Caputo Operator Research Articles

Finite Difference Method for Infection Model of HPV with Cervical Cancer under Caputo Operator

In this paper, a fractional model in the Caputo sense is used to characterize the dynamics of HPV with cervical cancer. Generalized mean value theorem has been used to examine whether the infection model has a unique positive solution. The model has two equilibrium points: the disease-free point and the endemic point. The examination of the system’s local and global stability is provided in terms of the basic reproductive number Rp°. The global stability analysis has been carried out using an appropriate Lyapunov function and the LaSalle invariant principle. The results demonstrate that in the infection model, if Rp°<1, then the solution converges to the disease-free equilibrium, which is both locally and globally asymptotically stable. Whilst Rp°>1, the endemic equilibrium is considered to exist. Simulations are implemented via a finite difference method with Grünwald-Letnikov discretization approach for Caputo derivative operator to define how changes in parameters impact the dynamic behavior of the system using Matlab.

Fractional mathematical modeling of breast cancer stages with true data from Saudi Arabia

This research focuses on the development, analysis, and simulation of fractional mathematical models to investigate the transmission dynamics of different phases of breast cancer. The suggested breast cancer model incorporates three often-used fractional operators in epidemiology: Caputo, Caputo–Fabrizio–Caputo, and Atangana–Baleanu–Caputo operators. In this study, the determination of the equilibrium point and its stability analysis is conducted using the Routh–Hurwitz criterion. Additionally, we examine the existence and uniqueness of solutions for the fractional system using Krasnoselskii’s and Banach fixed-point theory. Moreover, the global stability is discussed via the Ulam–Hyres criterion. Furthermore, the fractional models are being verified using reported occurrences of stage IV breast cancer among females in Saudi Arabia from 2004 to 2016. The real data is used to determine the values of the parameters that are fitted using the least squares error-minimizing methodology. Also, residuals and efficiency rates are computed for the integer as well as fractional-order models. Graphical representations are used to illustrate numerical results by examining different choices of fractional order parameters. Then, the dynamic characteristics of various stages of breast cancer are analyzed to demonstrate the impact of fractional order on breast cancer progression and how the rate of chemotherapy influences its behavior. We provide graphical results for a breast cancer model with effective parameters, resulting in fewer future incidences in the population of stages III and IV. Chemotherapy often raises the risk of cardiotoxicity, and our proposed model output reflects this. The goal of this study is to reduce the incidence of cardiotoxicity in chemotherapy patients while also increasing the pace of patient recovery. This research has the potential to significantly improve outcomes for patients and provide information on treatment strategies for breast cancer patients.

Fractional Analysis of Magnetohydrodynamics Maxwell Flow Over an Inclined Plate with the Effect of Thermal Radiation

This current study provides a comprehensive examination of a novel method for studying the dynamics of a fractionalized Maxwell flow near an inclined plate, considering non-uniform mass transfer through a permeable media. Through the use of partial differential equations, incorporating heat and mass movement effects, the study employs a combination of generalized Fick’s and Fourier’s law with the Caputo operator. Transforming the fractionalized model into dimensionless form using appropriate dimensionless values, semi-analytical solutions for the non-dimensional transmitted fractional model are obtained via the Laplace transformation method. Through graphical analysis, the precise contributions of key parameters such as heat generation, radiation, and chemical reactions are elucidated, including their impacts on the calculated heat generation parameter (Qo), radiation parameter (Nr), and others. The study’s significance lies in its implications for the design of efficient heat exchangers, fluid flow systems, and cooling components in complex engineering systems, including nuclear reactors and power generation plants. Furthermore, the fractional derivative approach offers a more accurate representation of the viscoelastic behavior of materials like polymers, crucial for optimizing fabrication processes such as extrusion and molding. The insights gained from this study extend to the realm of miniaturized fluidic devices, including bio-analysis tools, lab-on-a-chip technology, and microfluidic drug delivery systems, where improved performance and control need a grasp of Maxwell fluid dynamics. The physical outcome of this research lays the groundwork for future investigations that will maximize heat transfer efficiency in real-world systems and give insightful information on the behavior of complicated fluids. We compute and display the skin friction, mass and heat transfer rate in tabular form.

Fractional view analysis of coupled Whitham- Broer-Kaup and Jaulent-Miodek equations

In this study, I systematically investigate the fractional Whitham-Broer-Kaup (WBK) system and the fractional Coupled Jaulent-Miodek (CJM) equation under the Caputo fractional calculus. The nonlinear fractional differential equation systems are investigated via the Aboodh transform iteration method and the Aboodh residual power series method, thus offering a thorough analytical investigation. The dynamics of the fractional WBK system are accomplished using the Aboodh transform iteration method, and the Aboodh residual power series method is employed to study the behavior of the fractional CJM equation. Using established solutions, we completely analyze their dynamics employing both symbolic computations and numerical simulations. Consequently, novel solutions are identified, and the behavior of these fractional systems in the Caputo operator sense are clarified. The results obtained show good agreement of convergence of the numerical and analytical solutions, highlighting the effectiveness of the schemes adopted for unraveling the complex dynamics of fractional nonlinear systems.

Computational and stability analysis of Ebola virus epidemic model with piecewise hybrid fractional operator.

In this manuscript, we developed a nonlinear fractional order Ebola virus with a novel piecewise hybrid technique to observe the dynamical transmission having eight compartments. The existence and uniqueness of a solution of piecewise derivative is treated for a system with Arzel'a-Ascoli and Schauder conditions. We investigate the effects of classical and modified fractional calculus operators, specifically the classical Caputo piecewise operator, on the behavior of the model. A model shows that a completely continuous operator is uniformly continuous, and bounded according to the equilibrium points. The reproductive number R0 is derived for the biological feasibility of the model with sensitivity analysis with different parameters impact on the model. Sensitivity analysis is an essential tool for comprehending how various model parameters affect the spread of illness. Through a methodical manipulation of important parameters and an assessment of their impact on Ro, we are able to learn more about the resiliency and susceptibility of the model. Local stability is established with next Matignon method and global stability is conducted with the Lyapunov function for a feasible solution of the proposed model. In the end, a numerical solution is derived with Newton's polynomial technique for a piecewise Caputo operator through simulations of the compartments at various fractional orders by using real data. Our findings highlight the importance of fractional operators in enhancing the accuracy of the model in capturing the intricate dynamics of the disease. This research contributes to a deeper understanding of Ebola virus dynamics and provides valuable insights for improving disease modeling and public health strategies.

Solving the fractional Fornberg-Whitham equation within Caputo framework using the optimal auxiliary function method

In this work, we solve the fractional-order Fornberg-Whitham (FW) problem in the context of the Caputo operator by using the Optimal Auxiliary Function Method. Tables and figures showing full numerical findings indicate the correctness and efficacy of this strategy. The results provide insights into the solution behavior of the FW equation and demonstrate the applicability of the Optimal Auxiliary Function Method. By giving insight on the behavior of the FW equation in a fractional context, this research advances the use of fractional calculus techniques in the solution of complicated differential equations.

Mathematical insights of social media addiction: fractal-fractional perspectives

The excessive use of social media has become a growing concern in the current century, with dependence on these platforms developing into a complex behavioral addiction. Addressing this issue requires the employment of well-directed and inclusive efforts. In pursuit of continuous development in existent strategies, this article presents a non-linear deterministic mathematical model that encapsulates the dynamics of social media addiction within a population. The proposed model incorporates the fractal-fractional order derivative in the sense of the Caputo operator. The objectives of this research are attained by groping the dynamics of the social media addiction model through the stratification of the population into five compartments: susceptible individuals, exposed individuals, addicted individuals, recovery individuals, and those who have quit using social media. The validity of the devised model is established by proving the existence and uniqueness of the solution within the framework of the fixed-point theory. The Ulam-Hyer’s stability is established through nonlinear functional analysis, perturbing the problem with a small factor. Utilizing the Adam Bashforth numerical scheme, we obtain numerical solutions, which we validate through MATLAB simulations. Additionally, we explore the application of artificial neural networks (ANNs) to approximate solutions, presenting a significant innovation in this domain. We propose the adoption of this novel method for solving integral equations that elucidate the dynamics of social media addiction, surpassing traditional numerical methods. Numerical results are illustrated across various fractional orders and fractal dimensions, with comparisons made against integer orders. Our study indicates that ANN outperforms the Adams-Bashforth algorithm, offering a dependable approach to problem-solving. Throughout the article, we underscore the competitive advantage of our proposed strategy, providing a more nuanced understanding of the complex dynamics outlined in the model.

Ebola virus disease model with a nonlinear incidence rate and density-dependent treatment

This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change. The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment. Firstly, we provide a theoretical study of the nonlinear differential equations model obtained. More precisely, we derive the effective reproduction number and, under suitable conditions, prove the stability of equilibria. Afterwards, we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one. We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models. In fact, when a backward-bifurcation occurs, the disease-free equilibrium may be globally stable. Numerically, we use well-known standard tools to fit the model to the data reported for the 2018–2020 Kivu Ebola outbreak, and perform the sensitivity analysis. To control Ebola epidemics, our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy. Secondly, we propose and analyze a fractional-order Ebola epidemic model, which is an extension of the first model studied. We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme, and show its advantages.

On the approximations to fractional nonlinear damped Burger’s-type equations that arise in fluids and plasmas using Aboodh residual power series and Aboodh transform iteration methods

Damped Burger’s equation describes the characteristics of one-dimensional nonlinear shock waves in the presence of damping effects and is significant in fluid dynamics, plasma physics, and other fields. Due to the potential applications of this equation, thus the objective of this investigation is to solve and analyze the time fractional form of this equation using methods with precise efficiency, high accuracy, ease of application and calculation, and flexibility in dealing with more complicated equations, which are called the Aboodh residual power series method and the Aboodh transform iteration method (ATIM) within the Caputo operator framework. Also, this study intends to further our understanding of the dynamic characteristics of solutions to the Damped Burger’s equation and to assess the effectiveness of the proposed methods in addressing nonlinear fractional partial differential equations. The two proposed methods are highly effective mathematical techniques for studying more complicated nonlinear differential equations. They can produce precise approximate solutions for intricate evolution equations beyond the specific examined equation. In addition to the proposed methods, the fractional derivatives are processed using the Caputo operator. The Caputo operator enhances the representation of fractional derivatives by providing a more accurate portrayal of the underlying physical processes. Based on the proposed two approaches, a set of approximations to damped Burger’s equation are derived. These approximations are discussed graphically and numerically by presenting a set of two- and three-dimensional graphs. In addition, these approximations are analyzed numerically in several tables, including the absolute error for each approximate solution compared to the exact solution for the integer case. Furthermore, the effect of the fractional parameter on the behavior of the derived approximations is examined and discussed.

On a new version of Hermite–Hadamard-type inequality based on proportional Caputo-hybrid operator

In mathematics and the applied sciences, as a very useful tool, fractional calculus is a basic concept. Furthermore, in many areas of mathematics, it is better to use a new hybrid fractional operator, which combines the proportional and Caputo operators. So we concentrate on the proportional Caputo-hybrid operator because of its numerous applications. In this research, we introduce a novel extension of the Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator and establish an identity. Then, taking into account this novel generalized identity, we develop some integral inequalities associated with the left-side of Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator. Moreover, to illustrate the newly established inequalities, we give some examples with the help of graphs.

A comparative analytical investigation for some linear and nonlinear time-fractional partial differential equations in the framework of the Aboodh transformation

This article discusses two simple, complication-free, and effective methods for solving fractional-order linear and nonlinear partial differential equations analytically: the Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). The Caputo operator is utilized to define fractional order derivatives. In these methods, the analytical approximations are derived in series form. We calculate the first terms of the series and then estimate the absolute error resulting from leaving out the remaining terms to ensure the accuracy of the derived approximations and determine the accuracy and efficiency of the suggested methods. The derived approximations are discussed numerically using some values for the relevant parameters to the subject of the study. Useful examples are thought to illustrate the practical application of current approaches. We also examine the fractional order results that converge to the integer order solutions to ensure the accuracy of the derived approximations. Many researchers, particularly those in plasma physics, are anticipated to gain from modeling evolution equations describing nonlinear events in plasma systems.

Modeling and dynamics of measles via fractional differential operator of singular and non-singular kernels

Young children frequently die from measles, which is a major global health concern. Despite being more prevalent in infants, pregnant women, and people with compromised immune systems, it can infect anyone. Novel fractional operators, the constant-proportional Caputo operator, and the constant-proportional Atangana–Baleanu operator are used to create a hybrid fractional order model that helps analyze the dynamic transmission of the measles virus. We assess the measles-free and endemic equilibrium, reproductive number, biological viability, boundedness, well-posedness, and positivity of the model. We apply the Banach contraction principle to verify the uniqueness of the system’s solutions. The proposed system is confirmed to be Ulam–Hyres stable by using fixed point theory results. The aforementioned operators are further analyzed, and the Laplace-Adomian decomposition method is used to numerically simulate the system of fractional differential equations. To support our findings, the outcomes are graphically displayed. The efficacy and memory impact of fractional operators are illustrated through comparisons. Based on fractional parameter values, the study determines important disease-control strategies and shows that vaccinations greatly reduce the spread of measles. By reducing the number of infected people, increasing vaccination coverage lowers the burden of disease on the general population.

Solving Nonlinear Fractional Differential Equations by Using Shehu Transform and Adomian Polynomials

The current article provides a detailed analysis of the solution of non-linear ordinary differential equations of fractional and non-fractional order in series forms using the Shehu transform (ST) and the Adomian decomposition method (ADM), also known as the Shehu transform Adomian decomposition method (STADM). Previously, these methods were used to solve differential equations of integer order as well as a very small number of ordinary and fractional differential equations (FDEs). The Caputo's operator is used in a number of well-known FDEs, including the logistic equation, the Van der Pole equation, and other non-fractional order differential equations like the nonlinear Bratu type equation. It is noted that all of the example issues had series solutions thanks to STADM. Plotting the graph for several series terms of the series solution demonstrates how the approximate solution tends to the closed form solution. In some example problems the impact of α is shown.

Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks

Abstract Due to inherent operating constraints, wireless sensor networks (WSNs) need help assuring network security. This problem is caused by worms entering the networks, which can spread uncontrollably to nearby nodes from a single node infected with computer viruses, worms, trojans, and other malicious software, which can compromise the network’s integrity and functionality. This article discusses a fractional S E 1 E 2 I R {\mathsf{S}}{{\mathsf{E}}}_{1}{{\mathsf{E}}}_{2}{\mathsf{I}}{\mathsf{R}} model to explain worm propagation in WSNs. For capturing the dynamics of the virus, we use the Mittag–Leffler kernel and the Atangana–Baleanu (AB) Caputo operator. Besides other characteristics of the problem, the properties of superposition and Lipschitzness of the AB Caputo derivatives are studied. Standard numerical methods were employed to approximate the Atangana–Baleanu–Caputto fractional derivative, and a detailed analysis is presented. To illustrate our analytical conclusions, we ran numerical simulations.

Soret Effect on MHD Casson Fluid over an Accelerated Plate with the Help of Constant Proportional Caputo Fractional Derivative.

Non-Newtonian fluid flow is significant in engineering and biomedical applications such as thermal exchangers, electrical cooling mechanisms, nuclear reactor cooling, drug delivery, blood flow analysis, and tissue engineering. The Caputo operator has emerged as a prevalent tool in fractional calculus, garnering widespread recognition. This research aims to introduce a novel derivative by merging the proportional and Caputo operators, resulting in the fractional operator known as the constant proportional Caputo. In order to demonstrate this newly defined operator's dynamic qualities, it was employed in the analysis of the unsteady Casson flow model. In addition, the current work shows an analytical analysis to determine the Soret effect on the fractionalized MHD Casson fluid over an oscillating vertical plate. Fractional partial differential equations (PDEs) are used to formulate the problem along with IBCs. The introduction of appropriate nondimensional variables converts the PDEs into dimensionless form. The precise solutions to the fractional governing PDEs are then determined by the Laplace transform method. Velocity, concentration, and temperature profiles; the impacts of the Prandtl number; fractional parameter β and γ; and Soret and Schmidt numbers are graphically depicted. The profiles of temperature, concentration, and velocity rise with rising time and fractional parameters. Interestingly, as the Casson flow parameter is higher, fluid velocity decreases closest to the plate but increases away from the plate. Tables showing the findings for the skin-friction coefficient, Sherwood, and Nusselt numbers for a range of flow-controlling parameter values are provided. Furthermore, an investigation is undertaken to compare fractionalized and ordinary velocity fields. The results suggest that the fractional model employing a constant proportional derivative exhibits a quicker decay than the model incorporating conventional Caputo and Caputo-Fabrizio operators.

On fractional Caputo operator for the generalized glucose supply model via incomplete Aleph function

In this paper, we have attempted a fresh method to demonstrate how special functions and fractional calculus are used in real-world problems. Here, we have examined the glucose supply in human blood using the incomplete Aleph function (IAF) and the Caputo fractional operator. In this study, we used the incomplete Aleph function to find the blood glucose equation that is supplied to human blood. In terms of various hyper-geometric functions, we have also obtained several significant and unique results, and we have defined the blood glucose function in terms of IAF.

Global Stability with Lyapunov Function and Dynamics of SEIR-Modified Lassa Fever Model in Sight Power Law Kernel

Lassa fever is an acute viral hemorrhagic disease that affects humans and is endemic in various West African nations. In this study, a fractional-order model is constructed using the Caputo operator for SEIR-type Lassa fever transmission, including the control strategy. The proposed model examines the dynamics of Lassa fever transmission from rodents to humans and from person to person and in territories with infection in society. The model is analyzed both qualitatively and quantitatively. We examine the positively invariant area and demonstrate positive, bounded solutions to the model. We also show the equilibrium states for the occurrence and extinction of infection. The proposed nonlinear system is verified to be present, and a unique solution is shown to exist using fixed point theorems. Using the Volterra-type Lyapunov function, we investigate the global stability of the suggested system with a fractional Caputo derivative. To study the impact of the fractional operator through computational simulations, results are generated employing a two-step Lagrange polynomial in the generalized version of the power law kernel. A graphical evaluation is provided to show the simplicity and dependability of the model, and all rodents that could be source viruses are important in ecological research. The findings with a value equal to 1 are stronger, according to the comparison of outcomes with different fractional orders. The adverse effect of Lassa fever increases when all modes of transmission are taken into account, according to the study, with fractional-order findings indicating less detrimental effects on specific transmission routes.

Forecasting and dynamical modeling of reversible enzymatic reactions with a hybrid proportional fractional derivative

Chemical kinetics is a branch of chemistry that investigates the rates of chemical reactions and has applications in cosmology, geology, and physiology. In this study, we develop a mathematical model for chemical reactions based on enzyme dynamics and kinetics, which is a two-step substrate–enzyme reversible reaction, applying chemical kinetics-based modeling of enzyme functions. The non-linear differential equations are transformed into fractional-order systems utilizing the constant proportional Caputo–Fabrizio (CPCF) and constant proportional Atangana–Baleanu–Caputo (CPABC) operators. The system of fractional differential equations is simulated using the Laplace–Adomian decomposition method at different fractional orders through simulations and numerical results. Both qualitative and quantitative analyses such as boundedness, positivity, unique solution, and feasible concentration for the proposed model with different hybrid operators are provided. The stability analysis of the proposed scheme is also verified using Picard’s stable condition through the fixed point theorem.

Comparative analysis of classical and Caputo models for COVID-19 spread: vaccination and stability assessment

Several epidemiological models use the Caputo fractional-order differential operator without establishing its significance. This study verifies a Caputo operator-based fractional-order epidemiological model of the SAIVR type. COVID-19 kills. Infection weakens the immune system. The fractional Caputo operator describes COVID-19 immunization. Fundamental system characteristics are determined using fractional calculus. Our analysis included the fractional system’s Hyers–Ulam–Rassias stability and stable states. The uniqueness and existence of fractional Caputo system solutions are explored. The least-squares approach determines system parameters. The Caputo fractional-order α value is optimized to 6.757e−01\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$6.757\ ext{e}{-}01$\\end{document}, indicating that the system best fits real-life medical data for infection. Caputo and classical systems were compared for absolute mean errors. The Box-Whisker chart case summaries show the Caputo operator superiority. When α→1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha \\rightarrow 1$\\end{document}, the memory traces and hereditary traits are also observed. Finally, the Caputo fractional framework simulates COVID-19 using strong numerical methods.

New midpoint-type inequalities in the context of the proportional Caputo-hybrid operator

Fractional calculus is a crucial foundation in mathematics and applied sciences, serving as an extremely valuable tool. Besides, the new hybrid fractional operator, which combines proportional and Caputo operators, offers better applications in numerous fields of mathematics and computer sciences. Due to its wide range of applications, we focus on the proportional Caputo-hybrid operator in this research article. Firstly, we begin by establishing a novel identity for this operator. Then, based on the newfound identity, we establish some integral inequalities that are relevant to the left-hand side of Hermite–Hadamard-type inequalities for the proportional Caputo-hybrid operator. Furthermore, we show how the results improve upon and refine many previous findings in the setting of integral inequalities. Later, we present specific examples together with their related graphs to offer a better understanding of the newly obtained inequalities. Our results not only extend previous studies but also provide valuable viewpoints and methods for tackling a wide range of mathematical and scientific problems.