Effects Of Fractional Order Derivative Research Articles

Impressive Exact Solitons to the Space-Time Fractional Mathematical Physics Model via an Effective Method

A new class of truncated M-fractional exact soliton solutions for a mathematical physics model known as a truncated M-fractional (1+1)-dimensional nonlinear modified mixed-KdV model are achieved. We obtain these solutions by using a modified extended direct algebraic method. The obtained results consist of trigonometric, hyperbolic trigonometric and mixed functions. We also discuss the effect of fractional order derivative. To validate our results, we utilized the Mathematica software. Additionally, we depict some of the obtained kink, periodic, singular, and kink-singular wave solitons, using two and three dimensional graphs. The obtained results are useful in the fields of fluid dynamics, nonlinear optics, ocean engineering and others. Furthermore, these employed techniques are not only straightforward, but also highly effective when used to solve non-linear fractional partial differential equations (FPDEs).

Exploring the new soliton solutions to the nonlinear M-fractional evolution equations in shallow water by three analytical techniques

This paper explores the new soliton solutions of the evolution equations named as truncated M-fractional (1+1)-dimensional non-linear Kaup-Boussinesq system by utilizing the expa function, modified simplest equation and Sardar sub-equation techniques. This system is used in the analysis of long waves in shallow water. The attained results involving trigonometric, hyperbolic and exponential functions. The effect of fractional order derivative is also discussed. Obtained results are very close to the approximate results due to the use of M-fractional derivative. Achieved results are verified by Mathematica tool. Few of the gained results are also explained through 2-D, 3-D and contour graphs. At the end, these techniques are straight forward, useful and effective to deal with non-linear FPDEs.

An efficient computational technique for solving a fractional-order model describing dynamics of neutron flux in a nuclear reactor

This paper focuses on the development of a computational technique for numerical solution of the time-fractional neutron diffusion (FND) equation, which describes dynamics of neutron flux in a nuclear reactor. The time-fractional derivatives are discretized by means of L1 scheme, while the space derivative is discretized by employing a collocation method based on quartic B-spline (QUBS) basis functions. Numerical experiments are presented to illustrate the performance of proposed technique. Results reveal that the proposed technique yields a reliable approximation to the solution of the underlying model problem and has O(k+h4)−order convergence, with k and h representing the step sizes in time and space directions, respectively. It is important to point out that the numerical method for the underlying problem has not yet been discussed in the literature. The effect of fractional order derivative on the behavior of neutron flux is investigated.

Mathematical analysis of fractional-order Caputo’s derivative of coronavirus disease model via Laplace Adomian decomposition method

BackgroundThe world's survival ability has been threatened by the COVID-19 outbreak. The possibility of the virus reemerging in the future should not be disregarded, even if it has been confined to certain areas of the world after wreaking such havoc. This is because it is impossible to prove that the virus has been totally eliminated. This research attempts to investigate the spread and control of the COVID-19 virus in Nigeria using the Caputo fractional order derivative in a proposed model.ResultsWe proposed a competent nine-compartment model of Corona virus infection. It starts by demonstrating that the model is epidemiologically sound in terms of solution existence and uniqueness. The basic reproduction threshold R0 was determined using the next-generation matrix technique. We applied the Laplace-Adomian decomposition method to the fractional-order Caputo's derivative model of the Corona virus disease to produce the approximate solution of the model analytically. The obtained results, in the form of an infinite series, were simulated using the MAPLE 18 package to investigate the effect of fractional order derivative on the dynamics of COVID-19 transmission in the model and shed light on methods of eradication. The graphical interpretations of the simulation process were shown and discussed accordingly.ConclusionsThe study reveals the effect of the Caputo fractional order derivative in the transmission dynamics of the disease. Individual recovery was found to be greatest at an integer order, which represents the full implementation of other factors such as treatment, vaccination, and disease transmission reduction. Hence, we advised that researchers, government officials, and health care workers make use of the findings of this study to provide ways in which disease transmission will be reduced to a minimum to stop the prevalence of COVID-19 by applying the findings of this study.

Design and analysis of a high order computational technique for time‐fractional Black–Scholes model describing option pricing

This work deals with the construction and analysis of a high‐order computational scheme for a time‐fractional Black‐Scholes model that governs the European option pricing. The time‐fractional derivative is considered in the sense of Caputo and the L1 − 2 formula is employed to approximate the Caputo temporal‐fractional derivative of order α, where α ∈ (0, 1). A compact difference scheme is designed for discretization of space variable. The convergence of the method is discussed in detail. It is shown that the proposed method has fourth order accuracy in space and order accuracy in time. One numerical example with the known exact solution is considered to demonstrate the applicability and accuracy of present numerical scheme. Moreover, the suggested numerical scheme is employed for pricing three European option problems, namely European call option, European double barrier knock‐out option and European put option. The effect of fractional order derivative on option price profile is investigated. Furthermore, the effects of three relevant parameters, namely volatility, strike price and interest rate on the price of European double barrier knock‐out option are investigated. The computational time for the proposed method is provided.

Chaos in fractional system with extreme events.

Understanding extreme events attracts scientists due to substantial impacts. In this work, we study the emergence of extreme events in a fractional system derived from a Liénard-type oscillator. The effect of fractional-order derivative on the extreme events has been investigated for both commensurate and incommensurate fractional orders. Especially, such a system displays multistability and coexistence of multiple extreme events.

A compact finite difference scheme for fractional Black-Scholes option pricing model

In this paper, we present a numerical technique for solving the time-fractional Black-Scholes (TFBS) equation describing European options. The time-fractional derivative is described by means of Caputo and a compact finite difference method is employed for discretization of space derivative. Stability and convergence of the fully discrete scheme are studied. Two test problems with the known analytical solutions are considered to demonstrate the efficiency and accuracy of the method and validate the theoretical result. Further, in order to demonstrate the practicability of the method, the method is applied to three European options pricing problems governed by the TFBS equations, where analytical solutions to these problems are not known. We study the effect of fractional order derivative on the solution profile corresponding to option price.

A theoretical study of the Caputo–Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control

Mumps is the most common cause of acquired unilateral deafness in children, in which hearing loss occurs at all auditory frequencies. We use a box model to model hearing loss in children caused by the mumps virus, and since the fractional-order derivative retains the effect of system memory, we use the Caputo–Fabrizio fractional derivative in this modeling. In the beginning, we compute the basic reproduction number R0 and equilibrium points of the system and investigate the stability of the system at the equilibrium point. By utilizing the Picard–Lindelof technique, we prove the existence an unique solution for given fractional CF-system of hearing loss model and investigate the stability of iterative method by fixed point theory. The optimal control of the system is determined by considering the treatment as a control strategy to reduce the number of infected people. Using the Euler method for the fractional-order Caputo–Fabrizio derivative, the approximate solution of the system is calculated. We present a numerical simulation for the transmission of disease with respect to the transmission rate and the basic reproduction number in two cases R0<1 and R0>1. To investigate the effect of fractional order derivative on the behavior and value of each of the variables in Model 2, we calculate the results for several fractional order derivatives and compare the results. Also, considering the importance of reproduction number in the continuation of disease transmission, we analyze the sensitivity of R0 respect to each of the model parameters and determine the impact of each parameter.

Design and analysis of a numerical method for fractional neutron diffusion equation with delayed neutrons

The main purpose of this work is to construct and analyze an efficient numerical scheme for solving the fractional neutron diffusion equation with delayed neutrons, which describes neutron transport in a nuclear reactor. The L1 approximation is used for discretization of time derivative and finite difference method is used for discretization of space derivative. The stability and convergence analysis of the proposed method are studied. The method is shown to be second-order convergent in space and (2−2α)-th order convergent in time, where α is the order of fractional derivative. Numerical experiments are carried out to demonstrate the performance of the method and theoretical analysis. The effects of fractional order derivative, relaxation time and radioactive decay constant on the neutron flux behaviour are investigated. Moreover, the CPU time of the present method is provided.

Effects of fractional order derivative on the solution of time-fractional Cahn–Hilliard equation arising in digital image inpainting

The Cahn–Hilliard equation is a nonlinear partial differential equation which is used in digital image inpainting to restore damaged or missing parts of degraded text and high contrast images. Due to the nonlocal property of the fractional derivative, the fractional order Cahn–Hilliard equation can describe these physical processes in more flexible way. In this paper, the effects of fractional order derivative on the solutions of time-fractional Cahn–Hilliard equation are investigated using a new modification of the homotopy analysis method and a recently developed technique, namely residual power series method. The numerical and graphical illustrations depict the accuracy and reliability of the obtained results.

Exploring the potential of airborne hyperspectral image for estimating topsoil organic carbon: Effects of fractional-order derivative and optimal band combination algorithm

Estimating soil organic carbon (SOC) in topsoil can help improve soil quality and food production. This study aimed to explore the potential of airborne hyperspectral image to estimate the SOC of bare topsoil at an agricultural site located in the southeast part of Iowa State, United States. To magnify the subtle spectral signals concerning SOC, and accelerate calibration and improve predictive ability, we developed a framework to combine two advanced spectral algorithms, namely, fractional-order derivative (FOD) and optimal band combination algorithm for SOC predicting. Our case was based on 49 soil samples and a scattered airborne hyperspectral image. Random forest (RF) was utilized to establish SOC estimation models by incorporating the optimal spectral indices processed by different FOD transformations on the basis of the optimal band combination algorithm. Results indicated that when the fractional order increased, overlapping peaks and baseline drifts were gradually removed. However, the magnitude of spectral strength decreased concurrently. More detailed and abundant spectral variability was captured by FOD as compared with those by original reflectance and first and second derivatives. The estimation accuracies developed from the optimal band combination algorithm (cross-validation R2, 0.36–0.66) were generally better than those from full-spectrum data (cross-validation R2, 0.32–0.54). The RF model based on the combination of 0.75-order reflectance and optimal band combination algorithm obtained the highest estimation accuracy for SOC with cross-validation R2 of 0.66. This research provides guidance for future studies in selecting the most appropriate FOD transformation to preprocess spectral data and in using the optimal band combination algorithm to determine the spectral index. Airborne hyperspectral image-based modeling can be further used to map agricultural topsoil SOC to support local-scale agricultural planning.

Duhem histeresisin Riemann-Liouville kesirli türevi ile modellenmesi

In the view of memory effect of hysteresis, this work aims to interpret hysteresis nonlinearities in terms of Riemann-Liouville fractional derivative which is a singular operator with memory and hereditary properties. For this purpose, Duhem hysteresis, a model defined by a first order differential equation, is considered and adapted to a fractional order differential equation. Since the fractional order Duhem hysteresis cannot be solved by an analytical scheme, Grünwald-Letnikov approximation is used to obtain numerical solutions. Thus, the effect of fractional order derivative to Duhem hysteresis is demonstrated with graphics obtained by this approximation and plotting using MATLAB. As a result, it is observed that the fractional order model exhibits hysteresis behavior for the orders that are smaller than 1.

Dynamical analysis of a discrete conformable fractional order bacteria population model in a microcosm

In this paper, conformable fractional order differential equations with piecewise constant arguments are used for a modeling population density of a bacteria species in a microcosm. The discretization process of a differential equation with piecewise constant arguments gives us two dimensional discrete dynamical system in the interval t∈[nh,(n+1)h). By using the center manifold theorem and the bifurcation theory, it is shown that the discrete dynamical system undergoes flip and Neimark–Sacker bifurcation depending on the parameter r. It is observed that the model describes some biological phenomena for a bacteria population such as homogeneous bacteria distributions and inhomogeneous spatial population distributions. In addition, the effect of fractional order derivative on dynamic behavior of the system is investigated. Finally, all theoretical results are supported by numerical simulations.

Estimating lead and zinc concentrations in peri-urban agricultural soils through reflectance spectroscopy: Effects of fractional-order derivative and random forest

Heavy metal contamination of peri-urban agricultural soil is detrimental to soil environmental quality and human health. A rapid assessment of soil pollution status is fundamental for soil remediation. Heavy metals can be monitored by visible and near-infrared spectroscopy coupled with chemometric models. First and second derivatives are two commonly used spectral preprocessing methods for resolving overlapping peaks. However, these methods may lose the detailed spectral information of heavy metals. Here, we proposed a fractional-order derivative (FOD) algorithm for preprocessing reflectance spectra. A total of 170 soil samples were collected from a typical peri-urban agricultural area in Wuhan City, Hubei Province. The reflectance spectra and lead (Pb) and zinc (Zn) concentrations of the samples were obtained in the laboratory. Two calibration methods, namely, partial least square regression and random forest (RF), were used to establish the relation between the spectral data and the two heavy metals. In addition, we aimed to explore the use of spectral estimation mechanism to predict the Pb and Zn concentrations. Three model evaluation parameters, namely, coefficient of determination (R2), root mean squared error, and ratio of performance to inter-quartile range (RPIQ), were used. Overall, the spectral reflectance decreased with the increase in Pb and Zn contents. The FOD algorithm gradually removed spectral baseline drifts and overlapping peaks. However, the spectral strength slowly decreased with the increase in fractional order. High fractional-order spectra underwent more spectral noises than low fractional-order spectra. The optimal prediction accuracies were achieved by the 0.25- and 0.5-order reflectance RF models for Pb (validation R2 = 0.82, RPIQ = 2.49) and Zn (validation R2 = 0.83, RPIQ = 2.93), respectively. A spectral detection of Pb and Zn mainly relied on their covariation with soil organic matter, followed by Fe. In summary, our results provided theoretical bases for the rapid investigation of Pb and Zn pollution areas in peri-urban agricultural soils.

Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis, chaos control and synchronisation in its fractional-order form

In this paper, a Van der Pol–Duffing (VdPD) jerk oscillator is designed. The proposed VdPD jerk oscillator is built by converting the autonomous two-dimensional VdPD oscillator to a jerk oscillator. Dynamical behaviours of the proposed VdPD jerk oscillator are investigated analytically, numerically and analogically. The numerical results indicate that the proposed VdPD jerk oscillator displays chaotic oscillations, symmetrical bifurcations and coexisting attractors. The physical existence of the chaotic behaviour found in the proposed VdPD jerk oscillator is verified by using Orcad-PSpice software. A good qualitative agreement is shown between the numerical simulations and the PSpice results. Moreover, the fractional-order form of the proposed VdPD jerk oscillator is studied using stability theorem of fractional-order systems and numerical simulations. It is found that chaos, periodic oscillations and coexistence of attractors exist in the fractional-order form of the proposed jerk oscillator with order less than three. The effect of fractional-order derivative on controlling chaos is illustrated. It is shown that chaos control is achieved in fractional-order form of the proposed VdPD jerk oscillator only for the values of linear controller used. Finally, the problem of drive–response synchronisation of the fractional-order form of the chaotic proposed VdPD jerk oscillators is considered using active control technique.

A Study on Fractional Order Theory in Thermoelastic Half-Space under Thermal Loading

In this study, the effect of fractional order derivative on a two-dimensional problem due to thermal shock with weak, normal and strong conductivity is established. The governing equations are taken in the context of Green and Naghdi of type III model (GNIII model) under fractional order derivative. Based on the Laplace and exponential Fourier transformations with eigenvalues approach, the analytical solutions has been obtained. For weak, normal and strong conductivity, the numerical computations for copper-like medium have been done and the results are shown numerically. The graphical results indicate that the effect of fractional order parameter has a major role on all physical quantities involved in the problem.

Modelling of Piezothermoelastic Beam with Fractional Order Derivative

AbstractThis paper deals with the study of transverse vibrations in piezothermoelastic beam resonators with fractional order derivative. The fractional order theory of thermoelasticity developed by Sherief et al. [1] has been used to study the problem. The expressions for frequency shift and damping factor are derived for a thermo micro-electromechanical (MEM) and thermo nano-electromechanical (NEM) beam resonators clamped on one side and free on another. The effect of fractional order derivative on the derived expressions is observed analytically and shown graphically in the case of Lead Zirconate Titanate (PZT)-5A material. For α = 1, our results agree with those that are obtained by Grover and Sharma [20] and other particular cases of interest are also discussed.

Optimal design for fractional-order active isolation system

The optimal control of fractional-order active isolation system is researched based on the optimal control theory, and the effect of fractional-order derivative on passive isolation system is also analyzed. The mechanical model is established where viscoelastic features of isolation materials are described by fractional-order derivative. The viscoelastic property of the fractional-order derivative in dynamical system is studied and the fractional-order derivative could be divided into linear stiffness and linear damping. It is found that both the fractional coefficient and the fractional order could affect not only the resonance amplitude through the equivalent linear damping coefficient but also the resonance frequency by the equivalent linear stiffness. Based on optimal control theory, the feedback gain of fractional-order active isolation system under harmonic excitation is obtained, which is changed with the excitation frequency. The statistical responses of the displacement and velocity for passive and active vibration isolation systems subjected to random excitation are also presented, which further verifies the excellent performance of fractional-order derivative in vibration control engineering.