Fractional Model Research Articles

Viscoelastic study of cement and emulsified asphalt mortar under temperature variations based on a novel variable-order fractional Maxwell model

Cement and emulsified asphalt (CA) mortar is a critical viscoelastic component of ballastless track systems, exhibiting pronounced temperature sensitivity. However, accurately predicting the mechanical responses of CA mortar due to temperature fluctuations remains a challenge. This work proposes a novel variable-order fractional Maxwell (VOFM) model to investigate the viscoelastic properties of CA mortar under temperature variations. The VOFM model is numerically solved using a time-varying coefficient ladder (TVL) model. The VOFM model is validated by comparing with both numerical integration methods and experimental data. The basic properties of the VOFM model, the viscoelastic responses of CA mortar under varying temperatures and alternating stress are systematically investigated. The results show that the mechanical properties of VOFM model agree with the FM model when time-dependent parameters of the VOFM model correspond to that of the FM model. Moreover, high temperatures have a promoting effect on the deformation of CA mortar under both alternating and constant stress conditions. The peak amplitude of CA mortar strain under practical varying temperature and stress loading correlates with the trend of temperature changes.

Study of fractional order rabies transmission model via Atangana–Baleanu derivative

In this work, we aim at disentangling the theoretical contribution through mathematical modeling approach to advance the understanding of rabies dynamics and control in livestock population. A fractional order model of rabies, using Atangana–Baleanu fractional operator is created. The analysis of suggested system and its application are both conducted. The capacity of proposed model to forecast the disease can help researchers and livestock health care agencies to take preventive actions.

A novel hybrid method with convergence analysis for approximation of HTLV-I dynamics model

This paper presents a novel numerical approach for approximating the solution of the model describing the infection of CD4+T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$CD4^+T$$\\end{document}-cells by the human T-cell lymphotropic virus I (HTLV-I).The proposed method utilizes the operational matrix along with spectral method to convert the fractional model into a system of nonlinear algebraic equations. The Levenberg-Marquardt algorithm efficiently solves these equations. The study includes theoretical convergence analysis and error bounds to establish the validity of the proposed method. Through several test problems, we demonstrate the effectiveness and accuracy of the approach. We compare its performance and reliability to other existing methods in the literature. The results indicate that the proposed method is a reliable and efficient approach for solving the model.

Spatiotemporal dynamics in a fractional diffusive SIS epidemic model with mass action infection mechanism.

This paper is concerned with spatiotemporal dynamics of a fractional diffusive susceptible-infected-susceptible (SIS) epidemic model with mass action infection mechanism. Concretely, we first focus on the existence and stability of the disease-free and endemic equilibria. Then, we give the asymptotic profiles of the endemic equilibrium on small and large diffusion rates, which can reveal the impact of dispersal rates and fractional powers simultaneously. It is worth noting that we have some counter-intuitive findings: controlling the flow of infected individuals will not eradicate the disease, but restricting the movement of susceptible individuals will make the disease disappear.

The Investigation of Nonlinear Time-Fractional Models in Optical Fibers and the Impact Analysis of Fractional-Order Derivatives on Solitary Waves

In this article, we investigate a couple of nonlinear time-fractional evolution equations, namely the cubic-quintic-septic-nonic equation and the Davey–Stewartson (DS) equation, both of which have significant applications in complex physical phenomena such as fiber optical communication, optical signal processing, and nonlinear optics. Using a powerful technique named the extended generalized Kudryashov approach, we extract different rich structured soliton solutions to these models, including bell-shaped, cuspon, parabolic soliton, singular soliton, and squeezed bell-shaped soliton. We also study the impact of fractional-order derivatives on these solutions, providing new insights into the dynamics of nonlinear models. The results are compared with the existing literature, revealing novel and distinct solutions that offer a deeper understanding of these fractional models. The results show that the implemented approach is useful, reliable, and compatible for examining fractional nonlinear evolution equations in applied science and engineering.

Fractional Viscoelastic Analysis of Transversely Isotropic Surrounding Rock Along Shallow Buried Elliptical Tunnel

ABSTRACTThis study introduces the fractional order Merchant model to analytically solve the stress and displacement fields of the transversely isotropic viscoelastic surrounding rock along shallow elliptical tunnels. First, the stress and displacement solutions of fractional order viscoelastic and transversely isotropic half planes under arbitrary loads are obtained using the Laplace transform and the elastic‐viscoelastic correspondence principle. Second, based on the above half plane solution and the solution of the deep buried elliptical tunnel problem, the Schwarz alternating method is introduced to obtain the analytical solution of the shallow buried elliptical tunnel. A MATLAB program is developed, and the accuracy of the theory and program in this study is verified by comparing it with the results of ABAQUS. Finally, the effects of transversely isotropic parameters, tunnel burial depth, and viscoelastic parameters on the stress and displacement of tunnel surrounding rock are analyzed.

A-optimal experimental designs for Michaelis – Menten model

Kinetics of simple and complex kinetic reactions is usually described by exponential and rational models. The fractional rational models include the well-known Michaelis – Menten model of enzymatic kinetics. In the presented article A-optimal designs are proposed for the Michaelis – Menten model. The elimination method used to construct A-optimal designs, allowed us to determine the nodes and weights of A-optimal designs separately and thereby extend this approach to other criteria and models of the rational type. It is shown that the nodes of A-optimal designs determined by this method are the roots of the 4th degree algebraic equation with coefficients depending on the model parameters. If the nodes of the A-optimal design are already known, then the weights of the corresponding nodes are determined analytically by the Pukelsheim formula. The properties of the roots as well as the properties of the nodes of A-optimal designs were studied using the Sturm system constructed in the general form for the resulting equation. It is shown that with a certain combination of parameters of the Michaelis – Menten model, the degree of the resulting algebraic equation is reduced to three. A partition of the set of values of the parameters of the Michaelis – Menten model into two subsets has been found. In one of them, the A-optimal design is determined uniquely, whereas for the other one it is necessary to select the optimal node from two possible options. It is revealed that the degree of the algebraic equation is equal to three for points belonging to the curve which is the common boundary of the indicated subsets. Corresponding numerical examples are given to illustrate the results obtained.

A thermodynamically‐based fractional model combined viscoelastic‐viscoplastic‐ductile damage with application to fiber‐reinforced polymer composites

AbstractA thermodynamically‐based fractional viscoelastic‐viscoplastic‐damage constitutive model combined with continuous damage mechanics (CDM) theory was established, in order to describe the rate‐dependent nonlinear behavior of fiber‐reinforced polymer composites (FRPCs). The fractional Helmholtz free energy consists of four contributions: viscoelastic (VE), viscoplastic (VP), hardening and damage, in which the VE and VP parts are constructed by fractional Zener and Scott‐Blair (SB) element forms respectively. The constitutive equation is obtained through Helmholtz free energy for the fractional Zener model, and plastic flow and hardening evolution law are all derived in the process. The ductile damage, coupled to both VE and VP free energy parts, is introduced through fractional damage energy release rates to model the degradation of material properties. The corresponding strain energy release rate and dissipation contributions are also derived. The fractional implicit time integration algorithms of proposed model are presented. The model is applied to validate tests of FRPCs under various loading conditions. The model validation and comparison are presented by simulating experimental data and existing models in the literature. And the corresponding evolution of dissipated energy is discussed to further valid the characterization ability of the model.Highlights A thermodynamical fractional constitutive model was developed for FRPCs. The Helmholtz free‐energy potential for fractional Zener model is adopted. The physical significance of fractional order parameters is explored. Fractional implicit integration algorithm of proposed model is implemented. The validation and comparison of the model are presented under various loads.

Multiobjective linear fractional programming model with equality and inequality constraints under pentagonal intuitionistic fuzzy environment

Multiobjective linear fractional programming model with equality and inequality constraints under pentagonal intuitionistic fuzzy environment

Fractional Caputo Operator and Takagi–Sugeno Fuzzy Modeling to Diabetes Analysis

Diabetes is becoming more and more dangerous, and the effects continue to grow due to the population’s ignorance of the seriousness of this phenomenon. The studies that have been carried out have not been able to follow the phenomenon more precisely, which has led to the use of the fractional derivative tool, which has a very great capability to study real problems and phenomena but is somewhat limited on nonlinear models. In this work, we will develop a new fractional derivative model of a diabetic population, the Takagi–Sugeno fractional fuzzy model, which will enable us to study the phenomenon with these nonlinear terms in order to obtain greater precision in the results. We will study the existence and uniqueness of the solution using the Lipschizian theorem and then turn to the new fuzzy model, which leads us to four dynamical systems. The interpretation results show the quality of fuzzy membership in tracking the malleable phenomena of nonlinear terms existing in the system.

An Exact Stochastic Simulation Method for Fractional Order Compartment Models

An Exact Stochastic Simulation Method for Fractional Order Compartment Models

Nonlinear resonance of fractional order viscoelastic PET films under temperature loading

The effects of oven temperature during printing on nonlinear vibration for fractional-order PET films are considered in this paper. The effect of temperature, fractional order modelling and some other parameters are analysed with respect to the response of the resonance. Fractional order kelvin-Voigt ontological relationship is used to describe the characteristics of viscoelastic materials. The differential equations for nonlinear vibrations are inferred according to the second law of Newton and the theory of von Karman. Discretization for nonlinear equations on locomotion using the Bubnov–Galerkin method. Forced co-oscillatory amplitude-frequency response equations for thin-films systems under temperature loading were calculated using the multiple scales method. Results of numeral results show that temperature, and fractional-order visco-elastic modelling influence the membrane's response to resonance. These results provide a basis for studying fractional-order visco-elastic films vibrations and identifying regions of stable operation in moving systems to prevent divergent instabilities for flexible electronic device manufacturing.

A fractional order mathematical model for the omicron: a new variant of COVID-19

For the purpose of analyzing the Omicron pandemic, we build a novel SEIaIsIoHR mathematical model. The fundamental properties of the model are studied, including boundedness, uniqueness, and existence. The boundedness of the model’s solution is examined by solving the fractional Gronwall’s inequality using the Laplace transform method. Using the Picard-Lindelöf theorem, we verify the solution’s existence and uniqueness. The next-generation matrix is used to compute Ro (the basic reproduction number), which is significant in the mathematical modeling of infectious diseases. It is demonstrated that both the endemic and disease-free equilibrium solutions are globally and locally asymptotically stable. We also conducted a sensitivity analysis of the parameters based on Ro . Moreover, the model is extended by employing optimal control theory to examine the effects of certain control strategies. Finally, numerical simulations are performed to validate the model.

A finite-difference-based Adams-type approach for numerical solution of nonlinear fractional differential equations: A fractional Lotka-Volterra model as a case study

A finite-difference-based Adams-type approach for numerical solution of nonlinear fractional differential equations: A fractional Lotka-Volterra model as a case study

Exploring the dynamics of leprosy transmission with treatment through a fractal–fractional differential model

Leprosy continues to be a significant public health challenge in many parts of the world, necessitating novel approaches to understanding and controlling its transmission. The dynamics of leprosy are examined in this study by means of a caputo fabrizio fractal–fractional differential system model. Leprosy transmission and treatment are complex and non-linear processes that can be captured by fractal–fractional derivatives. The temporal progression of the disease is the primary focus of our mathematical analysis, which evaluates a variety of parameter values, including initial population densities and compartmental transitions. Through simulations, we examine the impact of critical parameters on the severity and spread of diseases, as well as the rates of recovery and treatment. Numerical results of the model offer valuable insights into the impact of these parameters on leprosy dynamics, which is beneficial for public health interventions. The stability analysis of the model identifies supplementary conditions that are necessary for the success of disease control. By incorporating these novel mathematical techniques, we hope to improve our understanding of leprosy transmission and ultimately contribute to more effective control strategies. Based on our findings, additional studies investigating the relation between population density, treatment accessibility, and recovery rates are warranted. We hope that by graphically representing the relationships between these factors, we can draw attention to the possibility of targeted interventions that can reduce the transmission of leprosy. This study provides a strong basis for future studies on infectious disease modeling and aids leprosy-affected communities in developing strategies to mitigate the disease’s impact.

Electric quadrupole transitions of 98–112Ru atomic nuclei via conformable fractional Bohr Hamiltonian

In this paper, we investigate the energy spectra, wave functions and the [Formula: see text] transition rates for [Formula: see text]Ru atomic nuclei, using the conformable fractional Bohr Hamiltonian model. For the [Formula: see text]-part of the potential, the newly proposed Yukawa plus modified exponential potential is considered and the harmonic oscillator potential in [Formula: see text]-part, with [Formula: see text] fixed around [Formula: see text]. By using the conformable fractional Nikiforov–Uvarov method, energy spectra and wave functions are obtained analytically. The sensitivity of the potential parameters and the spectra with respect to the fractional order parameter is investigated. The normalized fractional energies and fractional electric quadrupole transition rates are compared with the available experimental predictions and those from existing theoretical studies. Comparisons are made at different values of the fractional derivative order. The results are presented across a broader range of values of [Formula: see text], providing a systematic analysis of how variations in this parameter influence the energy spectra, wave functions, and [Formula: see text] transition rates in Ru nuclei. Overall, the comparison of our results with experimental data shows the good accuracy of our model, especially when the fractional parameter goes to lower values. It can be concluded that the order of fractional derivative plays a crucial role in refining theoretical predictions of electric quadrupole transition rates, especially for transitions involving higher multipolarities.

A study on existence and stability analysis of implicit k,Ψ$$ \left(k,\Psi \right) $$‐Hilfer fractional differential equations and application to RC$$ \mathcal{RC} $$ circuit model

In the present study, we establish the existence and uniqueness of solutions for nonlinear initial value problems and nonlocal boundary value problems associated with implicit fractional differential equations involving ‐Hilfer derivative operator. Furthermore, we explore the stability properties of these solutions in the sense of the Ulam–Hyers, Ulam–Hyers–Rassias, and their generalized stability concepts by proving and applying generalized Gronwall inequality. Lastly, we present the fractional electric circuit model as an example to show the practical applicability of our main results.

The soil-air interfacial migration process of volatile PFAS at the contaminated sites: Evidence from stable carbon isotopes with CSIA

Volatile per- and polyfluoroalkyl substances (PFAS) are prone to transport among various environmental media, with the soil-air interfacial migration process being an important pathway that significantly influences their environmental fate. To assess the migration and transformations of target volatile PFAS at contaminated site using compound-specific stable isotope analysis (CSIA), it is necessary to understand the isotopic fractionation that occurs during their transfer from soil to air. We have established methods for pre-treatment and GC/CSIA analysis methods of target volatile PFAS in soil and air samples and ensured the accuracy of carbon isotope analysis. GC/IRMS δ13C measurements showed optimal precision at instrumental response above 1.35–2.75 Vs, with recommended minimum on-column C levels of 1.67–5.00 nmol for target volatile PFAS. Stable carbon isotope fractionation factors related to the soil-air interfacial migration process for target volatile PFAS were determined by performing laboratory simulations. The observed εsoil-air values are all negative, suggesting that the soil-air interfacial migration process for target volatile PFAS is kinetic fractionation, the removal of molecules containing lighter isotopes. By comparing the simulated and experimentally observed δ13C (‰) values of target volatile PFAS, we found consistent trends in the soil and inverse trends in the air. These δ13C (‰) values and the related isotope fractionation model provide valuable insights into the isotopic behavior of target volatile PFAS during soil-air interfacial migration process, aiding in the assessment of their environmental fate.

A modified fuzzy goal programming procedure to solve fully quadratic fractional optimization model

A modified fuzzy goal programming procedure to solve fully quadratic fractional optimization model

Application of Fractional Modified Taylor Wavelets in the Dynamical Analysis of Fractional Electrical Circuits under Generalized Caputo Fractional Derivative

Abstract This study examines the application of fractional calculus in the analysis and modeling of electrical circuits of fractional order, highlighting its potential to explain the behaviour of complex electrical circuits accurately. In the domain of electrical circuits, fractional differential equations are employed in the analysis and simulation of systems that consist of resistors, capacitors and inductors. In the present paper, a novel approach utilizing fractional order modified Taylor wavelets is implemented to solve the fractional model of RL, LC, RC and RLC electrical circuits under generalized Caputo fractional derivative which offers precise and flexible modeling of non-locality and hereditary characteristics in complex systems. Furthermore, an additional parameter $\sigma$ (time scale parameter) is incorporated in fractional circuit dynamics to maintain the physical dimensionality. The considered wavelets with the collocation technique offer an efficient solution by converting the fractional model of electrical circuits into a set of algebraic equations which are further solved by using the Newton iteration method. Moreover, this study discusses the significance of Ulam-Hyers stability, emphasizing its role in ensuring stable and reliable circuit performance. The impact of fractional order on the dynamics of the electric circuit model is presented by tables and graphs. The approximate solutions obtained by the proposed method are well comparable with exact solutions and some other existing wavelet-based techniques. The residual errors are also evaluated under various model parameters for fractional orders. Furthermore, the graphs illustrate that the error progressively decreases as the number of wavelets basis increases.