullanan ağ erişim kontrolü modeli için kesirli yaklaşım

In this paper, we find the solution for the system of nonlinear ordinary differential equations having fractional-order arising in network access control using fractional natural decomposition method (FNDM). The consider a model which consists of a system of five nonlinear ordinary differential equations (NODEs), which illustrate the sensor networks are interesting essentials for malicious outbreaks that attack the network with the intention of reducing the integrity, availability and confidentiality. Further, we captured the nature of FNDM results for different value of fractional order in terms of the plots. The considered scheme highly effective and structured while examining nonlinear models and which can be observed and confirm from the obtained results. Further, the conspiracies cited in plots confirm the hired fractional operator and algorithm can help to exemplify the more fascinating properties of the nonlinear system associated real-world problems.

A study on fractional COVID-19 disease model by using Hermite wavelets.

The preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID‐19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID‐19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID‐19 model with time‐ arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary‐order problem into algebraic equations. The corrector scheme is also used for solving the COVID‐19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary‐order COVID‐19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet‐based results.

Pseudo-Differential (2 + α)-Order Butterworth Frequency Filter

This paper describes the design of analog pseudo-differential fractional frequency filter with the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(2+\alpha)$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0 &lt; \alpha &lt; 1$ </tex-math></inline-formula> . The filter operates in a mixed-transadmittance mode (voltage input, current output) and provides a low-pass frequency response according to Butterworth approximation. General formulas to determine the required transfer function coefficients for desired value of fractional order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> are also introduced. The designed filter provides the beneficial features of fully-differential solutions but with a less complex circuit topology. It is canonical, i.e. it employs a minimum number of passive elements, whereas all are grounded, and current conveyors as active elements. The proposed structure offers high input impedance, high output impedance, and high common-mode rejection ratio. By simple modification, voltage response can also be obtained. The performance of the proposed frequency filter is verified both by simulations and experimental measurements proving the validity of theory and the advantageous features of the filter.

Characterization of gas flow in backfill mining-induced coal seam using a fractional derivative-based permeability model

Backfill mining technique has become more and more popular in Chinese coal enterprises in recent years. When it is applied in high gassy mines, a modeling approach to gas flow in backfill mining-induced coal seam plays an important role in assessment of safety of the technological system. In order to establish an intrinsic relationship between post-peak damage of coal and its permeability, a fractional derivative permeability model based on the classic model is proposed by incorporating a damage variable into fractional derivative order. Since the coal body is damaged, its permeability increases rapidly, which can be represented by a increasing trend in the value of fractional order, indicating that the proposed model can effectively reflect the dramatic change in permeability caused by post-peak damage. By testing permeability during the loading and unloading process of coal samples, key parameters of the fractional permeability model were obtained to enter specific values into the model to represent in-situ conditions. Then, based on the Mazars damage evolution criterion and the fractional permeability model, stress field equations and seepage field equations of mining-induced coal are established and applied in a generalized numerical model to characterize gas flow in backfill mining-induced coal seam under different elastic foundation coefficients of filling body. The result shows that the damage scale and degree at coal seam in front of the longwall face, as well as its permeability, appear to be smaller with increase of filling body's elastic foundation coefficient. Along the direction of gas flow, as the elastic foundation coefficient increases, the gas flow velocity reduces, while the gas pressure and content go up correspondingly at the same position of the front coal body. It indicates that the ability for gas to flow is negatively correlated with the elastic foundation coefficient of filling body in backfill mining-induced coal seam.

A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials

AbstractEpidemiology is the glorious discipline underlying medical research, public health practice, and health care evaluation. Nowadays, research on disease models with anonymous parameters is a popular issue for researchers working in epidemiology. Due to popularity of this field, a new numerical method for the solution of the fractional SEIR epidemic of measles is introduced where fractional derivative is taken in Caputo sense. We have discussed about the framework of Genocchi wavelets for numerical simulations of above disease model. Furthermore, the operational matrix merged with the collocation method is used in order to convert fractional‐order problem into algebraic equations. The Adams‐Bashforth‐Moulton (ABM) numerical scheme is used to solve above disease model with various parameters. For we have compared the solutions with Adams‐Bashforth‐Moulton predictor corrector scheme for the accuracy and applicability of the Genocchi wavelets method (GWM). The behaviors of susceptible, exposed, infected, and recovered individuals are presented graphically at the value of various fractional order. The error and convergence analysis of the Genocchi wavelets has been discussed for the applicability of the present methods. Further, various numerical simulations have been carried out to justify our achieved finding.

Chaotic and spatiotemporal oscillations in fractional reaction-diffusion system

• Fractional derivative approximation in 2D. • Linear stability analysis for temporal system. • Alternating direction implicit finite difference method. • Numerical simulations. • Chaotic and spatiotemporal phenomenon. This paper focuses on the design and analysis of an efficient numerical method based on the novel implicit finite difference scheme for the solution of the dynamics of reaction-diffusion models. The work replaces the integer first order derivative in time with the Caputo fractional derivative operator. The dynamics of activator-inhibitor as encountered in chemistry, physics and engineering processes, and predator-prey models are two cases addresses in this study. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction-diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve cross-diffusion problem in two-dimensions. In the experimental results, a number of spatiotemporal and chaotic patterns that are related to Turing pattern are observed. It was discovered in the simulation experiments that the species predator-prey model distribute in almost same fashion, while that of the activator-inhibitor dynamics behaved differently regardless of the value of fractional order chosen.

Random Fractional Laplace Transform for Solving Random Time-Fractional Heat Equation in an Infinite Medium

The random Laplace and Fourier transforms are very important tools to solve random heat problems. Unfortunately, it is difficult to use these random integral transforms for solving the fractional random heat problems, where the mean square conformable fractional derivative is used to express for the time fractional derivative. Therefore, this work adopts the extension of the random Laplace transform into random fractional Laplace transform in order to solve this kind of heat problems. The stochastic process solution of the fractional random heat in an infinite medium is investigated by using random fractional Laplace transform together with random Fourier transform. The mean and the standard diffusion of the stochastic process solution is computed for different value of fractional order. When α = 1, the results show agree with the available results in the references context.

High-dimensional spatial patterns in fractional reaction-diffusion system arising in biology

Abstract The concept of fractional derivative has been demonstrated to be successful when applied to model a range of physical and real life phenomena, be it in engineering and science related fields. It is a known fact that reaction-diffusion equation permits the use of different numerical methods in space and time. As a result, we introduce the Fourier spectral method for the discretization of space fractional derivative and adapt the modified version of the exponential time-integrator to advance in time in attempt to explore the dynamic richness of fractional reaction-diffusion equations in two and three dimensions. This approach gives a full diagonal representation of the fractional derivative operator and yields a better spectral convergence irrespective of the value of fractional order chosen in the experiment. Recommendations are made based on some amazing results which arise from the computational experiments. We intend to answer the question ’why is wildlife animals going into extinction in Africa?’ to a reasonable extent. We believe that the spatial patterns obtained in the simulation framework to mimic the ones found in wildlife would provide a measure and serves as a good alternative to an act of killing of wildlife animals for ornamental and decorative purposes, also would serve as a guild to textile industries on pattern formations.

Suppression of Sommerfeld effect in a non-ideal discrete rotor system with fractional order external damping

Sommerfeld effect often causes dynamic instability in high speed rotors driven through energy source with limited power. As a result, the system dynamics changes in a very unusual manner while exceeding a critical value of input power with sudden nonlinear jump of rotor speed proximity to the system resonance. Such instability phenomena arising out of strong nonlinear rotor-drive interactions need to be attenuated to ensure smooth operation and industrial safety. A novel approach is presented here to study the attenuation of this kind of instability in an internally damped DC motor driven discrete rotor system with the adjustment of fractional order parameter in its external damping term. Using Lagrangian formulation, the equations of motion of the discrete rotor system are derived. The well-known Caputo model is adopted for the fractional order external damping modelling. Following, a characteristic equation in the form of a fifth order polynomial is obtained through steady-state energy balance approach which leads to obtain various amplitude frequency responses with a few values of fractional order of the external damping. As the fractional order increases gradually, the Sommerfeld effect is found to be attenuated accordingly. A few steady-state results are found to be in good agreement with some results reported earlier. Additionally, root-loci technique is employed to obtain a specific value of fractional order for which the complete disappearance of Sommerfeld effect is achieved through the merging of the break points. Finally, a transient analysis also confirms the steady-state results numerically.

Numerical study of unsteady natural convection of variable-order fractional Jeffrey nanofluid over an oscillating plate in a porous medium involved with magnetic, chemical and heat absorption effects using Chebyshev cardinal functions

The unsteady natural convection of an incompressible, magnetohydrodynamic (MHD), Jeffrey nanofluid in the vicinity of a vertical oscillating plate in a porous medium is examined. Several involving parameters such as thermal radiation, chemical reaction, Soret effect, thermal diffusion, magnetic field and heat absorption are taken into account. To gain an insight into the behavior of a memory-dependent fluid, the variable-order fractional calculus for heat, mass and concentration equations is implemented, which, to the best of the author's knowledge, has not been tackled by the researchers for the present comprehensive problem. The set of governing equations are reformed to the dimensionless form via invoking appropriate variables. A computational matrix method based on the Chebyshev cardinal functions (CCFs) is given to examine the behavior of the relevant problem. In fact, regarding to the established method, the unknown solutions are expanded by the CCFs. Then, the operational matrix of the variable-order fractional derivative is utilized to transfer the problem into solving an algebraic system of equations. The applicability and accuracy of the suggested method are investigated by solving some test problems. The numerical results confirm the high accuracy of the presented approach. The effect of several parameters such as Grashof, Hartmann, Prandtl, Soret and Schmidt numbers, in addition to oscillation frequency, retardation time, radiation, heat absorption and reaction rate are determined and presented graphically. According to the results, increasing the value of fractional order improves the natural convection mechanism as well as the peak value of the velocity profile.

A fractional-order hyper-chaotic economic system with transient chaos

We introduce for the first time a fractional-order hyperchaotic economic system. In this system, chaos generation depends upon the value of fractional-order. For certain fractional orders, a sustained regime of chaos is obtained. Also, the transient chaos phenomenon is detected for lower fractional orders. The dynamical behavior of the system is numerically investigated using bifurcations diagrams, basins of attraction, and Lyapunov exponents. Next, an adaptive terminal sliding mode control (ATSMC) with neural network estimator for finite-time stabilization and synchronization of the fractional-order system has been proposed. The radial basis function (RBF) neural network is used to achieve the estimation of the unknown function of the system. Also, the effects of external disturbances are fully taken into account with neural network estimator. The weights of the RBF neural network are updated based on the appropriate adaptation law. Using the fractional version of the Lyapunov stability theorem, the finite-time convergence of the closed-loop system has been proven. Finally, the new control technique is used for control and synchronization of the fractional-order hyperchaotic economic system. Simulation results illustrate the effectiveness of the proposed control scheme for uncertain fractional-order systems in the presence of external disturbances.