Fractional Order Research Articles

Hydrogen Oxidation and Evolution Reaction on Platinum in Alkaline Electrolyte: Fixed Reference vs. Overpotential and the Effect of H2 Concentration

Abstract Understanding the kinetics of the hydrogen oxidation and evolution reaction (HOR/HER) in alkaline media is of great interest, but the underlying mechanism and the dependency on the hydrogen pressure are still debated. Herein, we investigated the HOR/HER on polycrystalline platinum in alkaline electrolyte using a rotating disk electrode (RDE). Unlike typically done, we performed the data analysis on a fixed potential scale, which has implications for the assessment of RDE mass-transport corrections and the definition of reaction orders. Analyzing the kinetic currents on a fixed potential, we find hydrogen reaction orders of essentially one and zero for the HOR and HER, respectively, at larger overpotentials. In contrast, fitting the kinetic currents in a potential around equilibrium with a Butler-Volmer model yields differing kinetic parameters and fractional reaction orders. Our findings can be explained with a potential-dependent transition in the HOR mechanism from Tafel–Volmer at small overpotentials to Heyrovsky–Volmer at higher overpotentials, with a concomitant change in the H2 reaction order from fractional to unity, thus reconciling previous propositions on the alkaline HOR/HER mechanism. Our results demonstrate the strength of using a fixed potential scale, and we expect this approach to be beneficial for studies of other electrocatalytic reactions.

Direct Fractional-Order Adaptive Control Design for Cascaded Doubly-Fed Induction Generator (CDFIG) in a Wind Energy System

This paper is devoted to a fractional-order model reference adaptive control (FO-MRAC) synthesis for the independent control of the active and reactive power flows in the cascaded doubly fed induction generator (CDFIG) in wind energy systems. The proposed adaptive control law combines a second-order-like fractional reference model and a direct MIT adaptation law using a fractional order integrator. This generator configuration can be an interesting alternative to standard double-output wound rotor induction generators. It is made up of two identical wound rotor induction motors such that their rotors are mechanically and electrically coupled. Using two cascaded induction machines permits the elimination of the brushes and copper rings in the traditional doubly-fed induction generator DFIG, which makes the system more resistant and reduces maintenance costs. In the first step, we propose a classical PI controller synthesis to regulate the active and reactive power produced by CDFIG. Then, the FO-MRAC design is realized and a comparative study based on numerical simulations is performed between the classical regulators PI, MRAC, and FO- MRAC, to demonstrate the superiority of the proposed fractional-order adaptive controller relative to conventional integer order PI and MRAC controllers. These results illustrate the reliability and efficiency of the proposed adaptive control scheme.

Enhancing EMG signal classification using convolution neural network optimized with fractional order bat algorithm

Enhancing EMG signal classification using convolution neural network optimized with fractional order bat algorithm

Fractional-Order Mathematical Modeling of Methicillin- Resistant Staphylococcus aureus Transmission in Hospitals

This article investigates the environmental contamination and antibiotic exposure effect on the transmission dynamics of the Methicillin-Resistant Staphylococcus aureus (MRSA) model in hospitals using the fractional Adams–Bashforth–Moulton Method (ABMM). This epidemic model simulates the dynamics of patient populations, bacterial contamination, and healthcare worker safety under varying conditions. This model provides critical insights into the interactions between hospital practices, environmental factors, and infection dynamics, demonstrating the importance of symmetry in balancing hospital admission and discharge rates to manage infection spread effectively. The analysis extends to the impact of environmental bacterial density and hospital admission rates on patient colonization. Increasing admission rates introduce more susceptible patients, exacerbating infection spread when bacterial density is high. Conversely, lower admission rates and bacterial density result in a more controlled infection environment. The model further investigates how varying discharge rates influence colonization dynamics, highlighting that effective discharge practices can mitigate infection spread, especially in high-bacterial density scenarios. It must be noted that this model is studied fractionally for the first time. Overall, this model provides critical insights into the interactions between hospital practices, environmental factors, and infection dynamics, offering valuable guidance for infection control strategies and hospital policy formulation. By adjusting fractional order constant (σ) values and analyzing different scenarios, this research aids in understanding and managing bacterial infections in healthcare settings. The proposed method is able to provide the results presented in the figures within this study considering the influence of many factors.

Long time behavior for the two-dimensional magnetohydrodynamic flows in a general domain

By utilizing spectral analysis methods and properties of fractional order operators, some new estimates have been established for the nonlinear terms appearing from the two-dimensional non-stationary magnetohydrodynamic flows in a general domain. Based on these, a series of energy decay properties are given. Furthermore, the large time decay behavior of the solution and that of its gradient have also been addressed in the L∞ and L2 spaces, respectively.

New Solutions to the Fractional Perturbed Chen Lee Liu Model with Time-Dependent Coefficients: Applications to Complex Phenomena in Optical Fibersns

In this paper, we consider the fractional perturbed Chen Lee Liu model with time-dependent coefficients (FPCLLM-TDCs). We apply the mapping method in order to get hyperbolic, elliptic, trigonometric and rational fractional solutions. These solutions are vital for understanding some fundamentally complicated phenomena. The obtained solutions will be very helpful for applications such as fiber optics and plasma physics. Finally, we show how the conformable fractional derivative order affect the exact solutions of the FPCLLM-TDCs. Furthermore, we examine the effects of time-dependent coefficients when these coefficients take on special cases such as random variables, polynomials, and hyperbolic functions.

Gronwall-Type Inequalities and Qualitative Studies onHigher-Variable Orders of Atangana-Baleanu Fractional Operators via Increasing Functions

This paper introduces a novel extension of Caputo-Atangana-Baleanu and Riemann-Atangana-Baleanu fractional derivatives from constant to increasing variable order. We generalize the fractional order from a fixed value in (0, 1] to a time-dependent function in (k, k + 1], where k ≥ 0. The corresponding Atangana-Baleanu fractional integral is also extended. Key properties ofthese new definitions are explored, including a generalized Gronwall inequality. We then delve into the analysis of higher-variable initial fractional differential equations using the Caputo-Atangana-Baleanu operator with an increasing function, establishing existence and uniqueness results via Picard’s iterative method. The findings presented in this work are expected to stimulate further research on inequalities and fractional differential equations related to Atangana-Baleanu fractional calculus with respect to increasing functions. Concrete examples are provided to illustrate the practical applications of our results.

Comparative Real-Time Study of Three Enhanced Control Strategies Applied to Dynamic Process Systems

In this study, a comparative analysis of three different control methods for precise, real-time control of a complex dynamic double-tank liquid level process system was performed. Since the system in question has a time-delayed structure, feedforward proportional integral (FF-PI) control and cascaded nonlinear feedforward proportional integral delayed (CNPIR) controllers were tested on the process system. While the FF-PI controller improved the response time of the system, it showed limitations in handling external disturbances and nonlinearities. On the other hand, the CNPIR controller showed better improvements in control accuracy and lower overshoot compared to the FF-PI controller. Since the process system has a nonlinear model and is affected by external disturbances, these two controllers were inadequate in this study when compared to the fractional order adaptive proportional integral derivative sliding mode controller (FO-APIDSMC). The FO-APIDSMC controller provided fairly good performance in both tracking accuracy and disturbance rejection control for non-chattering, fast finite-time convergence, increased robustness, and uncertain dynamic processes. Experimental results reveal that the FO-APIDSMC controller achieves superior minimized tracking error and outperforms the FF-PI and CNPIR controllers by effectively handling uncertainties and external disturbances.

A fractional-order model of COVID-19 and Malaria co-infection

This paper explores the co-infection dynamics of coronavirus disease 2019 (COVID-19) and Malaria using Caputo-type fractional derivative to further understand the disease interactions and implement effective control strategies. We demonstrate the positivity and boundedness of the solution through Laplace transform techniques and establish the existence and uniqueness of the solution, showcasing model stability using fractional-order stability theory. Simulation experiments across varying fractional orders and disease classes offer insights into the co-infection dynamics. This is a new model and the findings underscore the potential impact of control measures on mitigating co-infection under endemic conditions. We conclude that infection with malaria does not guarantee immunity to COVID-19 and infection with COVID-19 as well does not guarantee immunity to malaria.

Multi-step Residual Power Series Method for Solving Stiff Systems

In this paper, an efficient algorithm based on the residual power series method (RPSM) is presented to solve stiff systems of Caputo fractional order. We apply the RPSM on subintervals to get approximate solutions of these types of systems. The RPSM has advantages that it is suitable to solve linear and nonlinear systems and it gives high accurate results. Modifying this technique to multi-step RPSM considerably reduces the number of arithmetic operations and so reduces the time, especially when dealing with Stiff systems. Several numerical examples are given to show the efficiency, simplicity and the accuracy of the proposed method. Comparing classical RPSM with the new multi-step scheme shows that multi-step RPSM controls the convergence behaviour of the stiff systems. That is, the comparison reveals that MS-FRPSM reduces both absolute and residual errors. More iterations and a smaller step size lead to higher accuracy. Moreover, in MS-FRPSM, the intervals of convergence for the series solution will increase.

Fractional-Order Mathematical Modeling of Breast Cancer: Comparing Adaptive Immune Responses and Estrogen Dynamics with Experimental Data

Breast cancer is a significant global health concern that requires innovative approaches to understand its behavior and improve treatment strategies. This paper introduces a new fractional-order mathematical model to explain the complex dynamics of breast cancer progression, including adaptive immune responses and estrogen dynamics. Utilizing Caputo fractional derivatives, our model reveals insights into the impact of fractional-order dynamics on cancer cell populations. Simulation results demonstrate a notable increase in cell populations with higher fractional orders, suggesting heightened aggressiveness, while lower orders correspond to subdued progression. Unlike traditional integer-order models, fractional-order derivatives offer a more nuanced depiction of nonlinear dynamics, crucial for capturing the complexities of cancer progression. Importantly, our findings underscore the potential clinical relevance of fractional-order models in informing personalized treatment strategies, particularly through the modulation of estrogen levels. By integrating treatment considerations, such as hormone therapy, our model holds promise for advancing precision medicine approaches tailored to individual patient characteristics.

Employing the Sadik Residual Power Series Method to Analyze a System of Nonlinear Caputo Time-Fractional Partial Differential Equations

The present study proposes an approximate analytical solution to a nonlinear system of time-fractional partial differential equations. The Sadik residual power series method, which integrates the two-part Sadik integral transform with the residual power series technique, is employed to solve the fractional differential equation in the Caputo sense. Nonlinear problems with known and unknown solutions are examined to demonstrate the capacity of the technique. Numerical simulations and 3D visualizations are conducted for various values of the fractional order to further understand the solution’s behavior. Additionally, the results are validated against exact solutions or existing methodologies to ensure their reliability and accuracy. A key advantage of the proposed method is its ability to generate results without the need for Adomian polynomials, perturbation techniques, discretization, or linearization, enabling a more efficient.

Asymmetry and Symmetry in New Three-Dimensional Chaotic Map with Commensurate and Incommensurate Fractional Orders

Asymmetry and Symmetry in New Three-Dimensional Chaotic Map with Commensurate and Incommensurate Fractional Orders

Stability and convergence computational analysis of a new semi analytical-numerical method for fractional order linear inhomogeneous integro-partial-differential equations

Abstract The motive behind this research work, is to originate a semi-analytical-numerical approach for the solution of fractional order linear integro partial differential equations (FOLIPDEs), especially in-homogeneous FOLIPDEs of different types, like fractional version of Fredholm and Volterra type IEs etc. To attain the said purpose, we will study some fractional formulations of linear model integral equations from the existing literature. Then we will describe the proposed semi analytical-numerical procedure alongwith its stability analysis and convergence properties. We will then prove with the aid of some particular numerical examples that the said approach is not only lucid and efficient but also precise. The outcomes of the proposed technique will show that this scheme has notable prospectives for solving a large class of FOLIEs. At the end, the contribution of this work in the advancements of semi analytical-numerical approaches for
the solution of fractional order linear integro partial differential equations will be laid down and discussion for future research in this field.

Advanced modelling techniques for magnetohydrodynamic Casson fluid squeezing flow via generalized fractional operators with neural network scheme

Abstract This paper aims to simulate and examine the unstable squeezed circulation of fractional-order (FO) magnetohydrodynamic (MHD) Casson fluid via a permeable medium. The Casson fluid system performs an essential role in comprehending the characteristics of non-Newtonian fluids, including toothpaste, condiments, printing substances and plasma circulation. The outcomes of this investigation are significant because previous research has not addressed the unsteady circulation of Casson fluid in a fractional nonsingular kernel and neural network-based stochastic context, considering the indicated consequences. An exceptionally dynamic ordinary differential equation is produced by using fractional calculus in combination with similarity transforms. After that, the predicted problem is addressed employing an amalgam of the Laplace transform in the Caputo-Fabrizio, modified Atangana-Baleanu-Caputo fractional derivatives operators, and the $q$-homotopy analysis transform method, accompanied by no-slip boundary requirements. The responses and oversights at various points in the FOs are scrutinized, along with previous findings, in order to ensure reliability. In terms of precision, $q$-HATM findings outperform other outcomes that are accessible in research. The focus of this research is on the influence of FOs on the velocity distribution, skin friction coefficient (SFC) and practices of relevant fluid factors. To find out how relevant fluid components affect the velocity distribution and SFC, an extensive, qualitative and visual evaluation is carried out. It was discovered through evaluation that the FO shows an analogous impact for both positive and negative squeezing numbers. Additionally, as the FO increases, SFC reduces. Analysis revealed that the FO exhibits a similar effect with regard to positive and negative compression numbers. Furthermore, SFC decreases with increasing FOs. Additionally, a highly effective stochastic method employing artificial neural networks (ANNs) and a back-propagated Levenberg-Marquardt (BPLM) procedure is generated to explore the effect of different parameter modifications on the SFC, velocity distribution, as well as various fluid factors. Multiple effectiveness measures were developed according to mean absolute deviations (MAD), erroneous Nash-Sutcliffe effectiveness (ENSE), and Theil's inequity coefficient (TIC) in order to verify the preciseness, productivity, and computing cost of the ANN-BPLM algorithms. The outlined scheme's analytical findings are verified through comparison using numerical outcomes obtained through the $q$-HATM, artificial intelligence strategies like NARX-LM, and the least squares methodology (LSM). The outcomes indicate the resilience and accuracy of the layout procedure by demonstrating that the average percentage of errors in our proposed outcomes in terms of ENSE, TIC, and MAD is nearly zero.

Numerical investigation of two-dimensional fractional Helmholtz equation using Aboodh transform scheme

Purpose This paper aims to present a numerical investigation for two-dimensional fractional Helmholtz equation using the Aboodh integral homotopy perturbation transform scheme (AIHPTS). Design/methodology/approach The proposed scheme combines the Aboodh integral transform and the homotopy perturbation scheme (HPS). This strategy is based on an updated form of Taylor’s series that yields a convergent series solution. This study analyzes the fractional derivatives in the context of Caputo. Findings This study illustrates two numerical examples and calculates their approximate results using AIHPTS. The derived findings are also presented in tabular form and graphical representations. Research limitations/implications In addition, He’s polynomials are calculated using HPS, so the minimal computational outcome is a defining feature of this method and gives a competitive advantage over other series solution techniques. Originality/value Numerical data and graphical illustrations for different fractional order levels confirm the proposed method’s successful performance. The results show that the proposed approach is speedy and straightforward to execute on fractional-ordered models.

Fractional‐order fast terminal sliding mode trajectory tracking control of quadrotor UAV based on finite time disturbance observer

AbstractIn this paper, a novel trajectory tracking control problem of the quadrotor UAV is addressed, which considers the system uncertainties and unknown external disturbances meanwhile. To ensure a rapid convergence rate of the states, fast terminal sliding mode control (FTSMC) algorithm is improved combining with a fractional order method. In addition, finite‐time observer is designed to handle disturbances and enhance the robustness of the system. The closed‐loop finite‐time stability of the quadrotor UAV system is established using Lyapunov theory in accordance with the novel control law. To evaluate the effectiveness of our proposed strategy, various simulations under different scenarios are conducted respectively. These simulation results clearly demonstrate the superiority of our control scheme, which we refer to as fractional‐order fast terminal sliding mode control (FOFTSMC).

FRACTIONAL ANALYSIS OF ONE-DIMENSIONAL SCHRÖDINGER MODEL IN THE CONTEXT OF CAPUTO-TYPE FRACTIONAL DERIVATIVES BASED ON RESIDUAL POWER SERIES METHOD

This research presents a novel analytical approach to explore the fractional analysis of the one-dimensional time-fractional Schrödinger model (TFSM) using Caputo fractional derivatives. By integrating the Mohand transform (MT) with the residual power series method (RPSM), we develop the Mohand residual power series method (MT-RPSM) that provides results in the form of convergent series without assumptions on variables. Initially, we employ MT to reduce the fractional order, and then we transfer the fractional problem into the Mohand space formulation. Second, we use the RPSM concept to derive the iterative series formula for the Mohand space formulation. We analyze these findings using visual layouts to show the physical representation of the TFSM, which matches the precise results very well. The results indicate that the proposed technique is a reliable and practical method for identifying and analyzing various nonlinear models of physical phenomena.

Fractional-Order Modeling and Stochastic Dynamics Analysis of a Nonlinear Rubbing Overhung Rotor System

To study the nonlinear dynamic behavior and system stability of a rubbing overhung rotor with viscoelastic and memory-effect damping and random uncertain parameters, this paper introduces a fractional-order modeling and stochastic dynamic analysis method for the nonlinear overhung rotor system with frictional impact faults. Firstly, the dynamic equations of the overhung rotor considering friction effect and fractional damping effect are established based on the transfer matrix method and fractional order derivative. Then, the time-domain response of the fractional-order dynamic equations is solved by combining the Runge–Kutta method and the continuous fractional expansion, and the steady-state response characteristics of different fractional damping are analyzed in the deterministic case. Finally, to analyze the response of the system under the effect of stochastic parameters, the sparse grid-based PCE metamodel of the system response is developed. Statistical moments, probability distributions, and sensitivity indices of the response of stochastic systems are revealed. The results of this paper provide a theoretical basis for efficient and accurate prediction of the stochastic response of nonlinear rubbing overhung rotor systems.

Chlamydia infection with vaccination asymptotic for qualitative and chaotic analysis using the generalized fractal fractional operator

In this work, we solve a system of fractional differential equations utilizing a Mittag-Leffler type kernel through a fractal fractional operator with two fractal and fractional orders. A six-chamber model with a single source of chlamydia is studied using the concept of fractal fractional derivatives with nonsingular and nonlocal fading memory. The fractal fractional model of the Chlamydia system can be solved by using the characteristics of a non-decreasing and compact mapping. A suggested model with the Lipschitz criteria and linear growth is studied both qualitatively and quantitatively, taking into account boundedness, uniqueness, and positive solutions at equilibrium points with Leray-Schauder results under time scale concepts. We examined the framework of local and global stability and insight into Lyapunov function properties for the infectious disease model. Chaos Control will employ the regulate for linear responses approach to stabilize the system following its equilibrium points. This will take into consideration a fractional order framework with a managed design, where solutions are bounded in the feasible domain and have a greater impact at the lower minimum infectious rate. To illustrate the implications of fractional and fractal dimensions with varying interest rate values through simulations with Newton’s polynomial method under the Mittag-Lefller kernel. Additionally, a comparative analysis of results is also derived by employing power and exponential decay kernels at various fractional orders.