Fractional Order Parameter Research Articles

Operational Temperature Optimization in Hydrogen Turbine Blades via Time-Fractional Conformable Sensitivity Analysis

This study focuses on optimizing the thermal performance of hydrogen turbine blades through a sensitivity analysis using generalized fractional calculus. The approach is designed to capture the transient temperature dynamics and optimize thermal profiles by analyzing the influence of a fractional-order parameter on the system’s behavior. The model was implemented in Python, using Monte Carlo simulations to evaluate the impact of the parameter on the temperature evolution in different thermal regimes. Three distinct regions were identified: the Quasi-Uniform Region (where fractional effects are negligible), the Sub-Classical Region (characterized by delayed thermal behavior), and the Super-Classical Region (exhibiting enhanced heat accumulation). Regression analyses reveal quadratic and cubic dependencies of blade temperature on the fractional-order parameter, confirming the robustness of the model with R2 values greater than 0.96. The study highlights the potential of using fractional calculus to optimize the thermal response of turbine blades, helping to identify the most suitable parameters for faster stabilization and efficient heat management in hydrogen turbines. Furthermore, it was found that by adjusting the fractional-order parameter, the system can be optimized to reach equilibrium more rapidly while achieving higher temperatures. Importantly, the equilibrium is not altered but rather accelerated based on the chosen parameter, ensuring a more efficient thermal stabilization process.

Exploring new exact solutions in the conformable time-fractional discrete coupled NLSE using a novel approach

This investigation focuses on the conformable time-fractional discrete coupled nonlinear Schrödinger system (CTFCDNLSEs). This system incorporates a fractional order represented as a conformable derivative. Through the application of the fractional transformation method (FTM), a set of novel analytical discrete solutions is derived. These solutions are characterized by an array of mathematical functions, including trigonometric, hyperbolic, and rational functions. Among these solutions, discrete fractional bright solitons, dark solitons, combined solitons, and periodic solutions stand out. To demonstrate the influence of the fractional-order parameter on the dynamics of fractional discrete solitons, graphical representations are provided. These findings are significant for exploring complex nonlinear discrete physical phenomena.

Soliton solutions for time-fractional (3+1)-dimensional nonlinear Schrödinger equation with cubic-quintic nonlinearity terms

This paper investigates the time-fractional extended (3+1)-dimensional nonlinear conformable Schrödinger equation in a dispersion and nonlinearity managed fiber laser. By utilizing the new direct mapping method and the unified Riccati equation technique, we derive various optical soliton solutions for the nonlinear Schrödinger equation with conformable derivative. The importance of these newly constructed soliton solutions is demonstrated through contour, three-dimensional, and two-dimensional graphs, demonstrating dark, bell-shaped, bright, and wave soliton formation. Further, the effect of the fractional order parameter and the temporal parameter on these solutions is analyzed, offering valuable insights into the conformable nonlinear Schrödinger model. The algorithms developed in this study hold promise for broader application across various nonlinear Schrödinger equations in fields such as nonlinear optics and applied mathematics. Ultimately, this study deepens our understanding of nonlinear optics by uncovering new soliton dynamics and their governing mechanisms within dispersion and nonlinearity-managed fiber laser systems.

Enhanced Control of BLDC Motor Using FOPID Comparative Analysis and Optimization

ABSTRACT This paper explores the design and implementation of a Fractional Order Proportional-Integral Derivative (FOPID) controller for Brushless DC (BLDC) motors, specifically aimed at improving electric vehicle (EV) performance. BLDC motors are known for their efficiency, compact size, and precise control. However, their nonlinear dynamics and sensitivity to load variations present significant challenges for traditional Proportional Integral-Derivative (PID) controllers, which often struggle under dynamic conditions. The proposed FOPID controller incorporates fractional-order parameters, offering greater flexibility and control precision. A detailed mathematical model of the BLDC motor is developed, including power balance equations and current analysis, to establish a solid foundation for control design. Using MATLAB/Simulink, the system’s performance is simulated and evaluated under conditions like step inputs, load disturbances, and gradual speed changes. The FOPID controller consistently demonstrates better results than conventional PID methods, including reduced overshoot, faster settling times, and improved torque and speed stability. This work addresses key challenges such as parameter tuning and system stability, providing a scalable and efficient solution for BLDC motor control in real-world EV applications. These findings support the development of more energy-efficient and reliable motor control strategies for the next generation of electric vehicles. Keywords: Fractional Order PID (FOPID), BLDC motor, Electric vehicles (EV), Nonlinear dynamics, PID controllers.

Fractional order modeling of hepatitis B virus transmission with imperfect vaccine efficacy

This study aims to develop and analyze a model of hepatitis B virus transmission dynamics using integer and fractional derivatives in the Caputo sense. After formulating the models, we conduct an asymptotic stability analysis of the disease-free equilibrium point of both models. The Lyapunov technique demonstrates that under specific conditions, the disease-free equilibrium point in both models remains globally asymptotically stable. The study demonstrates that both models can have at least one endemic equilibrium when 1$$\\end{document}]]>, using the vaccination coverage parameter to identify positive equilibrium points. The Banach contraction principle is used to establish the uniqueness and existence of each fractional model’s solutions, followed by demonstrating their global stability using the Ulam-Hyers technique. The model is calibrated using reported hepatitis B cases in Nigeria, allowing for parameter estimations. The study indicates that the disease is endemic in this country, as , indicating a higher level of endemicity. The Adams-Bashforth approach is used to develop a numerical scheme, which is then validated through numerical simulations and evaluated under fractional order parameter variations.

Optical solutions to the extended (3+1)-dimensional cubic-quintic nonlinear conformable Schrödinger equation via two effective algorithms

This study explores the extended (3+1)-dimensional cubic and quartic conformable nonlinear Schrödinger equation, which is characterized by two distinct forms of nonlocal nonlinearities and has applications in optical fibre communications. Using the Kudryashov auxiliary equation approach and the simplest equation technique, a variety of innovative optical soliton solutions have been developed. The relevance and distinct features of these solutions are highlighted through visual representations such as contour diagrams, three-dimensional models, and two-dimensional plots. The effects of the fractional order and temporal parameter are investigated in detail, providing a deeper understanding of the conformable nonlinear Schrödinger equation's dynamics. Such models have significant applications in optical fibre communications, where solitons serve as stable carriers for high-speed data transmission over long distances with minimal distortion. The algorithms developed herein are applicable to other classes of nonlinear Schrödinger equations, making them valuable tools in nonlinear optics and applied mathematics.

THE EFFECT OF FRACTIONAL ORDER DERIVATIVE ON HUMAN HEAD SKIN TISSUE IN RESPONSE TO THERMAL DIFFUSION

In this manuscript, the Caputo-Fabrizio fractional order model is used to study the effects of thermal diffusion in human head skin tissue. This leads to analyzing variations in the concentration distribution of the diffusive material in skin tissue and the temperature distribution function with relaxation time due to the chemical potential function, which depends on time and thermal diffusion on the bounding plane. The Laplace transform technique is employed to compute the results, which show a significant effect of the Caputo-Fabrizio fractional order parameter, distance, and time. The procured results have been illustrated graphically using MATLAB, where the influence of a non-singular memory kernel has been observed, which implies a more realistic description of the thermal process in skin tissue. The numerical results for concentration and temperature distribution functions are obtained using the method of numerical inversion based on the Tzou method. The validity of this intended model is assessed by comparing it with previously published results. Significantly, the infinite memory effects have been avoided by introducing a smooth, non-singular kernel. The accumulation of heat by the skin tissue over time is realistically modeled. Observing the more realistic treatment of memory effects, according to the authors, the Caputo-Fabrizio approach provides a more refined and realistic model, which is explicitly suited for precision heat application in thermal therapies like cryosurgery and hyperthermia treatments for tumors, also enhancing the accuracy of simulations related to thermal damage, skin burn, and pain response in heterogeneous tissue.

The Improved Conformable Fractional Grey Model Using ADMM and HOA Algorithm

The grey model (GM) is a forecasting model known for its ability to provide accurate predictions for small sample data, but it struggles with complex nonlinear time series. This paper introduces a fractional calculus prediction model, utilizing Conformable Fractional Difference (CFD) and Conformable Fractional Accumulation (CFA) to calculate the new time series. The traditional least squares method for parameter estimation is transformed into a Least Absolute Shrinkage and Selection Operator (LASSO) problem and is solved using the dual form of the Alternating Direction Method of Multipliers (ADMM). Additionally, the Hiking Optimization Algorithm (HOA) is employed to optimize the fractional order parameter. As a result, the improved model demonstrates greater accuracy than the original model when dealing with small sample data.

QUALITATIVE AND QUANTITATIVE ANALYSES OF THE DYNAMICS OF A VECTOR-BORNE DISEASE USING A FRACTIONAL FRAMEWORK

Vector-borne diseases pose substantial risks to public health and have a high impact on different sectors of life around the world. Hence, it is valuable to construct a conceptual framework for comprehending the dynamics of these infections to introduce efficacious public health control strategies. In this paper, we structure a model for a vector-borne infection to conceptualize the complex dynamics of dengue, including drug resistance and insecticide. The recommended model of dengue fever is presented through a non-integer derivative to capture the role of memory, drug resistance and insecticide. We examine the positivity and boundedness of the solution using analytic skills and the threshold parameter is determined via the next-generation matrix approach. The well-known theorem of fixed-point is utilized to evaluate the solution’s existence and uniqueness. We have provided sufficient conditions for the Ulam–Hyers stability. To delve further into the intricacies, we employ a numerical technique to elucidate how various input factors influence the infection dynamics. The impact of fractional-order parameters, vaccination rates, the rate of immunity waning, biting rates, resistance to drugs and insecticides is visualized. Our analysis has underscored the pivotal significance of certain parameters of the system, which can exert substantial influence over the intensity of dengue fever. In contrast, our findings suggest that parameters such as vaccination rates, the memory index, and fractional order parameters, along with treatment and insecticide policies, hold promise as effective control measures in mitigating the spread and impact of dengue infection.

Exploring Climate-Induced Oxygen–Plankton Dynamics Through Proportional–Caputo Fractional Modeling

In this work, we develop and analyze a novel fractional-order framework to investigate the interactions among oxygen, phytoplankton, and zooplankton under changing climatic conditions. Unlike standard integer-order formulations, our model incorporates a Proportional–Caputo (PC) fractional derivative, allowing the system dynamics to capture non-local influences and memory effects over time. Initially, we rigorously verify that a unique solution exists by suitable fixed-point theorems, demonstrating that the proposed fractional system is both well-defined and robust. We then derive stability criteria to ensure Ulam–Hyers stability (UHS), confirming that small perturbations in initial states lead to bounded variations in long-term behavior. Additionally, we explore extended UHS to assess sensitivity against time-varying parameters. Numerical simulations illustrate the role of fractional-order parameters in shaping oxygen availability and plankton populations, highlighting critical shifts in system trajectories as the order of differentiation approaches unity.

Soliton solutions to the time-fractional Kudryashov equation: Applications of the new direct mapping method

In this paper, we analyze the dynamic characteristics of the well-known Kudryashov equation with a conformable derivative in the context of pulse propagation within optical fibers. To investigate the equation, we utilize the modified direct method, a robust technique capable of deriving various soliton solutions. The study includes two-dimensional, three-dimensional, and contour plots that illustrate bell-shaped, wave, W-shaped, and mixed dark-bright soliton solutions, highlighting the significance of these novel optical soliton solutions. These forms of the wave soliton and the W-shaped soliton are constructed for the first time for the Kudryashov equation with a conformable derivative. The structure of the new soliton solutions can be used to transmit two signals simultaneously, potentially increasing transmission efficiency. A key contribution of the study is its analysis of the impact of the fractional order parameter and the time parameter on these optical solutions, illustrating how fractional calculus affects soliton behavior. This highlights the significance of the conformable nonlinear Schrödinger model and its relevance to understanding dispersion effects in optical fibers, advancing prior research in the field.

Analysis Modulation Instability and Parametric Effect on Soliton Solutions for M-Fractional Landau–Ginzburg–Higgs (LGH) Equation Through Two Analytic Methods

This manuscript studies the M-fractional Landau–Ginzburg–Higgs (M-fLGH) equation in comprehending superconductivity and drift cyclotron waves in radially inhomogeneous plasmas, especially for coherent ion cyclotron wave propagation, aiming to explore the soliton solutions, the parameter’s effect, and modulation instability. Here, we propose a novel approach, namely a newly improved Kudryashov’s method that integrates the combination of the unified method with the generalized Kudryashov’s method. By employing the modified F-expansion and the newly improved Kudryashov’s method, we investigate the soliton wave solutions for the M-fLGH model. The solutions are in trigonometric, rational, exponential, and hyperbolic forms. We present the effect of system parameters and fractional parameters. For special values of free parameters, we derive some novel phenomena such as kink wave, anti-kink wave, periodic lump wave with soliton, interaction of kink and periodic lump wave, interaction of anti-kink and periodic wave, periodic wave, solitonic wave, multi-lump wave in periodic form, and so on. The modulation instability criterion assesses the conditions that dictate the stability or instability of soliton solutions, highlighting the interplay between fractional order and system parameters. This study advances the theoretical understanding of fractional LGH models and provides valuable insights into practical applications in plasma physics, optical communication, and fluid dynamics.

Staged Parameter Identification Method for Non-Homogeneous Fractional-Order Hammerstein MISO Systems Using Multi-Innovation LM: Application to Heat Flow Density Modeling

For the non-homogeneous fractional-order Hammerstein multiple input single output (MISO) system, a method for identifying system coefficients and fractional-order parameters in stages is proposed. The coefficients of the system include the coefficients of nonlinear terms and the coefficients of the transfer function. In order to estimate them, we derived the coupling auxiliary form between the original system coefficients, developed a multi-innovation principle combined with the LM (Levenberg–Marquardt) parameter identification method, and introduced a decoupling strategy for the coupling coefficients. The entire identification process of fractional orders is split into three stages. The division of stages is based on assuming that the system is of different fractional order types, including global homogeneous fractional-order systems, local homogeneous fractional-order systems, and non-homogeneous fractional-order systems. Except for the first stage, the estimated initial value of the fractional order in each stage is derived from the estimated value of the fractional order in the previous stage. The fractional order iteration will re-drive the iteration of the system coefficients to achieve the purpose of alternate estimation. To validate the proposed algorithm, we modeled the fractional-order system of heat flow density through a two-layer wall system, demonstrating the algorithm’s effectiveness and practical applicability.

Stat-Space Approach to Three-Dimensional Thermoelastic Half-Space Based on Fractional Order Heat Conduction and Variable Thermal Conductivity Under Moor–Gibson–Thompson Theorem

This study presents a mathematical model of a three-dimensional thermoelastic half-space with variable thermal conductivity under the definition of fractional order heat conduction based on the Moor–Gibson–Thompson theorem. The non-dimensional governing equations using Laplace and double Fourier transform methods have been applied to a three-dimensional thermoelastic, isotropic, and homogeneous half-space exposed to a rectangular thermal loading pulse with a traction-free surface. The double Fourier transforms and Laplace transform inversions have been computed numerically. The numerical distributions of temperature increment, invariant stress, and invariant strain have been shown and analysed. The fractional order parameter and the variability of thermal conductivity significantly influence all examined functions and the behaviours of the thermomechanical waves. Classifying thermal conductivity as weak, normal, and strong is crucial and closely corresponds to the actual behaviour of the thermal conductivity of thermoelastic materials.

Integrating fractional-order SEI1I2I3QCR model with awareness and non-pharmaceutical interventions for optimal COVID-19 pandemic

Infectious diseases like COVID-19 continue to pose critical challenges globally, underscoring the need for effective control strategies that go beyond traditional vaccinations and treatments. This study introduces an advanced SEI1I2I3QCR model, uniquely incorporating fractional-order delay differential equations to account for latency periods and dynamic transmission patterns of COVID-19, improving accuracy in capturing disease progression and peak oscillations. Stability analyses of the model reveal the critical role of delay and fractional order parameters in managing disease dynamics. Additionally, we applied optimal control theory to simulate non-pharmaceutical interventions, such as quarantine and awareness campaigns, demonstrating a notable reduction in infection rates. Numerical simulations align the model closely with real-world COVID-19 data from China, validating its utility in guiding pandemic response strategies. Our findings emphasize the significance of integrating time-delay factors and fractional calculus in epidemic modeling, offering a novel framework for pandemic management through targeted, cost-effective control measures.

Directed transport of particles in coupled fractional-order systems excited by Lévy noise.

This paper investigates the directed transport of particles in a coupled fractional-order system excited by Lévy noise. Numerical simulations reveal the effects of fractional order, Lévy noise and coupling coefficients on the directed transport. It is found that there exists an optimal fractional order, which maximizes the directed transport of particles. The optimal fractional order for the directed transport shifts to the left or right with different noise parameters, which means that the appropriate fractional order and noise parameters should be taken into account to maximize the directed transport. Meanwhile, the increase of the scale and symmetry parameters intensifies the directed transport of the particles, while the increase of the stability index suppresses the directed transport, so appropriate Lévy noise parameters will effectively amplify the directed transport. In addition, strong coupling can also effectively promote the directed transport of particles. These studies may provide a theoretical basis for the design of nanomachines, improving drug delivery across cell membranes and treating diseases of the nervous system.

Unitary description of the Jaynes-Cummings model under fractional-time dynamics.

The time-evolution operator corresponding to the fractional-time Schrödinger equationis nonunitary because it fails to preserve the norm of the vector state in the course of its evolution. However, in the context of the time-dependent non-Hermitian quantum formalism applied to the time-fractional dynamics, it has been demonstrated that a unitary evolution can be achieved for a traceless two-level Hamiltonian. This is accomplished by considering a dynamical Hilbert space embedding a time-dependent metric operator concerning which the system unitarily evolves in time. This allows for a suitable description of a quantum system consistent with the standard quantum mechanical principles. In this work, we investigate the Jaynes-Cummings model in the fractional-time scenario taking into account the fractional-order parameter α and its effect in unitary quantum dynamics. We analyze the well-known dynamical properties, such as the atomic population inversion and the atom-field entanglement, when the atom starts in its excited state and the field in a coherent state.

Artificial neural network with incompressible smoothed particle hydrodynamics for exothermic chemical reaction on heat and mass transfer in a rectangular annulus

This work aims to simulate the impacts of exothermic reaction and Soret–Dufour numbers on the double diffusion of Nano Enhanced Phase Change Materials (NEPCM) inside a porous annulus. The complex rectangular annulus contains two ellipses and two triangles on the walls’ vertical sides. The complex proposals of closed domains during heat/mass transfer of NEPCM can be used in energy savings, cooling electronic devices, and heat exchangers. The fractional-time derivative of the governing systems is solved numerically based on the ISPH method. The artificial neural network (ANN) is combined with the ISPH results to predict the average Nusselt number Nu¯ and Sherwood number Sh¯. The main objective of establishing the ANN model in this investigation is to create a reliable predictive instrument capable of estimating the values of Nu¯ and Sh¯. The results described the impacts of dimensionless Frank-Kamenetskii number (Fk = 0–1), Darcy number (Da = 10−2–10−5), Dufour number (Du = 0–0.1), buoyancy ratio (N = − 2 to 5), Rayleigh number (Ra = 103–106), Lewis number (Le = 1–20), Soret number (Sr = 0–0.2), fusion temperature (θf = 0.05–0.9), and fractional order parameter (α = 0.9–1) on thermosolutal convection of a suspension. The overall heat/mass transition as well as the velocity field are dramatically enhanced when Ra and N were boosted. The fractional time derivative helps reach a steady state in less time instants. The phase change material (PCM) is always changed when temperature distribution changes and is controlled by a fusion temperature. The porous struggled with nanofluid flow at a lower Darcy number. Frank-Kamenetskii number is a promising factor in enhancing the temperature distributions in an annulus. As a result, this work may be applied in various engineering and industrial fields because it contains significant terms in improving heat/mass transmission as well as a phase change material. The ANN model introduced a precise agreement of the prediction values with the actual values of Nu¯ and Sh¯. Then, the present ANN model can accurately estimate the Nu¯ and Sh¯ values.

A Vieta–Lucas collocation and non-standard finite difference technique for solving space-time fractional-order Fisher equation

The purpose of the article is to analyze an accurate numerical technique to solve a space-time fractional-order Fisher equation in the Caputo sense. For this purpose, the spectral collocation technique is used, which is based on the Vieta–Lucas approximation. By using the properties of Vieta–Lucas polynomials, this technique reduces the nonlinear equations into a system of ordinary differential equations (ODEs). The non-standard finite difference (NSFD) method converts this system of ODEs into algebraic equations which have been solved numerically. Moreover, the error estimate is investigated for the proposed method. To show the accuracy and efficiency of the technique, the obtained numerical results are compared with the analytical results and existing results of the particular forms of the considered fractional order models through error analysis. The important feature of this article is the exhibition of variations of the field variable for various values of spatial and temporal fractional order parameters for different particular cases.

Analysis of RL electric circuits modeled by fractional Riccati IVP via Jacobi-Broyden Newton algorithm.

This paper focuses on modeling Resistor-Inductor (RL) electric circuits using a fractional Riccati initial value problem (IVP) framework. Conventional models frequently neglect the complex dynamics and memory effects intrinsic to actual RL circuits. This study aims to develop a more precise representation using a fractional-order Riccati model. We present a Jacobi collocation method combined with the Jacobi-Newton algorithm to address the fractional Riccati initial value problem. This numerical method utilizes the characteristics of Jacobi polynomials to accurately approximate solutions to the nonlinear fractional differential equation. We obtain the requisite Jacobi operational matrices for the discretization of fractional derivatives, therefore converting the initial value problem into a system of algebraic equations. The convergence and precision of the proposed algorithm are meticulously evaluated by error and residual analysis. The theoretical findings demonstrate that the method attains high-order convergence rates, dependent on suitable criteria related to the fractional-order parameters and the solution's smoothness. This study not only improves comprehension of RL circuit dynamics but also offers a solid numerical foundation for addressing intricate fractional differential equations.