Values Of Fractional Order Research Articles

Derivation of optical solitons for perturbed highly dispersive conformable fractional nonlinear Schrödinger equation with sextic-power law refractive index

In this article, the modified extended direct algebraic method is applied for the perturbed highly dispersive nonlinear Schrödinger equation with conformable fractional derivative and sextic-power law refractive index. Various types of solutions are extracted such as bright solitons, dark solitons, combo bright-dark solitons, singular solitons, singular periodic wave solutions, exponential wave solutions and rational solutions. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order.

Fractional mathematical model for the transmission dynamics and control of Lassa fever

This paper aims to investigate various epidemiological aspects of Lassa fever viral infection using a fractional-order mathematical model so as to assess the impacts of treatment and vaccination on the spread of Lassa fever transmission dynamics. Initially, the model employs integer-order nonlinear differential equations, incorporating imperfect vaccination and treatment as control measures for the human population. Subsequently, the model is redefined using a fractional order derivative with power law to enhance understanding of disease dynamics. The paper establishes conditions ensuring the existence and uniqueness of the model’s solution in the fractional case, alongside presenting stability analysis of the endemic equilibrium using the Lyapunov function approach. Numerical simulations are performed using the fractional Adams–Bashforth–Moulton method, shedding light on the influence of model parameters and fractional order values on Lassa fever dynamics and control. Additional numerical simulations conducted utilizing surface and contour plots revealed that elevating the contact rates and diminishing the efficacy of vaccines would escalate the prevalence of Lassa fever among the human populace. It was also observed that optimizing treatment and enhancing vaccination strategies would ultimately mitigate the prevalence of Lassa fever within the human population.

Stability analysis of fractional epidemic model for two infected classes incorporating hospitalization impact

This article presents an epidemic disease propagation mathematical model in fractional order. The epidemiological characteristics are presented based on the susceptible, exposed, unknown infected, known infected, hospitalized population and the population in the secure zone. Both the disease endemic equilibrium and the disease-free equilibrium’s stability characteristics have been examined using the basic reproduction number. Variation of basic reproduction number based on the different sensitive parameters has been discussed. It has been disputed whether the fractional model provides a uniform, reliable solution. An analysis of the time history of unknown and known infected populations, hospitalized populations and recovered populations at different values of various sensitive parameters has been carried out. To support the key theoretical conclusions, some numerical simulations are completed using MATLAB. The impact of various populations on the propagation of the illness has also been investigated, as well as how specific state variables change over time for various fractional order values.

Fractional-Order Dynamics in Epidemic Disease Modeling with Advanced Perspectives of Fractional Calculus

Piecewise fractional-order differential operators have received more attention in recent years because they can be used to describe various evolutionary dynamical problems to investigate crossover behaviors. In this manuscript, we use the aforementioned operators to investigate a mathematical model of COVID-19. By utilizing fractional calculus, our approach aims to capture the crossover dynamics of disease spread, considering heterogeneity and transitions between epidemic phases. This research seeks to develop a framework using specialized mathematical techniques, such as the Caputo fractional derivative, with the potential to investigate the crossover dynamical behaviors of the considered epidemic model. The anticipated contribution lies in bridging fractional calculus and epidemiology, offering insights for both theoretical advancements and practical public health interventions. In order to improve our understanding of epidemic dynamics and support, we used MATLAB to simulate numerical results for a visual representation of our findings. For this interpretation, we used various fractional-order values. In addition, we also compare our simulated results with some reported results for infected and death classes to demonstrate the efficiency of our numerical method.

Fractional dynamics and sensitivity analysis of measles epidemic model through vaccination

Measles is a highly contagious disease that mainly affects children worldwide. Even though a reliable and effective vaccination is available, there were 140,000 measles deaths worldwide in 2018, and most of them were children under the age five years. In this paper, we comprehensively investigate a novel fractional SVEIR (Susceptible-Vaccinated-Exposed-Infected-Recovered) model of the measles epidemic powered by nonlinear fractional differential equations to understand the epidemic’s dynamical behaviour. We use a non-singular Atangana-Baleanu fractional derivative to analyze the proposed model, taking advantage of non-locality. The existence, uniqueness, positivity and boundedness of the solutions are shown via concepts of fixed point theory, and we also perform the Ulam-Hyers stability of the considered model. The parameter sensitivity is discussed in the context of the variance with each parameter using 3-D graphics based on the basic reproduction number. Moreover, with the Atangana-Toufik numerical scheme, numerical findings are depicted for different fractional-order values. The presented approach produce results that are efficiently consistent and in excellent agreement with the theoretical results.

Physiological and chaos effect on dynamics of neurological disorder with memory effect of fractional operator: A mathematical study

Background and objective:To study the dynamical system, it is necessary to formulate the mathematical model to understand the dynamics of various diseases that are spread worldwide. The main objective of our work is to examine neurological disorders by early detection and treatment by taking asymptomatic. The central nervous system (CNS) is impacted by the prevalent neurological condition known as multiple sclerosis (MS), which can result in lesions that spread across time and place. It is widely acknowledged that multiple sclerosis (MS) is an unpredictable disease that can cause lifelong damage to the brain, spinal cord, and optic nerves. The use of integral operators and fractional order (FO) derivatives in mathematical models has become popular in the field of epidemiology. Method:The model consists of segments of healthy or barian brain cells, infected brain cells, and damaged brain cells as a result of immunological or viral effectors with novel fractal fractional operator in sight Mittag Leffler function. The stability analysis, positivity, boundedness, existence, and uniqueness are treated for a proposed model with novel fractional operators. Results:Model is verified the local and global with the Lyapunov function. Chaos Control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium so that solutions are bounded in the feasible domain. To ensure the existence and uniqueness of the solutions to the suggested model, it makes use of Banach’s fixed point and the Leray Schauder nonlinear alternative theorem. For numerical simulation and results the steps Lagrange interpolation method at different fractional order values and the outcomes are compared with those obtained using the well-known FFM method. Conclusion:Overall, by offering a mathematical model that can be used to replicate and examine the behavior of disease models, this research advances our understanding of the course and recurrence of disease. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies from our justified outcomes.

Dynamic Fractional-Order Grey Prediction Model with GWO and MLP for Forecasting Overseas Talent Mobility in China

International students play a crucial role in China’s talent development strategy. Thus, predicting overseas talent mobility is essential for formulating scientifically reasonable talent introduction policies, optimizing talent cultivation systems, and fostering international talent cooperation. In this study, we proposed a novel fractional-order grey model based on the Multi-Layer Perceptron (MLP) and Grey Wolf Optimizer (GWO) algorithm to forecast the movement of overseas talent, namely MGDFGM(1,1). Compared to the traditional grey model FGM(1,1), which utilizes the same fractional order at all time points, the proposed MGDFGM(1,1) model dynamically adjusts the fractional-order values based on the time point. This dynamic adjustment enables our model to better capture the changing trends in the data, thereby enhancing the model’s fitting capability. To validate the effectiveness of the MGDFGM(1,1) model, we primarily utilize Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE) as the evaluation criteria for the prediction accuracy, as well as standard deviation (STD) as an indicator of the model stability. Furthermore, we perform experimental analysis to evaluate the predictive performance of the MGDFGM(1,1) model in comparison to NAÏVE, ARIMA, GM(1,1), FGM(1,1), LSSVR, MLP, and LSTM. The research findings demonstrate that the MGDFGM(1,1) model achieves a remarkably high level of prediction accuracy and stability for forecasting overseas talent mobility in China. The implications of this study offer valuable insights and assistance to government departments involved in overseas talent management.

On the [formula omitted]-Caputo fractional Bloch equations and analytical solutions

Bloch equations (BEs) are widely applied in physics, chemistry, magnetic resonance imaging (MRI), and nuclear magnetic resonance (NMR); these equations determine the dynamic equivalence among externally applied magnetic fields and internal model relaxation times. In this article, we generalize the fractional Bloch equations (FBEs) by using a fractional derivative of a function with respect to another function (Ψ-Caputo derivative) and obtain Ψ-Caputo FBEs; then, we use the generalized Laplace transform method (GLTM) for solving the FBEs analytically. We compared the analytical solutions of the Ψ-Caputo FBEs with several functions of Ψ(t) and different values of fractional orders. According to the results, we have shown that the presented method is efficient.

Solving Linear and Nonlinear Fuzzy Fractional Volterra-Fredholm Integro Differential Equations Using Shehu Adomian Decomposition Method

Objectives: In applied sciences and engineering, fuzzy fractional differential equations (FFDEs) and fuzzy fractional integral equations (FFIEs) are a crucial topic. The main objective of this work is to discover an analytical approximate solution for the fuzzy fractional Volterra-Fredholm integro differential equations (FFVFIDE). In the Caputo concept, fractional derivatives are regarded. Methods: The Shehu transform is challenging to exist for nonlinear problems. So, the Shehu transform is combined with the Adomian decomposition method is called the Shehu Adomian decomposition method (SHADM) and has been proposed to solve both linear and nonlinear FFVFIDEs. Findings: Both linear and nonlinear FFVIFIDEs can be solved using this technique. For nonlinear terms, Adomian polynomials have been used. The main benefit of this approach is that it converges quickly to the exact solution. Figures and numerical examples demonstrate the expertise of the suggested approach. Novelty: The comparison between the exact solution and numerical solution is shown in figures for various values of fractional order . The numerical evolution demonstrates the efficiency and reliability of the proposed SHADM. The proposed approach is rapid, exact, and simple to apply and produce excellent outcomes. Keywords: Fractional calculus, fuzzy number, Mittag ­ Leffler function, Shehu Adomian decomposition method, fuzzy fractional Volterra­-Fredholm integro differential equation

On theoretical and numerical analysis of fractal--fractional non-linear hybrid differential equations

Abstract Recently, fractals and fractional calculus have received much attention from researchers of various fields of science and engineering. Because the said area has been found applicable in modeling various real-world processes and phenomena. Hybrid differential equations (HDEs) play significant roles in mathematical modeling of various processes because the aforesaid equations incorporate different dynamical systems as specific cases. For instance, it is possible to model and describe non-homogeneous physical phenomena on using the said equations. Therefore, this research work is concerned with studying a class of nonlinear hybrid fractal–fractional differential equations. We develop the existence result for the qualitative study using a hybrid fixed point theorem. For the mentioned goal, a fixed point theory for the product of two operators is applied to deduce appropriate conditions for the existence of exactly one solution. Additionally, the stability result based on Ulam–Hyers is also deduced. The said stability results play an important role in numerical investigations. In addition, a numerical method based on Euler procedure is utilized to approximate the solution of the proposed problems. Various computational test problems are given to demonstrate the results. Also, using various fractal–fractional order values, several graphical presentations are given for the examples. The concerned analysis will help in investigating many real-world problems modeled using HDEs with fractal–fractional orders in the near future.

Noval soliton solution, sensitivity and stability analysis to the fractional gKdV-ZK equation

This work examines the fractional generalized Korteweg-de-Vries-Zakharov-Kuznetsov equation (gKdV-ZKe) by utilizing three well-known analytical methods, the modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\frac{G^{'}}{G^2}\\right)$$\\end{document}-expansion method, 1G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( \\frac{1}{G^{'}}\\right)$$\\end{document}-expansion method and the Kudryashov method. The gKdV-ZK equation is a nonlinear model describing the influence of magnetic field on weak ion-acoustic waves in plasma made up of cool and hot electrons. The kink, singular, anti-kink, periodic, and bright soliton solutions are observed. The effect of the fractional parameter on wave shapes have been analyzed by displaying various graphs for fractional-order values of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta$$\\end{document}. In addition, we utilize the Hamiltonian property to observe the stability of the attained solution and Galilean transformation for sensitivity analysis. The suggested methods can also be utilized to evaluate the nonlinear models that are being developed in a variety of scientific and technological fields, such as plasma physics. Findings show the effectiveness simplicity, and generalizability of the chosen computational approach, even when applied to complex models.

A Fractional-Order Mathematical Model of Banana Xanthomonas Wilt Disease Using Caputo Derivatives

This article investigates a fractional-order mathematical model of Banana Xanthomonas Wilt disease while considering control measures using Caputo derivatives. The proposed model is numerically solved using the L1-based predictor-corrector method to explore the model’s dynamics in a particular time range. Stability and error analyses are performed to justify the efficiency of the scheme. The non-local nature of the Caputo fractional derivative, which includes memory effects in the system, is the main motivation for incorporating this derivative in the model. We obtain varieties in the model dynamics while checking various fractional order values.

Usage of the Fuzzy Adomian Decomposition Method for Solving Some Fuzzy Fractional Partial Differential Equations

In this study, we examine the numerical solutions of nonlinear fuzzy fractional partial differential equations under the Caputo derivative utilizing the technique of fuzzy Adomian decomposition. This technique is used as an alternative method for obtaining approximate fuzzy solutions to various types of fractional differential equations and also investigated some new existence and uniqueness results of fuzzy solutions. Some examples are given to support the effectiveness of the proposed technique. We present the numerical results in graphical form for different values of fractional order and uncertainty γ∈0,1.

The fractional soliton solutions: shaping future finances with innovative wave profiles in option pricing system

<p>Financial engineering problems hold considerable significance in the academic realm, where there remains a continued demand for efficient methods to scrutinize and analyze these models. Within this investigation, we delved into a fractional nonlinear coupled system for option pricing and volatility. The model we examined can be conceptualized as a fractional nonlinear coupled wave alternative to the governing system of Black-Scholes option pricing. This introduced a leveraging effect, wherein stock volatility aligns with stock returns. To generate novel solitonic wave structures in the system, the present article introduced a generalized Ricatti mapping method and new Kudryashov method. Graphical representations, both in 3D and 2D formats, were employed to elucidate the system's response to pulse propagation. These visualizations enabled the anticipation of appropriate parameter values that align with the observed data. Furthermore, a comparative analysis of solutions was presented for different fractional order values. Additionally, the article showcases the comparison of wave profiles through 2D graphs. The results of this investigation suggested that the proposed method served as a highly reliable and flexible alternative for problem-solving, preserving the physical attributes inherent in realistic processes. To sum up, the main objective of our work was to conceptualize a fractional nonlinear coupled wave system as an alternative to the Black-Scholes option pricing model and investigate its implications on stock volatility and returns. Additionally, we aimed to apply and analyze methods for generating solitonic wave structures and compare their solutions for different fractional order values.</p>

Modified fractional order social media addiction modeling and sliding mode control considering a professionally operating population

<abstract><p>With the advancement of technology, social media has become an integral part of people's daily lives. This has resulted in the emergence of a new group of individuals known as "professional operation people". These individuals actively engage with social media platforms, taking on roles as content creators, influencers, or professionals utilizing social media for marketing and networking purposes. Therefore, in this article, we designed a six-dimensional fractional-order social media addiction model (FOSMA) in the sense of Caputo, which took into account the professional operations population. Initially, we established the positivity and boundedness of the FOSMA model. After that, the basic regeneration number and the equilibrium points (no addiction equilibrium point and addiction equilibrium point) were computed. Then, the local asymptotic stability of the equilibrium points were proved. In order to investigate the bifurcation behavior of the model when $ R_0 = 1, $ we extended the Sotomayor theorem from integer-order to fractional-order systems. Next, by the frequency analysis method, we converted the fractional order model into an equivalent partial differential system. The tanh function was introduced into the scheme of sliding mode surface. The elimination of addiction was achieved by the action of the fractional order sliding mode control law. Finally, simulation results showed that fractional order values, nonlinear transmission rates, and specialized operating populations had a significant impact on predicting and controlling addiction. The fractional-order sliding mode control we designed played an important role in eliminating chatter, controlling addiction, and ensuring long-term effectiveness. The results of this paper have far-reaching implications for future work on modeling and control of fractional-order systems in different scenarios, such as epidemic spread, ecosystem stabilization, and game addiction.</p></abstract>

Investigation and application of a classical piecewise hybrid with a fractional derivative for the epidemic model: Dynamical transmission and modeling.

In this research, we developed an epidemic model with a combination of Atangana-Baleanu Caputo derivative and classical operators for the hybrid operator's memory effects, allowing us to observe the dynamics and treatment effects at different time phases of syphilis infection caused by sex. The developed model properties, which take into account linear growth and Lipschitz requirements relating the rate of effects within its many sub-compartments according to the equilibrium points, include positivity, unique solution, exitance, and boundedness in the feasible domain. After conducting sensitivity analysis with various parameters influencing the model for the piecewise fractional operator, the reproductive number R0 for the biological viability of the model is determined. Generalized Ulam-Hyers stability results are employed to preserve global stability. The investigated model thus has a unique solution in the specified subinterval in light of the Banach conclusion, and contraction as a consequence holds for the Atangana-Baleanu Caputo derivative with classical operators. The piecewise model that has been suggested has a maximum of one solution. For numerical solutions, piecewise fractional hybrid operators at various fractional order values are solved using the Newton polynomial interpolation method. A comparison is also made between Caputo operator and the piecewise derivative proposed operator. This work improves our knowledge of the dynamics of syphilis and offers a solid framework for assessing the effectiveness of interventions for planning and making decisions to manage the illness.

Analytical fractional reduced-order model identification method for processes with overdamped and underdamped response

In this paper, an analytical procedure to identify a fractional-first order plus dead time (FFOPDT) model is presented. This procedure is applied to processes with overdamped or underdamped step response, i.e., fractional order values in the range 0.5 ≤ α ≤ 2.0 can be estimated. The required process information is obtained through a simple open-loop step experiment and is based on fitting three arbitrary points (x1-x2-x3%) on the process reaction curve. Some numerical examples are proposed to show the effectiveness and applicability of this procedure for the identification of fractional-order models. The main feature of this approach is that it estimates the FFOPDT model of overdamped and underdamped processes.

Analysis of fractal-fractional Alzheimer's disease mathematical model in sense of Caputo derivative

<abstract> <p>Alzheimer's disease stands as one of the most widespread neurodegenerative conditions associated with aging, giving rise to dementia and posing significant public health challenges. Mathematical models are considered as valuable tools to gain insights into the mechanisms underlying the onset, progression, and potential therapeutic approaches for AD. In this paper, we introduce a mathematical model for AD that employs the fractal fractional operator in the Caputo sense to characterize the temporal dynamics of key cell populations. This model encompasses essential elements, including amyloid-<italic>β</italic> ($\mathbb{ A_\beta }$), neurons, astroglia and microglia. Using the fractal fractional operator, we have established the existence and uniqueness of solutions for the model under consideration, employing Leray-Schaefer's theorem and the Banach fixed-point methods. Utilizing functional techniques, we have analyzed the proposed model stability under the Ulam-Hyers condition. The suggested model has been numerically simulated by using a fractional Adams-Bashforth approach, which involves a two-step Lagrange polynomial. For numerical simulations, different ranges of fractional order values and fractal dimensions are considered. This new fractal fractional operator in the form of the Caputo derivative was determined to yield better results than an ordinary integer order. Various outcomes are shown graphically by for different fractal dimensions and arbitrary orders.</p> </abstract>

Analysis and chaotic behavior of a fish farming model with singular and non-singular kernel

Fish farming, a potential activity for the production of protein with high nutritional value, has emerged as a substantial source of income in the modern world. The degradation of natural materials uses oxygen to reduce them below the life-support stage. A huge area with an excessive amount of water swamp for oxygen exchange makes for the ideal ecosphere for fish farming, yet the mussel population may grow swiftly in a really modest water swapping farm producing a lot of feed. In this study, we examine the dynamics and chaotic behavior of a fish farm model containing populations of nutrients and mussels with different kernels insight of fractal-fractional operator. We examined the positively invariant area as well as demonstrate positive, bounded solutions of the model. We also show the equilibrium states for the occurrence and extinction of species. We prove the existence and uniqueness of positive solutions through fixed-point theorems and verified the Ulam–Hyers stability for proposed model. To study the impact of the fractional operator through computational simulations, results are generated employing a two-step Lagrange polynomial in the generalized version for the various kernels, which include the power-law and Mittag–Leffler kernel. We develop some numerical simulations to test our theoretical conclusions with chaotic behavior at different fractional-order values.

A neural network computational structure for the fractional order breast cancer model

The current study provides the numerical performances of the fractional kind of breast cancer (FKBC) model, which are based on five different classes including cancer stem cells, healthy cells, tumor cells, excess estrogen, and immune cells. The motive to introduce the fractional order derivatives is to present more precise solutions as compared to integer order. A stochastic computing reliable scheme based on the Levenberg Marquardt backpropagation neural networks (LMBNNS) is proposed to solve three different cases of the fractional order values of the FKBC model. A designed dataset is constructed by using the Adam solver in order to reduce the mean square error by taking the data performances as 9% for both testing and validation, while 82% is used for training. The correctness of the solver is approved through the negligible absolute error and matching of the solutions for each model’s case. To validates the accuracy, and consistency of the solver, the performances based on the error histogram, transition state, and regression for solving the FKBC model.