Fractional Order Derivative Research Articles

Solving Undamped and Damped Fractional Oscillators via Integral Rohit Transform

Background: The dynamics of fractional oscillators are generally described by fractional differential equations, which include the fractional derivative of the Caputo or Riemann-Liouville type. These equations induce classical oscillator equations like the harmonic oscillator equation, to include fractional order derivatives. Solving fractional differential equations numerically can be challenging due to the non-local nature of fractional derivatives. Objective: In this paper, a recently developed integral Rohit transform is utilized for solving systems of undamped and damped fractional oscillators characterized by differential equations of fractional or non-integral order involving the Caputo-fractional derivative operator. The solutions of fractional systems which include undamped-simple fractional oscillators, undamped-driven fractional oscillators, damped-driven fractional oscillators, and damped-fractional oscillators are obtained. Methods: by applying the integral Rohit transform, also written as RT. Differential equations of fractional or non-integral order are generally solved by utilizing methods which include the fractional variational iteration approach, the homotopy-perturbation method, the equivalent linearized method, the Adomian decomposition method, etc. Results: This paper demonstrates the effectiveness, reliability, and efficiency of the integral Rohit transform in solving fractional systems, which include undamped-simple fractional oscillators, undamped-driven fractional oscillators, damped-driven fractional oscillators, and damped-fractional oscillators and are characterized by differential equations of fractional or non-integral order involving the Caputo-fractional derivative operator. Conclusions: The Rohit transform brought the progressive principles or methodologies that offer new insights or views on the problems examined in the paper, distinguishing itself from existing methods and doubtlessly beginning up new research instructions. It provided precise results for the specific problems discussed in the paper, surpassing the capabilities of other methods in terms of decision, constancy, or robustness to noise and disturbances.

Analytical solutions of higher-dimensional coupled system of nonlinear time-fractional diffusion-convection-wave equations

This paper discusses how to generalize the invariant subspace method for deriving the analytical solutions of the multi-component [Formula: see text]-dimensional coupled nonlinear time-fractional PDEs (NTFPDEs) in the sense of Caputo fractional-order derivative for the first time. Specifically, we describe how to systematically find different invariant product linear spaces with various dimensions for the considered system. Also, we observe that the obtained invariant product linear spaces help to reduce the multi-component [Formula: see text]-dimensional coupled NTFPDEs into a system of fractional-order ODEs, which can then be solved using the well-known analytical methods. More precisely, we illustrate the effectiveness and importance of this developed method for obtaining a long list of invariant product linear spaces for the multi-component [Formula: see text]-dimensional coupled nonlinear time-fractional diffusion-convection-wave equations. In addition, we have shown how to find different kinds of generalized separable analytical solutions for a multi-component [Formula: see text]-dimensional coupled nonlinear time-fractional diffusion-convection-wave equations along with the initial and boundary conditions using invariant product linear spaces obtained. Finally, we provide appropriate graphical representations of some of the derived generalized separable analytical solutions with various fractional-order values.

New Event-Triggered Synchronization Criteria for Fractional-Order Complex-Valued Neural Networks with Additive Time-Varying Delays

This paper explores the synchronization control issue for a class of fractional-order Complex-valued Neural Networks (FOCVNNs) with additive time-varying delays (TVDs) utilizing a sampled-data-based event-triggered mechanism (SDBETM). First, an innovative free-matrix-based fractional-order integral inequality (FMBFOII) and an improved fractional-order complex-valued integral inequality (FOCVII) are proposed, which are less conservative than the existing classical fractional-order integral inequality (FOII). Secondly, an SDBETM is inducted to conserve network resources. In addition, a novel Lyapunov–Krasovskii functional (LKF) enriched with additional information regarding the fractional-order derivative, additive TVDs, and triggering instants is constructed. Then, through the integration of the innovative FOCVII, LKF, SDBETM, and other analytical methodologies, we deduce two criteria in the form of linear matrix inequalities (LMIs) to ensure the synchronization of the master–slave FOCVNNs. Finally, numerical simulations are illustrated to confirm the validity of the proposed results.

THERMAL BEHAVIOUR OF A CIRCULAR PLATE UNDER CAPUTO-FABRIZIO FRACTIONAL IMPACT WITH SECTIONAL HEATING

Recent advances in the understanding of the precise physical thermal behaviour of various solids under the effect of fractional-order derivatives have boosted the study of thermoelasticity, which is primarily important in various industrial designs of usable structural materials. We investigated a thin circular plate that was subjected to additional sectional heating on its top and lower surfaces while creating thermal insulation around its outside border. In this work, we maintained the heat transfer equation while accounting for the impact of Caputo-Fabrizio fraction-order derivatives. According to specified boundary constraints, the integral transformation approach is used to assess the analytical solution of the displacement, temperature change, and thermal stresses. Furthermore, various functions and fractional parameters are computed using the material properties of aluminium metal plates for numerical purposes.

Extension of Meir-Keeler-Khan (ψ − α) Type Contraction in Partial Metric Space

In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type (ψ−α)-contraction mapping and propose fixed point results in partial metric spaces. Our proposed results extend, unify, and generalize existing findings in the literature. In regards to applicability, we provide evidence for the existence of a solution for the fractional-order differential operator. In addition, the solution of the integral equation and its uniqueness are also discussed. Finally, we conclude that our results are superior and generalized as compared to the existing ones.

MFFGD: An adaptive Caputo fractional-order gradient algorithm for DNN

As a primary optimization method for neural networks, gradient descent algorithm has received significant attention in the recent development of deep neural networks. However, current gradient descent algorithms still suffer from drawbacks such as an excess of hyperparameters, getting stuck in local optima, and poor generalization. This paper introduces a novel Caputo fractional-order gradient descent (MFFGD) algorithm to address these limitations. It provides fractional-order gradient derivation and error analysis for different activation functions and loss functions within the network, simplifying the computation of traditional fractional order gradients. Additionally, by introducing a memory factor to record past gradient variations, MFFGD achieves adaptive adjustment capabilities. Comparative experiments were conducted on multiple sets of datasets with different modalities, and the results, along with theoretical analysis, demonstrate the superiority of MFFGD over other optimizers.

Effect of slip on MHD flow of fluid with heat and mass transfer through a plate

Fractional order derivatives are recognized as advanced mathematical tools with broad applications in physics and engineering for deriving real-world solutions. This study examines a fractional order model of a non-linear Casson fluid, highlighting the widespread utility of fractional derivatives. Additionally, the problem incorporates the Dufour and slip effects. Specifically, the constant proportional Caputo fractional model is formulated using generalized Fick's and Fourier's laws. Initially, the governing equations are transformed into a non-dimensional form and subsequently solved using the Laplace transform. Analysis of the figures indicates that the Casson parameter reduces fluid motion, while the diffusion thermo effect enhances it. Furthermore, the study includes a comparison between fractionalized and ordinary velocity fields. Highlight: Introduction of novel CPC fractional derivative model of Casson fluid flow. Semi-analytically solutions using Laplace transform method for accurate fluid behavior representation. comprehensive parametric analysis linking theoretical findings. Temperature contour affected by significant values of Du?

Global solutions for time-space fractional fully parabolic Keller–Segel system

We show the existence of a global solution to time-space fractional fully parabolic Keller–Segel system: \left\{ \begin{align*}{}_{0}^{c}D_{t}^{\beta} u+(-\Delta)^{\alpha/2}u+\nabla\cdot(u \nabla v)&=0,&&x\in\mathbb{R}^{n},\ t>0,\\{}_{0}^{c}D^{\beta}_{t} v+(-\Delta)^{\alpha/2}v-u&=0,&&x\in\mathbb{R}^{n},\ t>0,\\ u(x,0)=u_{0}(x),\quad v(x,0)&=v_{0}(x),&&x\in\mathbb{R}^{n},\end{align*}\right. under the smallness condition on the initial data, where 0<\beta<1 , 1<\alpha\leq 2 and n\geq 2 , u and v denote the cell density and the concentration of the chemoattractant, respectively, and {}_{0}^{c}D^{\beta}_{t} denotes the Caputo fractional derivative of order \beta with respect to time t . The nonlocal operator (-\Delta)^{\alpha/2} , defined with respect to the space variable x , is known as the Laplacian of order \frac{\alpha}{2} . We establish the existence of weak solution to the above system by fixed-point arguments under suitable conditions on u_{0} and v_{0} .

Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations

A semilinear initial–boundary value problem with a Caputo time derivative of fractional order α∈(0,1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For an L2-type discretization of order 3−α, we give sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading.

Hyperspectral Estimation of Chlorophyll Content in Wheat under CO2 Stress Based on Fractional Order Differentiation and Continuous Wavelet Transforms

The leaf chlorophyll content (LCC) of winter wheat, an important food crop widely grown worldwide, is a key indicator for assessing its growth and health status in response to CO2 stress. However, the remote sensing quantitative estimation of winter wheat LCC under CO2 stress conditions also faces challenges such as an unclear spectral sensitivity range, baseline drift, overlapping spectral peaks, and complex spectral response due to CO2 stress changes. To address these challenges, this study introduced the fractional order derivative (FOD) and continuous wavelet transform (CWT) techniques into the estimation of winter wheat LCC. Combined with the raw hyperspectral data, we deeply analyzed the spectral response characteristics of winter wheat LCC under CO2 stress. We proposed a stacking model including multiple linear regression (MLR), decision tree regression (DTR), random forest (RF), and adaptive boosting (AdaBoost) to filter the optimal combination from a large number of feature variables. We use a dual-band combination and vegetation index strategy to achieve the accurate estimation of LCC in winter wheat under CO2 stress. The results showed that (1) the FOD and CWT methods significantly improved the correlation between the raw spectral reflectance and LCC of winter wheat under CO2 stress. (2) The 1.2-order derivative dual-band index (RVI (R720, R522)) constructed by combining the sensitive spectral bands of the CO2 response of winter wheat leaves achieved a high-precision estimation of the LCC under CO2 stress conditions (R2 = 0.901). Meanwhile, the red-edged vegetation stress index (RVSI) constructed based on the CWT technique at specific scales also demonstrated good performance in LCC estimation (R2 = 0.880), verifying the effectiveness of the multi-scale analysis in revealing the mechanism of the CO2 impact on winter wheat. (3) By stacking the sensitive spectral features extracted by combining the FOD and CWT methods, we further improved the LCC estimation accuracy (R2 = 0.906). This study not only provides a scientific basis and technical support for the accurate estimation of LCC in winter wheat under CO2 stress but also provides new ideas and methods for coping with climate change, optimizing crop-growing conditions, and improving crop yield and quality in agricultural management. The proposed method is also of great reference value for estimating physiological parameters of other crops under similar environmental stresses.

Discussion on the existence and controllability of Hilfer fractional stochastic differential equation with non-dense domain

This article studies the approximate controllability for a class of fractional control system with semigroup operators governed by differential equations with Hilfer fractional derivatives of order α ∈ ( 0 , 1 ) and type β ∈ [ 0 , 1 ] in Hilbert spaces. First, we establish a set of sufficient conditions for the approximate controllability for Hilfer fractional stochastic differential equation with non-dense domain in Hilbert spaces. The existence results are obtained by using the fractional calculus, semi-group theory, Wiener process and fixed point technique to prove our main results. Further, we extend the result to study the approximate controllability concept. An example is provided to illustrate the application of the obtained theory.

Dynamics of a two-neuron hopfield neural network: Memristive synapse and autapses and impact of fractional order

There are numerous studies on Hopfield neural networks with electromagnetic induction using memristors in either autaptic or synaptic connections. In this study, we explore a novel scenario where all connections are influenced by electromagnetic induction. We investigate and compare the network’s dynamics with one memristive autapse, two memristive autapses, and a memristive synapse. The results indicate that having two memristive autapses instead of one increases the dynamical range, leading to chaotic dynamics in unequal autaptic strengths. In contrast, in the presence of the memristive synapse, chaos can emerge only in very strong synaptic strength. Using fractional-order derivatives can transform the periodic attractor of the integer-order model into a chaotic one in some parameters. Furthermore, incorporating more memristors leads to chaos at lower fractional orders.

A spatio-temporal infection epidemic model with fractional order, general incidence, and vaccination analysis

The paper investigates a spatio-temporal SEIR epidemic model on a global level with fractional order. It employs four partial differential equations incorporating fractional derivatives and account for diffusion to characterize infection dynamics. By applying fixed-point theorem results, the paper establishes the existence, uniqueness, and boundedness of the solution. Equilibrium points are determined based on vaccination values and the basic reproduction number R0. Global stability of each equilibrium is confirmed using the Lyapunov direct method. Through simulations with a predictor–corrector algorithm, key insights into epidemiological dynamics are provided, elucidating the impact of vaccination on reducing disease transmission and altering R0. Trajectories of various compartments closely align with theoretical equilibrium points, affirming the model’s predictive precision. Furthermore, simulations indicate the potential for attaining disease-free equilibria with heightened vaccination rates, underscoring the pivotal role of vaccination strategies in epidemic control and disease eradication. It was shown that the fractional derivative order has no effect on the equilibrium stability but rather only on the convergence speed towards the equilibrium points.

Analytical approximate solutions of some fractional nonlinear evolution equations through AFVI method

In this article, we construct and investigate a new fractional variational iteration technique which is named as AFVI method. After that, we formulate some IVPs corresponding to the fractional nonlinear evolution equations NLTFFWE, mNLTFFWE, TFmCHE and TFmDPE. The Caputo fractional order derivative has been used to fractionalize the considered NLEEs. Then, we solve the considered IVPs by applying the formulated AFVI method. Lastly, we check the accuracy of the attained AASs by making some graphical and numerical comparisons with their corresponding exact solutions and other existing equivalent AASs. The result of this article confirm that the efficiency, appropriateness and time spent capability of the newly constructed AFVI method are better than that of the other existing analogous fractional analytical approximation methods. Here, we apply the Maple 2021 programming software to obtain the AASs of the formulated IVPs and drawing the 3D graphs of the attained AASs. Finally, in this article we validate the applicability of the Caputo fractional order derivative to form a novel analytical approximation method as well as to fractionalize some essential NLEEs of Mathematical physics.

Fractional-order memristive dynamics in colloidal graphitic carbon nitride systems.

We report on the synthesis and characterization of a colloidal graphitic carbon nitride (g-C_{3}N_{4}) system exhibiting complex memfractance behavior. The g-C_{3}N_{4} colloid was prepared through thermal polymerization of urea, followed by dispersion in deionized water. X-ray diffraction and scanning electron microscopy confirmed the successful synthesis of g-C_{3}N_{4}. Electrical characterization revealed nonpinched hysteresis loops in current-voltage curves, indicative of memristive behavior with additional capacitive components. The device demonstrated stable resistive switching between high (∼50kΩ) and low (∼22kΩ) impedance states over 500 cycles, as well as synaptic plasticity-like conductance modulation. To capture these complex dynamics, we employed a generalized memfractance model that interpolates between memristive, memcapacitive, and second-order memristive elements. This model, employing fractional-order derivatives, accurately fitted the experimental data, revealing the device's memory effects. The emergence of memfractance in this colloidal system opens new avenues for neuromorphic computing and unconventional information processing architectures, leveraging the unique properties of liquid-state memory devices.

Smart waterborne disease control for a scalable population using biodynamic model in IoT network

We propose a biodynamic model for managing waterborne diseases over an Internet of Things (IoT) network, leveraging the scalability of LoRa IoT technology to accommodate a growing human population. The model, based on fractional order derivatives (FOD), enables smart prediction and control of pathogens that cause waterborne diseases using IoT infrastructure. The human-pathogen-based biodynamic FOD model utilises epidemic parameters (SVIRT: susceptibility, vaccination, infection, recovery, and treatment) transmitted over the IoT network to predict pathogenic contamination in water reservoirs and dumpsites in Iji-Nike, Enugu, the study community in Nigeria. These pathogens contribute to person-to-person, water-to-person, and dumpsite-to-person transmission of disease vectors. Five control measures are proposed: potable water supply, treatment, vaccination, adequate sanitation, and health education campaigns. A stable disease-free equilibrium point is found when the effective reproduction number of the pathogens, R0eff<1 and unstable if R0eff>1. While other studies showed a 98.2% reduction in infections when using IoT alone, this paper demonstrates that combining the SVIRT epidemic control parameters (such as potable water supply and health education campaign) with IoT achieves a 99.89% reduction in infected human populations and a 99.56% reduction in pathogen populations in water reservoirs. Furthermore, integrating treatment with sanitation results in a 99.97% reduction in infected populations. Finally, combining these five control strategies nearly eliminates infection and pathogen populations, demonstrating the effectiveness of multifaceted approaches in public health and environmental management. This study provides a blueprint for governments to plan sustainable smart cities for a growing population, ensuring potable water free from pathogenic contamination,in line with the United Nations Sustainable Development Goals #6 (Clean Water and Sanitation) and #11 (Sustainable Cities and Communities).

Mainardi smoothing homotopy method for solving nonlinear optimal control problems

This paper proposes a new type of smoothing homotopy method for solving general nonlinear optimal control problems via the indirect method, leveraging the Mainardi kernel as the smoothing kernel. The Mainardi kernel is firstly derived from the fundamental solution of the time-fractional diffusion-wave equation, representing a generalized form of the Gaussian kernel. By altering the fractional derivative order, the kernel can seamlessly switch between non-Gaussian and Gaussian forms. Then, parts of the two-point boundary value problems are convolved with the smoothing kernel, and the resulting surrogates are incorporated into the necessary conditions, replacing the terminal state and costate variables. The homotopy process is used for the smoothing parameter: increasing this parameter enhances the smoothing effect, simplifying the homotopy problems, while a parameter value of zero implies no smoothing, reverting to the original problems. Additionally, explicit expressions for the Mainardi kernel at specific derivative orders are derived, avoiding the inefficiencies associated with fractional derivatives. Simulation examples demonstrate that the proposed method offers significant advantages in flexibility and convergence compared to the traditional Gaussian smoothing homotopy method.

A virtual element scheme for the time-fractional parabolic PDEs over distorted polygonal meshes

We extend the virtual element method to the two-dimensional time-fractional parabolic PDE, characterized by a fractional derivative of order α∈(0,1) in time. To illustrate the working of this fractional virtual element scheme, a numerical investigation of the following time-fractional problem over distorted polygonal meshes is conducted. cDtαu(z,t)−Δu=f(z,t)inz∈Ω,t∈(0,T],where Ω is a spatial domain, α is fractional order, and t is time variable. Our methodology is based on the fundamental technical component, fractional version of the Grunwald–Letnikov approximation. We prove the method’s well-posedness, that is the approximate solution’s existence and uniqueness. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L2-norm and H1-norm over distorted mesh configuration is validated by numerical results, underlining the practical effectiveness of the proposed method.

Shifted Legendre Gauss‐Lobatto collocation scheme for solving nonlinear coupled space‐time fractional reaction‐advection diffusion equations

AbstractThis work aims to develop and apply an algorithm for solving space‐time fractional reaction‐advection diffusion (STFRAD) nonlinear coupled equations with specified initial and boundary conditions by employing a Shifted Legendre Gauss‐Lobatto collocation (SLGLC) scheme. The fractional derivatives of time and space are defined in Caputo sense. The approximate solutions have been constructed with the help of shifted Legendre polynomials. The proposed numerical scheme reduces the coupled fractional order differential equations into a system of algebraic equations which can then be easily solved. The error analysis through the two test problems is carried out to validate the efficiency and efficacy of the method. The effect of advection and reaction terms, and various space‐time fractional order derivatives on the solution profiles have been studied and are shown graphically for different particular cases. It has been observed that the species whose transport is simulated using STFRADE shows higher concentration at a specific distance in porous media in case of integer order differential equations as compared to fractional order equations demonstrating overestimation of its impact in case of integer order differential equation. This is extremely important observation with respect to applications in the field of groundwater pollution management. The salient feature of the article is the stability and convergence analyses of the proposed scheme on the concerned model, which demonstrates the effectiveness and capabilities of the developed method.

Damping efficiency of the fractional Duffing system and an assessment of its solution accuracy

Dynamical systems with high damping efficiency in a wide frequency band can be useful on a small scale – for harvesting energy from ambient vibrations, and on a large scale – for damping harmful vibrations of mechanical structures. In this paper we present an assessment of the quality of solutions and damping efficiency of systems with fractional order derivatives. To simulate the fractional system the fourth-order Runge–Kutta method and Grünwald–Letnikov methods are used. We propose a coefficient for assessing the quality of solutions to fractional systems by reference to the quality of the calculated energy balance. As an exemplary system we study the Duffing model with embedded additional fractional-order derivative terms. Based on this coefficient, intervals of key numerical simulation parameters are determined to ensure the appropriate quality for the calculations of energy flows and energy balance. The determined values of these parameters are then used in tests of the damping efficiency of the studied system. Our results show that by modifying the fractional terms it is possible to configure a system that exhibits a “broadband effect”, i.e. a system that is characterized by high-amplitude vibrations and, consequently, high energy efficiency in a wide range of excitation frequencies.