Nonlinear Fractional Model Research Articles

Fuzzy computational study on the generalized fractional smoking model with caputo gH-type derivatives

This work considers a generalized fuzzy fractional smoking model with Caputo [Formula: see text]-types fractional derivatives upon considering the case of uncertainty quantification. The disease-free equilibrium point and stability of the equilibrium point have been discussed for the fuzzy nonlinear fractional smoking model. The analytical proofs for the existence and uniqueness of the proposed model are concerned with the help of the fixed-point theorem, Banach contraction, and Schauder theorem. A robust double parametric approach with a generalized transform is used to study the behavior of the fuzzy fractional model in an uncertain context and obtain the convergence analysis of the study in a crisp context. Finally, the obtained results of the proposed model have been validated with the Runge–Kutta method of fourth order in crisp case [Formula: see text].

Refined nonlinear fractional derivative model of vehicle-track coupling dynamics

The coupled vehicle-track system (CVTS) dynamics have been extensively investigated for decades. However, the calculation accuracy of prevailing vehicle-track coupling models needs to be improved in the high frequency range due to the inappropriate model simplification and neglect of material nonlinearity. In this study, we propose a refined numerical model of the CVTS that considers the nonlinear properties of the railpads and primary suspension using the fraction derivative Zener model. Furthermore, we more realistically simulate the wheelset, rail and railpad configuration with the elastic axle, solid finite element and surface-support models, respectively, and improve the computation efficiency by employing the mode superposition method. The results demonstrate that the refined CVTS model is more accurate than the classical model in simulating vehicle-track coupling dynamics above 2 kHz. In particular, there are significant differences in the dynamic response of the elastic wheelset model compared to the rigid model over a broad frequency range, with an 11% difference in the bogie acceleration response at the first dominant frequency. When the railpads are modeled using the surface-support model, the rail acceleration differences exceed 41% near 1 kHz and 44% near 2650 Hz, compared to the point-support model. Additionally, the rail response at various locations across the rail cross section can be calculated using the finite element method in this refined model. Overall. the proposed CVTS model provides high accuracy and efficiency for random vibration analysis, especially in the high frequency domain.

Stability and numerical analysis via non-standard finite difference scheme of a nonlinear classical and fractional order model

In this paper, we develop a new mathematical model for an in-depth understanding of COVID-19 (Omicron variant). The mathematical study of an omicron variant of the corona virus is discussed. In this new Omicron model, we used idea of dividing infected compartment further into more classes i.e asymptomatic, symptomatic and Omicron infected compartment. Model is asymptotically locally stable whenever R0<1 and when R0≤1 at disease free equilibrium the system is globally asymptotically stable. Local stability is investigated with Jacobian matrix and with Lyapunov function global stability is analyzed. Moreover basic reduction number is calculated through next generation matrix and numerical analysis will be used to verify the model with real data. We consider also the this model under fractional order derivative. We use Grunwald–Letnikov concept to establish a numerical scheme. We use nonstandard finite difference (NSFD) scheme to simulate the results. Graphical presentations are given corresponding to classical and fractional order derivative. According to our graphical results for the model with numerical parameters, the population’s risk of infection can be reduced by adhering to the WHO’s suggestions, which include keeping social distances, wearing facemasks, washing one’s hands, avoiding crowds, etc.

Exploring the wave solutions of a nonlinear non-local fractional model for ocean waves

In this research, analytical and semi-analytical soliton solutions for the nonlinear fractional (2 + 1)-dimensional integrable Calogero–Bogoyavlenskii–Schiff equation (FCBSE) in the non-local form are obtained using recent computational and numerical methods. The FCBSE is a significant model for investigating various phenomena, such as internal ocean waves, tsunamis, river tidal waves, and magneto-sound waves in plasma. The constructed solution helps in understanding the interaction between a long wave moving along the x-axis and a Riemann wave propagating along the y-axis. Various analytical solutions, such as exponential, trigonometric, and hyperbolic, have been formulated differently for this model, which is a specific derivation of the well-known Korteweg–de Vries equation. Density charts in two and three dimensions are used to visualize the behavior of a single soliton in reality through simulations. The results demonstrate the effectiveness of the employed numerical scheme and various methods to ensure the consistency of computational and approximation answers. Overall, this study demonstrates the potential of recent computational and numerical techniques for solving nonlinear mathematical and physical problems.

Application of the Optimal Homotopy Asymptotic Approach for Solving Two-Point Fuzzy Ordinary Differential Equations of Fractional Order Arising in Physics

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method’s efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis.

Implementation of non-linear mixed effects models defined by fractional differential equations.

Fractional differential equations (FDEs), i.e. differential equations with derivatives of non-integer order, can describe certain experimental datasets more accurately than classic models and have found application in pharmacokinetics (PKs), but wider applicability has been hindered by the lack of appropriate software. In the present work an extension of NONMEM software is introduced, as a FORTRAN subroutine, that allows the definition of nonlinear mixed effects (NLME) models with FDEs. The new subroutine can handle arbitrary user defined linear and nonlinear models with multiple equations, and multiple doses and can be integrated in NONMEM workflows seamlessly, working well with third party packages. The performance of the subroutine in parameter estimation exercises, with simple linear and nonlinear (Michaelis-Menten) fractional PK models has been evaluated by simulations and an application to a real clinical dataset of diazepam is presented. In the simulation study, model parameters were estimated for each of 100 simulated datasets for the two models. The relative mean bias (RMB) and relative root mean square error (RRMSE) were calculated in order to assess the bias and precision of the methodology. In all cases both RMB and RRMSE were below 20% showing high accuracy and precision for the estimates. For the diazepam application the fractional model that best described the drug kinetics was a one-compartment linear model which had similar performance, according to diagnostic plots and Visual Predictive Check, to a three-compartment classic model, but including four less parameters than the latter. To the best of our knowledge, it is the first attempt to use FDE systems in an NLME framework, so the approach could be of interest to other disciplines apart from PKs.

Two effective methods for solving nonlinear coupled time-fractional Schrödinger equations

The objective of this work is to implement two efficient techniques, namely, the Laplace Adomian decomposition method (LADM) and the modified generalized Mittag–Leffler function method (MGMLFM) on a system of nonlinear fractional partial differential equations (NFPDEs) to get an analytic-approximate solution. The nonlinear time-fractional Schrödinger equation (TFSE) and coupled fractional order Schrödinger-Korteweg-de Vries (Sch-KdV) equation are found in various areas such as quantum mechanics and physics. These equations describe different types of wave propagation like dust-acoustic waves, Langmuir and electromagnetic waves in plasma physics. Using the proposed methods, a convenient solution is established for the considered nonlinear fractional order models. The obtained analytic-approximate travelling-waves solutions and the effect of the fractional order α on the behaviour of these projected solutions are presented in some figures and tables along with the exact solution. We compare the approximate values with their corresponding values of the known exact solution and compute the absolute error. Consequently, we can deduce that the used methods are very efficient, reliable and simple to construct a series form that rapidly convergent to the exact solution, which indicates the advantages of the methods.

Abundant solitary wave solutions for space-time fractional unstable nonlinear Schrödinger equations and their applications

The unstable nonlinear Schrödinger models illustrate the time evolution of disturbances in marginally stable or unstable media. In this article, an improved auxiliary equation method is proposed for conformable fractional-order nonlinear evolution equations to obtain the novel wave solutions of space–time fractional unstable and modified unstable nonlinear Schrödinger equations. Several types of novel wave solutions are constructed in forms of some special Jacobi elliptic functions, rational, exponential, trigonometric and hyperbolic functions in which various are unique and having key utilization in quantum physics. With the aid of appropriate values to parameters, different shapes of bright, dark, kink type soliton, multi-peak solitons, periodic solitary waves etc. are depicted. These distinct physical structures are helpful in the understanding the complex physical interpretation of unstable dynamical models. Furthermore, this article gives an idea, how can reduce the conformable fractional order unstable nonlinear Schrödinger equations into an ODE of one variable to obtain the exact solutions. The numerous obtained waves and other solutions reveal the usefulness of the proposed technique which can be utilized to other nonlinear fractional models arising in quantum physics and other fields of applied sciences.

A fractional order age-specific smoke epidemic model

This paper presents a nonlinear fractional mathematical model for the smoke epidemic that includes two age groups. To solve the smoke epidemic, the Atangana-Baleanu-Caputo fractional derivative is used. The Banach and Krasnoselskii type fixed point theorem is used to determine existence and uniqueness. We explored model stability using the Hyers-Ulam form of stability. Using Lagrange interpolation, the behaviour of the smoke epidemic of the 2-age group model is generated. The numerical simulation shows that the model has potential for both groups, and that age-specific interventions can be used to reduce smoking rates in the general population.

Nonlinear Fractional Order Model Identification of the Voltage Source Inverter Fed Induction Motor

This paper presents a new nonlinear fractional-order model identification of a two-level voltage source inverter fed induction motor (VSI-IM). The nonlinearities present in inverter and IM cause the failure of the sensorless controls and model predictive controls, which mainly depends on the VSI-IM model. In the proposed work, an attempt has been made to identify the exact nonlinear behavior of the VSI-IM model using the PWM signal injection method. The identified fractional-order VSI-IM model is helpful to provide more insight into the dynamic behavior of the IM drive in the speed controller design (sensorless and sensored) and in predictive controls (model-based and model-free). The fractional VSI-IM models are identified at low, medium, and high frequencies. The best fit of these models is compared to the integer-order identified VSI-IM models. The experimental study clearly shows that the fractional-order VSI-IM identified model is superior to the integer-order VSI-IM model in terms of best fit with the actual hardware speed response.

A fractional mathematical model for listeriosis infection using two kernels

Listeriosis is a life-threatening illness that can be spread through contaminated food. The bacteria responsible for this disease is Listeria monocytogenes. The annual incidence rate ranges from less than one to more than ten per one million persons in different parts of the world. Despite the low incidence of listeriosis, the high mortality rate linked with the disease makes it a major public health problem. In this paper, we present an investigation of a nonlinear fractional mathematical model of listeriosis infection using two fractional kernels. To analyse the existence of the fractional model, we employ the fixed point technique, whereas stability is determined using the Hyers–Ulam analysis. The fractional model is numerically investigated using the Caputo–Fabrizio and Atangana-Baleanu-Caputo fractional kernels. The numerical results reveal that the Mittag- Leffler kernel captures fewer human infections than the exponential kernel. In other simulations, we presented the trajectory impacts of varying environmental transmission rates, the rate of infected human recovery, the rate at which uncontaminated food becomes contaminated, and the net growth rate of Listeria monocytogenes on all the affected compartments. We observed that these parameters in the model give distinct disease pathways (a unique rate of convergence), unlike the integer model. Furthermore, the exponential kernel shows more dynamics of contaminated foods than the Mittag-Leffler kernel, which indicates that an increase in contaminated food could lead to a high spread of listeriosis. Finally, we noticed the reduction in the rate of uncontaminated food becoming contaminated increases the number of recovered individuals.

Optical solitonic structures with singular and non-singular kernel for nonlinear fractional model in quantum mechanics

Optical solitonic structures with singular and non-singular kernel for nonlinear fractional model in quantum mechanics

Fractional evaluation of Kaup-Kupershmidt equation with the exponential-decay kernel

&lt;abstract&gt;&lt;p&gt;In this paper, we investigate the semi-analytical solution of Kaup-Kupershmidt equations with the help of a modified method known as the new iteration transformation technique. This method combines the Yang transform and the new iteration technique. The nonlinear terms can be calculated straightforwardly by a new iteration method. The numerical simulation results have been presented to demonstrate the reliability and validity of the proposed approach. The result confirms that the suggested technique is the best tool for dealing with any nonlinear problems arising in technology and science. In addition, in terms of figures for varying fractional order, the physical behavior of new iteration transformation technique solutions has been shown and the numerical simulation is also exhibited. The solutions of the new iteration transformation technique reveal that the projected technique is reliable, competitive and powerful for studying complex nonlinear fractional type models.&lt;/p&gt;&lt;/abstract&gt;

Modeling the impact of public health education on tungiasis dynamics with saturated treatment: Insight through the Caputo fractional derivative.

Public health education is pivotal in the management and control of infectious and non-infectious diseases. This manuscript presents and analyses a nonlinear fractional model of tungiasis dynamics with the impact of public health education for the first time. The human population is split into five classes depending on their disease status. The infected population is split into two subgroups; infected but unaware and infected but aware. The model focuses on the impacts of public health education, contact and treatment contact on tungiasis transmission dynamics. Notably, public health education is important for containing as well as reducing disease outbreaks in communities. The Caputo fractional derivative is utilised in defining the model governing equations. Model equilibrium points existence and stability are investigated using simple matrix algebra. Model analysis shows that tungiasis is contained when the reproduction number is less than unity. Otherwise, if it is greater than unity, the disease persists and spread in the population. The generalised Adams-Bashforth-Moulton approach is utilised in solving the derived tungiasis model numerically. The impacts of public health education, treatment and contact rate on overall disease dynamics are discussed through numerical simulations. From the simulations, we see that for given fractional order, public health education and treatment increase the quality of life plus reduce equilibrium numbers of tungiasis-infected individuals. We observe that population classes converge quicker to their steady states when α is increased. Thus, we can conclude that the derivative order α captures the role of experience or knowledge that individuals have on the disease's history.

Error analysis of a Haar wavelets-based numerical method with its application to a nonlinear fractional dengue model

Millions of people are infected by Dengue every year. So to understand its mechanism, we have considered one of the recent fractional and nonlinear Dengue models proposed by Diethelm. However, the problem is that its exact solution is not known. Therefore, an extended Haar wavelets based numerical method has been implemented to solve it. Although Haar wavelets-based numerical methods have been used by researchers in recent years its theoretical error analysis over the extended interval , , has never been done before for fractional dynamical systems. It helps to show the convergence of a sequence of approximate solutions to the exact solution. Also, the derived error bound gives a guarantee that error in the considered Haar approximate solution cannot exceed its bound. And any desired accuracy can be obtained by increasing the number of Haar wavelets in the approximate solution.

Excitation system voltage regulator modeling with the use of fractional calculus

The article presents a new and measurement-verified approach in the modeling of a voltage regulator, which is used in the power industry and applied for the purpose of regulation of generating units. The voltage regulator extended model, proposed in this paper, involves nonlinearities and the application of fractional calculus. The addition of these features has not only significantly increased the number of parameters determining the model response, it has also established a challenge of developing a solver capable of computing the resulting complex system of fractional differential–algebraic equations (FDAE). The mathematical formulation of a single simulation of a transient process for the model has been described. The application of the Oustaloup Recursive Approximation (ORA) of the fractional derivative has been considered in order to increase the speed of the model evaluation. Generalized equations leading to an approximating system of Differential–Algebraic Equations (DAE) have been presented in detail. In order to obtain the parameters for the fractional nonlinear model, a parameter estimation procedure has been applied, which consists of implementations of several algorithms that are executed sequentially. The usefulness and drawbacks of the application of the ORA has also been discussed.

A Generalized Approach of the Gilpin–Ayala Model with Fractional Derivatives under Numerical Simulation

In this article, we study the existence and uniqueness of multiple positive periodic solutions for a Gilpin–Ayala predator-prey model under consideration by applying asymptotically periodic functions. The result of this paper is completely new. By using Comparison Theorem and some technical analysis, we showed that the classical nonlinear fractional model is bounded. The Banach contraction mapping principle was used to prove that the model has a unique positive asymptotical periodic solution. We provide an example and numerical simulation to inspect the correctness and availability of our essential outcomes.

On the Solution of the Variable Order Time Fractional Schrödinger Equation

In the present paper, we used the Adomian decomposition method to obtain solutions of linear and nonlinear variable order fractional Schrödinger equations. Twelve illustrative applications have been presented. When the fractional order is unity, the results are the same as those obtained by the standard integer order derivative. Some of the obtained results are discussed graphically. Some of the obtained results 1st appear in the literature. This technique can be applied to other linear or nonlinear fractional differential models.&#x0D; HIGHLIGHTS&#x0D; &#x0D; We study a class of time variable-order fractional nonlinear Schrödinger equations (tVOFNLSEs)&#x0D; We introduce a new procedure for solving the tVOFNLSE and tVOFSE&#x0D; We used the Adomian decomposition method to obtain solutions of linear and nonlinear variable order fractional equations&#x0D; The calculations are performed in the frame of Caputo sense&#x0D; &#x0D; GRAPHICAL ABSTRACT

A SARS-CoV-2 Fractional-Order Mathematical Model via the Modified Euler Method

This article develops a within-host viral kinetics model of SARS-CoV-2 under the Caputo fractional-order operator. We prove the results of the solution’s existence and uniqueness by using the Banach mapping contraction principle. Using the next-generation matrix method, we obtain the basic reproduction number. We analyze the model’s endemic and disease-free equilibrium points for local and global stability. Furthermore, we find approximate solutions for the non-linear fractional model using the Modified Euler Method (MEM). To support analytical findings, numerical simulations are carried out.

Effective Analytical Computational Technique for Conformable Time-Fractional Nonlinear Gardner Equation and Cahn-Hilliard Equations of Fourth and Sixth Order Emerging in Dispersive Media

The aim of this paper is to investigate the approximate solutions of nonlinear temporal fractional models of Gardner and Cahn-Hilliard equations. The fractional models of the Gardner and Cahn-Hilliard equations play an important role in pulse propagation in dispersive media. The time-fractional derivative is observed in the conformable framework. In this orientation, a reliable computationally algorithm is designed and developed by following a residual error and multivariable power series expansion. Basically, the approximate solutions of pulse wave function of the fractional higher-order Gardner and Cahn-Hilliard equations are obtained in the form of a conformable convergent fractional series. Relevant consequences are theoretically and numerically investigated under the conformable sense. Besides, the analysis of the error and convergence of the developed technique are discussed. Some of the unidirectional homogeneous physical applications of the posed models in a finite compact regime are tested to confirm the theoretical aspects, demonstrate different evolutionary dynamics, and highlight the superiority of the novel developed algorithm compared to other existing analytical methods. For this purpose, associated graphs are displayed in two and three dimensions. Growing and decaying modes of the fractional parameters are analyzed for several α values. From a numerical viewpoint, the simulations and results declare that the proposed iterative algorithm is indeed straightforward and appropriate with efficiency for long-wavelength solutions of nonlinear partial differential equations.