Fractional Order Derivative Research Articles

Chimera states in fractional-order coupled Rayleigh oscillators

Chimera states, characterized by the coexistence of synchronized and desynchronized oscillators, have been extensively described via integer differential equations over the past few decades. However, it is still unclear whether and under which conditions chimera states will appear in terms of the coupled systems involving fractional derivatives. In this work, we investigate the chimera states of Rayleigh oscillators in the fractional-order systems with attractive and repulsive couplings. Firstly, we introduce the repulsive control factor and show the various transition routes from the desynchronized state to the incoherent oscillation death by altering this factor and the fractional-order derivatives. Then, it is demonstrated that as the coupling strength varies, a vast repertoire of interesting collective behaviors will emerge, such as synchronization, solitary state, traveling wave, traveling chimera, oscillatory cluster, imperfect traveling chimera state, imperfect amplitude chimera state, and chimera death. Finally, we discuss the effect of attractive and repulsive couplings and find that decreasing the repulsive control factor can reduce the repulsive interaction, allowing the attractive coupling to dominate and promote the emergence of the coherent state.

On the construction of various soliton solutions of two space-time fractional nonlinear models

Abstract In this article, we investigate a couple of nonlinear fractional models of eminent interests subsequently the conformable derivative sense is used to designate the fractional order derivatives. The given structures are transformed into nonlinear ordinary differential equations of integer order, and the extended simple equation technique is then employed to solve the resulting equations. Initially, the nonlinear space time fractional Klein–Gordon equation is considered emerging from quantum and classical relativistic mechanics, which have application in plasma physics, dispersive wave phenomena, quantum field theory, and optical fibres. Later, the (2 + 1)-dimensional time fractional Zoomeron equation is analysed which is convenient to explore the innovative phenomena related to boomerons and trappons. As a result, various new soliton solutions are successfully established. The reported results offer a key implementation for analysing the soliton solutions of nonlinear fractional models which are extremely encouraging arising in the recent era of science and engineering. The 3D simulations have been carried out to demonstrate dynamics of the various soliton solutions for a given set of parameters.

A Two-Temperature Nonlocal Poro-Thermoelastic Solid Via Higher-Order Time-Derivatives Model with Phase Lag

Abstract Purpose The present study has depicted the effect of gravitational on a two-temperature nonlocal poro-thermoelastic solid. Methods The multi-phase-lag model and fractional derivatives are used to tackle the issue. Through normal mode analysis, analytical formulas for the variable fields are derived. Using appropriate boundary conditions, the physical fields are calculated and the numerical computations have been carried out with the help of MATLAB programming. Results Through a careful comparison of the numerical data, the impacts of gravity, fractional derivative order, and locatiy on the behavior of physical fields are described. Conclusion Physical variables are affected by nonlocal thermoelasticity, fractional derivative order as well as the gravity field.

Estimating soil salinity in mulched cotton fields using UAV-based hyperspectral remote sensing and a Seagull Optimization Algorithm-Enhanced Random Forest Model

Xinjiang is a crucial production base for mulched cotton fields but faces increasingly severe challenges of soil salinization. Under the current drive towards precision agriculture, high precision monitoring of agricultural conditions at the field scale has become critical. Traditional remote sensing technologies are limited by insufficient spatial and spectral resolution, making field monitoring under mulched conditions exceptionally challenging. This study introduces a novel approach combining drone-based hyperspectral remote sensing with the Seagull Optimization Algorithm-Enhanced Random Forest Model (SOA-RF). After employing the preprocessing techniques of standard normal variable (SNV) and fractional order derivative (FOD), we employed the model to assess the level of field-scale salinization under optimization of feature selection using separate dimensionality reduction algorithms (sequential projection algorithm (SPA), optimal band combination analysis (OBCA)) and a combined approach (SPA-OBCA). The following main results were found: (1) SNV and FOD enhanced the spectral characteristics of cotton canopies, with the FOD technique performing better in detecting subtle signal changes in the positive and negative peaks of the spectrum. (2) SPA and OBCA were effective in screening the soil salt-sensitive bands in cotton canopies but the combined SPA-OBCA method provided a better selective preference for hyperspectral bands, with an optimal correlation of 0.926. (3) The model combined the advantages of SOA with the power of Random Forest to optimize the parameters and improve the estimation accuracy (R2 = 0.925, RPD = 4.128). We used unmanned aerial vehicle hyperspectral imagery to create a 5-cm resolution map of soil salinity distribution in cotton fields that allowed detection of differences in soil salinity between mulches, thus providing a scientific basis for soil salinity management and the development of precision agriculture.

Generalization of KCC-theory to fractional dynamical systems and application to viscoelastic oscillations

As a geometrical method for analyzing the stability of dynamical systems, Kosambi–Cartan–Chern (KCC)-theory determines the Jacobi stability of a perturbation given to a trajectory based on the deviation curvature tensor. In this study, KCC-theory is generalized to dynamical systems with fractional derivatives, and the deviation curvature tensor is obtained from the fractional differential equation under the condition that the order of the fractional derivatives is close to an integer. Fractional KCC-theory is then applied to a one-dimensional spring–mass–damper model in which the damping force is expressed as a fractional derivative. In this case, three solution types are obtained depending on the damping coefficient and the fractional derivative order. One is damped oscillation with overshoot, which also appears in models with integer order derivatives, and another is damped oscillation without overshoot. The remaining one is overdamping, which neither oscillates nor overshoots. The deviation curvature tensor obtained from the fractional oscillatory model shows that both the damped oscillation without overshoot and the overdamping are Jacobi stable. When a discontinuous Jacobi instability appears on the solution curve, the solution type is classified as damped oscillation with overshoot. This result indicates that damped oscillation with overshoot and other solutions can be classified according to Jacobi stability.

On the existence and uniqueness results for fuzzy fractional boundary value problem involving Caputo iractional derivative

In this paper, we investigate the existence and uniqueness of solutions for fuzzy boundary value problems involving fuzzy Caputo fractional derivatives of order q 2 (2, 3). As a preliminary step, we construct a generic structure of the solution associated with our proposed model by utilizing the Green’s function. We establish the existence of a unique solution of the proposed model paired with the given initial conditions by using Banach fixed point theorem. At last, as application, an illustrative example is given to show the applicability of our theoritical results.

Dynamical behavior of fractal-fractional order monkeypox virus model

A virus that infects both humans and animals, monkeypox is common in many West African countries and has sporadically spread to other parts of the world. There are serious public health concerns around the globe as a result of the recent spike in cases of monkeypox among both endemic and non-endemic populations. This paper proposes to use the Caputo sense of fractal fractional-order derivatives to examine the dynamics of monkeypox transmission. The study uses Schauder’s fixed point theorem to evaluate the solutions qualitatively and establish their uniqueness inside the model. The next generation matrix technique is used to compute the essential reproduction number. The stability of equilibrium points is further investigated in the research, and a sensitivity analysis of model parameters is carried out. When the basic reproduction number R0 is less than 1, the equilibrium without infections is locally stable. Also, when R0 exceeds 1, this equilibrium becomes unstable. The proposed model incorporates Ulam–Hyers stability through nonlinear functional analysis. To estimate solutions for the fractal-fractional order monkeypox model, the Lagrange’s interpolation method is utilized. In addition, numerical simulations are carried out to examine the influence of some parameters on the overall dynamics of the model. Numerical simulations are performed using MATLAB software to exemplify the model behavior in the context of the Nigerian case study. The graphical representations suggest that the fractal fractional order affects the dynamics of the monkeypox. The findings indicate that isolation of infected individuals in the human population helps to reduce disease transmission.

Impulsive fractional dynamic equation with non-local initial condition on time scales

In this manuscript we investigate the existence and uniqueness of an im-pulsive fractional dynamic equation on time scales involving non-local initial condition with help of Caputo nabla derivative. The existency is based on the Scheafer’s fixed point theorem along with the Arzela-Ascoli theorem and Banach contraction theorem. The comparison of the Caputo nabla derivative and Riemann-Liouvile nabla derivative of fractional order are also discussed in the context of time scale.

Analyzing the time-fractional (3 + 1)-dimensional nonlinear Schrödinger equation: a new Kudryashov approach and optical solutions

The paper focuses on investigating the time-fractional (3 + 1)-dimensional cubic and quantic nonlinear Schrödinger equation. We adopt the novel Kudryashov method to generate a distinct class of optical solutions for the current conformable fractional derivative problem. Our method explores various solution forms, including dark, wave, mixed dark-bright, and singular solutions. The soliton solutions we construct are visually represented to illustrate the influence of the fractional order derivative. Further, we elucidate the influence of solution parameters on the wave envelope, providing clear interpretations through 2D graphics presentations. The results underscore the efficacy of our approach in discovering exact solutions for nonlinear partial differential equations, especially in cases where alternative methods prove ineffective. The significance of the present paper lies in its contribution to advancing the understanding of the behavior of optical solutions in nonlinear systems, providing valuable insights for both theoretical and practical applications. In the field of nonlinear optics, this equation can describe the propagation of optical pulses in nonlinear media. It helps in understanding the behavior of intense laser beams as they propagate through materials exhibiting nonlinear optical effects such as self-focusing, self-phase modulation, and optical solitons.

Optimizing Variational Problems through Weighted Fractional Derivatives

In this article, we explore a variety of problems within the domain of calculus of variations, specifically in the context of fractional calculus. The fractional derivative we consider incorporates the notion of weighted fractional derivatives along with derivatives with respect to another function. Besides the fractional operators, the Lagrange function depends on extremal points. We examine the fundamental problem, providing the fractional Euler–Lagrange equation and the associated transversality conditions. Both the isoperimetric and Herglotz problems are also explored. Finally, we conclude with an analysis of the variational problem, incorporating fractional derivatives of any positive real order.

Convergence Analysis of Novel Fractional-Order Backpropagation Neural Networks With Regularization Terms.

Fractional-order derivatives have the potential to improve the performance of backpropagation (BP) neural networks. Several studies have found that the fractional-order gradient learning methods may not converge to real extreme points. The truncation and the modification of the fractional-order derivative are applied to guarantee convergence to the real extreme point. Nonetheless, the real convergence ability is based on the assumption that the algorithm is convergent, which limits the practicality of the algorithm. In this article, a novel truncated fractional-order BP neural network (TFO-BPNN) and a novel hybrid TFO-BPNN (HTFO-BPNN) are designed to solve the above problem. First, to avoid overfitting, a squared regularization term is introduced into the fractional-order BP neural network. Second, a novel dual cross-entropy cost function is proposed and employed as a loss function for the two neural networks. The penalty parameter helps to adjust the effect of the penalty term and further alleviates the gradient vanishing problem. In terms of convergence, the convergence ability of the two proposed neural networks is first proved. Then, the convergence ability to the real extreme point is further analyzed theoretically. Finally, the simulation results effectively illustrate the feasibility, high accuracy, and good generalization ability of the proposed neural networks. Comparative studies among the proposed neural networks and some related methods further substantiate the superiority of the TFO-BPNN and the HTFO-BPNN.

A fractional order model for the transmission dynamics of shigellosis

Shigellosis, a highly contagious bacterial infection causing diarrhea, fever, and abdominal pain, necessitates a deep understanding of its transmission dynamics to devise effective control measures. Our study takes a novel approach, employing a fractional order framework to explore the influence of memory and control measures on Shigellosis transmission dynamics, thereby making a unique contribution to the field. The model is presented as a system of Caputo fractional differential equations capturing time constant controls. The Caputo derivatives are chosen for their inherent benefits. The qualitative features of the model, such as the solutions' existence and uniqueness, positivity, and boundedness, are thoroughly investigated. Moreover, the equilibria of the model are derived and analyzed for their stability using suitable theorems. In particular, local stability was proved through Routh's criteria, while global stability results were established in the Ulam-Hyers sense. The model is then solved numerically with the help of the predict-evaluate-correct-evaluate method of Adams-Bashforth-Moulton. The numerical results underscore the significant impact of memory on disease evolution, highlighting the novelty of integrating memory-related aspects into the meticulous planning of effective disease control strategies. Using fractional-order derivatives is more beneficial for understanding the dynamics of Shigellosis transmission than integral-order models. The advantage of fractional derivatives is their ability to provide numerous degrees of freedom, allowing for a broader range of analysis of the system's dynamic behaviour, including nonlocal solutions. Also, an investigation on the impacts of control measures via parameter variation is done; the findings show that applying treatment and sanitation minimizes disease eruption.

A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations

In this paper, the tempered Ψ–Riemann–Liouville fractional derivative and the tempered Ψ–Caputo fractional derivative of order n−1<α<n∈N are introduced for Cn−1–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered Ψ–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered Ψ–Caputo fractional differential equations are proved. Illustrative examples are given.

Analysis and Applications of the Chaotic Hyperbolic Memristor Model with Fractional Order Derivative

This work presents a new five-dimensional fractional-order chaotic system that includes a feedback memristor. A Lorenz-Stenflo-based fractional-order chaotic system with five dimensions that contains feedback memristor dynamics is used to do this. This work focuses on chaotic models through the ABC fractional derivative, a concept we rigorously examine. We designed and usedsophisticated numerical algorithms to model fractional-order dynamics with the utmost precision to understand this system’s complex characteristics. These numerical approaches excel at non-integer order differentiation, which standard numerical methods struggle with. Our research uses fractional calculus to increase the system’s complexity and robustness, revealing its hyperchaotic nature. We demonstrate that the system is chaotic and stable over fractional orders using eigenvalues, Lyapunov exponents, Kaplan-Yorke dimensions, maximal exponents, phase portraits, and equilibrium points. Our conclusions are strengthened by using these numerical systems, intended for this work, and analytical tools. We improve our understanding of fractional-order chaotic systems with our research. It illuminates how to improve electrical integrations and create more sophisticated nonlinear dynamical systems. This work establishes the approach and insights for future research, guiding the development of new systems with enhanced dynamical features

Enhancing image denoising: A novel non-local anisotropic diffusion framework based on Caputo derivatives and Gaussian convolution for the Perona–Malik model

In the field of image restoration, denoising is considered one of the most important techniques. It is a pre-processing approach aims to refine image clarity and enhance its overall quality by effectively reducing noise present in the image. The aim is to obtain good-quality images from a version degraded by additive noise or convolutional noise that introduces blur. As a result, more advanced treatments can be performed on the resulting image. In order to remove Gaussian noise from input images, we propose the following methodology using a fractional differential equation in time-space based on Gaussian convolution, where the integer and fractional order derivatives of Caputo can be discretized using finite difference and L1−approximations. Once the equation is solved numerically, the scheme is applied to grayscale digital images using the presented algorithm. The parameters must be optimized and adjusted. As a result of testing with natural images, we are able to successfully suppress the noise present in the images. Aside from that. Our model demonstrates strong visual quality, as verified by the calculation of indexes such as Peak Signal-To-Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM). Our denoising technique demonstrates its effectiveness in mitigating noise present in both MRI and X-ray images, resulting in improved image quality and diagnostic accuracy for these vital medical imaging modalities.

Generalization of Direct Adaptive Control Using Fractional Calculus Applied to Nonlinear Systems

This paper presents a new direct adaptive control (DAC) technique using Caputo’s definition of the fractional-order derivative. This is the first time a fractional-order adaptive law is introduced to work together with an integer-order stable manifold for approximating the uncertainty of a class of nonlinear systems. The DAC approach uses universal function approximators such as multi-layer perceptrons with one hidden layer or fuzzy systems to approximate the controller. This paper presents a new lemma, which elucidates and clarifies the link between the Caputo and the Riemann–Liouville definitions. The introduced lemma is useful in developing a Lyapunov candidate to prove the stability of using the proposed fractional-order adaptive law. This is further explained by a numerical example, which is provided to elucidate the practicality of using the fractional-order derivative for updating the approximator parameters. The main novelty of the results in this paper is a rigorous stability proof of the fractional DAC approach for a class of nonlinear systems that is subjected to unstructured uncertainty and deals with the adaptation mechanism using a traditional integer-order stable manifold. This makes the control scheme easier to implement in practice. The fractional-order adaptation law provides greater degrees of freedom and a potentially larger functional control structure than the conventional adaptive control. Finally, the paper demonstrates that traditional integer-order DAC is a special case of the more general fractional-order DAC scheme introduced here.

Adapting Laplace residual power series approach to the Caudrey Dodd Gibbon equation

In real-life applications, nonlinear differential equations play an essential role in representing many phenomena. One well-known nonlinear differential equation that helps describe and explain many chemicals, physical, and biological processes is the Caudrey Dodd Gibbon equation (CDGE). In this paper, we propose the Laplace residual power series method to solve fractional CDGE. The use of terms that involve fractional derivatives leads to a higher degree of freedom, making them more realistic than those equations that involve the derivation of an integer order. The proposed method gives an easy and faster solution in the form of fast convergence. Using the limit theorem of evaluation, the experimental part presents the results and graphs obtained at several values of the fractional derivative order.

A study on the fractional Black–Scholes option pricing model of the financial market via the Yang-Abdel-Aty-Cattani operator

PurposeFinancial mathematics is one of the most rapidly evolving fields in today’s banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF).Design/methodology/approachThe homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity.FindingsIn this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function.Originality/valueThe Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives.

MHD flow of blood-based hybrid nanofluid through a stenosed artery with thermal radiation effect

The MHD flow of a blood-based hybrid nanofluid with thermal radiation and stenosis effects is explored in this study. Gold (Au) and Copper (Cu) hybrid nanoparticles are assumed to be distributed in the blood. Relevant governing equations are established and appropriately normalized. The Caputo fractional-order derivative is used to convert the transient terms of normalized equations into time-fractional forms, which are then semi-analytically solved using the Laplace transform method to yield complex forms of modified Bessel functions. The concentrated matrix exponential (CME) approach is used to numerically determine approximate inverse Laplace transforms of the updated equations. The distribution patterns of velocity, temperature, and concentration are evaluated graphically. The results showed that the fractional-order derivative and the height of stenosis greatly influence the velocity, temperature and concentration profiles. We also observed that both velocity and temperature decreased with increasing values of Ha. Temperature rises with increasing values of the thermal radiation parameter (Ra) and the Eckert number (Ec) whilst concentration is enhanced with rising values of the chemical reaction parameter (Cr) and the Schmidt number (Sc). Our numerical results show that relevant parameters have considerable influences on skin friction, Nusselt number, and Sherwood number, which are important for understanding blood mobility, heat and mass transfer dynamics, and treating cardiovascular illnesses. Our results are also essential for controlling blood flow, temperature and delivery of drugs to diseased arteries in the human body.

The analytical analysis of fractional differential system via different operators and normalization functions

This article is related to the utilization of the Laplace Adomian Decomposition Method (LADM) to analyze the solution of a fractional order system under different operators and normalized functions. In this connection, the three new and updated operators i.e., Caputo, Caputo–Fabrizio, and Atangana–Baleano are used to define the fractional order derivative with the help of four different normalization functions. It is observed that, among the four normalized functions, the fourth kind of normalized function is the most suitable function for the above-mentioned operators. The LADM series solutions are obtained for both integer and fractional orders, implying that the proposed approach has a greater accuracy and convergence rate. Figures and Tables are constructed to verify the validity and efficiency of the suggested technique. LADM simulations confirmed, that the more iterations of LADM reduced the associated absolute error. Moreover, fractional order solutions are obtained which are convergent towards integer order solutions.