Caputo Fractional Derivative Of Order Research Articles

Mathematical analysis of a within-host dengue virus dynamics model with adaptive immunity using Caputo fractional-order derivatives

AbstractDengue fever poses a significant global health threat, with over 50 million annual infections spanning more than 100 countries. Given the absence of a specific treatment, medical intervention primarily targets symptom alleviation. The present study utilizes a Caputo-type fractional-order derivative operator to investigate and analyze the dynamics of dengue virus spread within a host with adaptive immune responses. The developed model describes and analyzes the dynamics of immune cells, free dengue particles, infected monocytes, and susceptible monocytes in the presence of cytotoxic T-Lymphocytes. A range of analytical methods is employed to probe the fractional-order within-host model. The application of the generalized mean value theorem aids in investigating the model’s solutions, employing positivity and boundedness theory. Furthermore, the Banach fixed-point approach is utilized to establish the existence and uniqueness of solutions. Employing the normalized forward sensitivity approach, the fractional-order system’s response to various model parameters is scrutinized. The study reveals that the dynamics of the viral model are significantly influenced by the transmission rate and parameters representing adaptive immune responses. Numerical simulations underscore the critical role of transmission rates and adaptive immune responses in the model. Additionally, the study examines the impact of memory on the density of susceptible monocytes, infected monocytes, free dengue particles, and immune cells to optimize immune responses. Through simulations, the study illustrates the influence of memory on immune dynamics.

Error analysis of a fully discrete method for time-fractional diffusion equations with a tempered fractional Gaussian noise

We develop and analyze a fully discrete method, in order to solve numerically time-fractional diffusion equations driven by additive tempered fractional Gaussian noise. The stochastic model involves a Caputo fractional derivative of order α∈(0,1). We discuss the regularity results of the solution by using the inversion of Laplace transform, the Wright function and covariance function of stochastic process. A spectral Galerkin method is applied to approximate the stochastic model in space. Time discretization is achieved by using firstly a Riemann–Liouville derivative to reformulate the spatial semi-discrete form, and then applying the Grünwald–Letnikov formula. We derive the error estimates of the discrete methods, based on regularity results of the solution. Finally, extensive numerical experiments are presented to confirm our theoretical analysis.

Fractional order iterative boundary value problem

In the present paper, we establish the existence and uniqueness solution for an iterative differential equation involving caputo fractional derivative of order $1\alpha2$. The existence results is proved by using Schauder fixed point. To prove the extremal solution, we demonstrate some fractional inequalities. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained result.

Advanced medical image encryption techniques using the fractional-order Halvorsen circulant systems: dynamics, control, synchronization and security applications

In this paper, we describe the Halvorsen circulant system (HCS) with a fractional-order Caputo derivative and its qualitative properties. The numerical solution of the fractional order Halvorsen circulant system (FO-HCS) is proposed based on the Adomian decomposition method (ADM). The ADM method is used to solve fractional-order systems. Then, dynamics is analyzed using different methods including Lyapunov exponents, bifurcation diagrams, complexity, and phase diagrams. This paper also investigates the stabilization and synchronization of identical FO-HCS, and stability theory proves adaptive feedback control and synchronization. In addition, using the fractional-order system’s remarkable properties to develop the image encryption technique using the extended fractional sequences. The proposed method uses a keystream generator for high security based on the enhanced fractional Halvorsen circulant chaotic behavior. The simulation results confirm that it can resist various attacks, including statistical analysis, differential attacks, brute-force attacks, known plaintext attacks, and chosen plaintext attacks, with high security, and low computational complexity. Finally, the results of the simulation and its performance prove that it's effective and secure in image data.

INITIAL-BOUNDARY VALUE PROBLEMS TO THE TIME-NONLOCAL DIFFUSION EQUATION

This article investigates a fractional diffusion equation involving Caputo fractional derivative and Riemann-Liouville fractional integral. The equation is supplemented by initial and boundary conditions in the domain defined by the interval by space 0<x<1 and interval by time 0<t<T. The fractional operators are defined rigorously, utilizing the Caputo fractional derivative of order β and the Riemann-Liouville fractional integral of order α, where 0<α<β≤1. The main results include the presentation of well-known properties associated with fractional operators and the establishment of the unique solution to the given problem. The key findings are summarized through a theorem that provides the explicit form of the solution. The solution is expressed as a series involving the two-parameter Mittag-Leffler function and orthonormal eigenfunctions of the Sturm-Liouville operator. The uniqueness of the solution is proven, ensuring that the problem has a single, well-defined solution under specific conditions on the initial function. Furthermore, the article introduces and proves estimates related to the Mittag-Leffler function, providing bounds crucial for the convergence analysis. The convergence of the series is investigated, and conditions for the solution to belong to a specific function space are established. The uniqueness of the solution is demonstrated, emphasizing its singularity within the given problem. Finally, the continuity of the solution in the specified domain is confirmed through the uniform convergence of the series.

Rich phenomenology of the solutions in a fractional Duffing equation

In this paper, we characterize the chaos in the Duffing equation with negative linear stiffness and a fractional damping term given by a Caputo fractional derivative of order α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} ranging from 0 to 2. We use two different numerical methods to compute the solutions, one of them new. We discriminate between regular and chaotic solutions by means of the attractor in the phase space and the values of the Lyapunov Characteristic Exponents. For this, we have extended a linear approximation method to this equation. The system is very rich with distinct behaviours. In the limits α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} to 0 or α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} to 2, the system tends to basically the same undamped system with a behaviour clearly different from the classical Duffing equation.

A novel finite difference scheme for numerical solution of fractional order population growth model

In this paper, we propose a new scheme based on the implicit finite difference method for solving the fractional population growth model (FPGM). We use the well-known L1 finite difference method to approximate the Caputo fractional derivative of order 0 < α ≤ 1, and the linear interpolation to approximate the integral part. We provide a study on the stability and convergence of the scheme. We present the numerical solution of the proposed method and compare it with three other methods to demonstrate its validity and efficiency.

Finite-time stability of equilibrium point of a class of fractional-order nonlinear systems

This paper studies the finite-time stability (FTS) of equilibrium point of a class of fractional-order nonlinear systems (FONSs) described by Caputo fractional-order derivative (CFOD). The definition of finite-time equilibrium point (FTEP) is proposed for the FONSs, and it is proved that the CFOD of the FTEP is not constantly equal to 0. A sufficient and necessary condition on the FTEP is proposed. Based on the asymptotical stability and the method of the integral of unbounded integrand, two sufficient conditions on the FTS of the equilibrium point are provided. Consequently, the existence of finite-time stable equilibrium points in the FONSs is confirmed. Two examples are presented to illustrate the proposed results.

Influence of induced magnetic field and chemically reacting on hydromagnetic Couette flow of Jeffrey fluid in an inclined channel with variable viscosity and convective cooling: A Caputo derivative approach

In this study, the influence of induced magnetic field and chemically reacting on hydromagnetic generalized Couette flow of Jeffrey fluid in an inclined channel through a porous medium with variable viscosity and convective cooling has been investigated using the Caputo fractional order derivative operator. The mathematical formulation used for the hydromagnetic Couette flow of Jeffrey fluid takes into account the effects of viscous dissipation, Soret, and Dufour. The system of nonlinear partial differential equations governing the flow were solved numerically using the explicit finite difference method. The numerical results for the behavior of various physical parameter on the flow variables are obtained and represented graphically. Moreover, effects of the flow parameters on heat and mass transfer rates are obtained and discussed numerically through tabular forms. The graphical findings show that velocity, concentration, induced magnetic field, and thermal field profiles decline with progressively increment of Jeffrey parameter. The velocity, and temperature of the fluid decline with higher values of fluid viscosity parameter. An increase in the chemical reaction parameter recede the concentration field profiles while increase with raises the values of Soret number. Increasing Biot, and Dufour numbers significantly grows the thermal field profiles. Induced magnetic field grows with larger values of fluid viscosity parameter. The findings of this study are important due to its application in magnetohydrodynamics pumps, polymer manufacturing, fins designs, and food processing.

Numerical analysis of COVID-19 model with Caputo fractional order derivative

This paper focuses on the numerical solutions of a six-compartment fractional model with Caputo derivative. In this model, we obtain non-negative and bounded solutions, equilibrium points, and the basic reproduction number and analyze the stability of disease free equilibrium point. The existence and uniqueness of the solution are proven by employing the Picard–Lindelof approach and fixed point theory. The product–integral trapezoidal rule is employed to simulate the system of FODEs (fractional ordinary differential equations). The numerical results are presented in the form of graphs for each compartment. Finally, the sensitivity of the most important parameter (β) and its impact on COVID-19 dynamics and the basic reproduction number are reported.

Dynamical study of fractional order Leslie-Gower model of predator-prey with fear, Allee effect, and inter-species rivalry

Employing an improved Leslie-Gower model, the Allee impact on the predator population and the fear influence on the prey population, a mathematical model of the interaction involving two populations, prey and predator, has been explored in the current work. The influence of the memory effect on the evolving behaviour of the model is integrated using a well-known fractional operator known as the Caputo fractional derivative of order (0, 1]. We were inspired to do this study because further research is desired to fully comprehend the dynamics of the Allee impact, fear, and Caputo fractional order. The innovative aspect of the current work is the consideration of the Caputo fractional derivative, which enhances the consistency domain of the system. To support the model's biological resilience and accuracy, the existence, uniqueness, positivity, and boundedness of the solution are provided. On the origin, axial, and interior, four distinct types of equilibrium points are distinguished along with the prerequisites for their existence. We additionally stated about the Hopf bifurcation of our proposed model at the interior equilibrium point in terms of fractional order and the fear influence as the bifurcation factors. To illustrate how various biological characteristics affect the dynamics of the solutions, numerical simulations are offered.

Mathematical modeling and stability analysis of the novel fractional model in the Caputo derivative operator: A case study

The fundamental goal of this research is to suggest a novel mathematical operator for modeling visceral leishmaniasis, specifically the Caputo fractional-order derivative. By utilizing the Fractional Euler Method, we were able to simulate the dynamics of the fractional visceral leishmaniasis model, evaluate the stability of the equilibrium point, and devise a treatment strategy for the disease. The endemic and disease-free equilibrium points are studied as symmetrical components of the proposed dynamical model, together with their stabilities. It was shown that the fractional calculus model was more accurate in representing the situation under investigation than the classical framework at α = 0.99 and α = 0.98. We provide justification for the usage of fractional models in mathematical modeling by comparing results to real-world data and finding that the new fractional formalism more accurately mimics reality than did the classical framework. Additional research in the future into the fractional model and the impact of vaccinations and medications is necessary to discover the most effective methods of disease control.

Improving passengers’ attitudes toward safety and unreliable train operations: analysis of a mathematical model of fractional order

In this study, we aimed to explore the dynamics of rail passengers’ negative attitudes that can be influenced by safety concerns and unreliable train operations. We mainly formulated and analyzed a mathematical model of fractional order and derived an optimal control problem considering the Caputo fractional order derivative. In the analysis part of the model, we proved that the solutions of the model for the dynamical system are non-negative and bounded, and determined the passengers’ negative attitude-free and negative attitude persistence equilibrium points of the model. Both the local and global stabilities of these equilibrium points were examined. Furthermore, we verified the conditions necessary for the existence of optimal control strategies. We then proceeded to analyze the proposed control strategies, which aim to prevent negative attitudes and improve the attitudes of passengers who have already developed negative attitudes. Finally, we conducted numerical simulations to examine the effects of these control strategies. The results revealed that protecting passengers from developing negative attitudes and improving the attitudes of those who have already developed such attitudes are crucial for improving the overall attitude of railway passengers. These measures can effectively address any negative experiences caused by safety concerns and unreliable train operations.

Modeling and Analysis of Fractional Order Logistic Equation Incorporating Additive Allee Effect

This study investigates the logistic model of a single species incorporating the additive Allee effect using Caputo fractional order differential equations. The Allee effect describes a positive correlation between individual fitness and population density at low densities. Populations subjected to the strong Allee effect can move towards extinction when their population is below a critical level. This study calculates the threshold level of the population suffering from the strong Allee effect. Various published studies are showing that fractional order models are more appropriate for explaining real-world phenomena than ordinary integer-order systems; therefore, this study involves the use of the Caputo fractional order derivative. Single-species models have been extensively used in mathematical biology, such as insect control, optimal biological resource planning, epidemic avoidance and control, and cell growth regulation. This study can help save vulnerable species from extinction and eliminate unwanted species by subjecting them to a strong Allee effect using artificial strategies.

Numerical investigations of the fractional order derivative-based accelerating universe in the modified gravity

In this work, a Liouville–Caputo fractional order (FO) derivative for the mathematical system based on the accelerating universe in the modified gravity (AUMG), i.e. FO-AUMG is proposed to get more accurate solutions. The nonlinear dynamics of the FO-AUMG is classified into five dynamics. The performances of the designed nonlinear FO-AUMG are numerically stimulated with the stochastic procedures of Levenberg–Marquardt backpropagated (LMB) scheme-based neural networks. The statics for FO-AUMS is used for the nonlinear FO-AUMG as 72%, 16% and 12% for training, authorization, and testing. Twenty neurons in hidden layers have been used to approximate the solution of the nonlinear FO-AUMS. The comparison of three different cases of the nonlinear FO-AUMS is performed with dataset generated by Adams method. To validate the uniformity, legitimacy, precision, and competence of LMB-based adaptive neural networks, the outcomes of the state transitions parameters, regression, correlation, error-histogram plots have been exploited.

Stability Analysis and Simulation of Diffusive Vaccinated Models

This paper begins by analyzing the key mathematical properties of diffusive vaccinated models, including existence, uniqueness, positivity, and boundedness. Equilibria are identified, and the basic reproductive number is calculated. The Banach contraction mapping principle is applied to rigorously establish the solution existence and uniqueness. In order to understand the disease’s time transmission, it is important to examine the global stability of the equilibrium points. Disease‐free equilibrium and endemic equilibrium are the two equilibria in this model. Here, we demonstrate that the endemic equilibrium is worldwide asymptotic stable when the basic reproductive number is greater than 1, and the disease‐free equilibrium is globally asymptotic stable whenever the basic reproductive number is less than 1. Moreover, based on the Caputo fractional derivative of order and the implicit Euler’s approximation, we offered an unconditionally stable numerical solution for the resultant system. This work explores the solution of some significant population models of noninteger order using an approach known as the iterative Laplace transform. The proposed methodology is developed by effectively combining Laplace transformation with an iterative procedure. A series form solution that exhibits some convergent behavior towards the precise solution can be attained. It is noted that there is a close contact between the obtained and precise solutions. Moreover, the suggested method can handle a variety of fractional order derivative problems because it involves minimal computations. This information will be helpful in further studies to determine the ideal strategy of action for preventing or stopping the spread disease transmission.

Exponential Time Differencing Scheme for Fractional Plasma Oscillations

Plasma represents one of the four basic states of matter, along with the solid, liquid, and gas states. Plasma oscillations represent a fundamental concept in plasma physics, and their understanding is crucial in various applications. Fractional order Lorentz models for plasma oscillations offer a more flexible framework for describing the complex dynamics and memory effects that are often observed in plasma systems. In this paper, we consider a fractional-order model for plasma oscillations. The model involves Caputo fractional derivative of order between one and two. An exponential time differencing method is developed to approximate the motion of electrons driven by zero, static, and oscillating electric fields.

Mathematical Analysis of Fractional Diabetes Model via an Efficient Computational Technique

Diabetes is referred to a chronic metabolic disease signalized by elevated levels of blood glucose (also known as blood sugar level), which results over time in serious damage to the heart, blood vessels, eyes, kidneys, and nerves in the body. A mathematical assessment of the diabetes model using the Caputo fractional order derivative operator is given in this research paper. The concept of a Caputo fractional order derivative is a novel class of non-integer order derivative that has many applications in real-life scenarios. The proposed model is represented by a set of fractional ordinary differential equations. The authors employed the Sumudu Transform Homotopy Perturbation Method (STHPM) for finding the series solutions of the model being studied. By giving various numerical values to the respective model parameters, graphical analysis is also performed. It is observed in the numerical discussion that a decrease in both fractional order and leads to decrease in the number of diabetic people.

INVESTIGATION OF FRACTIONAL-ORDERED TUMOR-IMMUNE INTERACTION MODEL VIA FRACTIONAL-ORDER DERIVATIVE

Dynamical analysis of system of ordinary differential equations (ODEs) is an interesting topic among researchers and several tools have been discovered since so far to deal with its several other features. There are many built-in functions designed in MATLAB that researchers can use as a tool for the system of ODEs with integer order but in the similar systems have fractional-order derivative is a difficult task. Therefore, the purpose of this work is to use the Caputo fractional-order derivative to qualitatively analyze and then numerically simulate the tumor-immune system interaction model. Moreover, the Schauder fixed point theorem is used to establish the condition for the existence of at least one solution, while the Banach fixed point theorem is utilized to guarantee the existence of a unique solution. Moreover, the stability in the trajectories of considered system achieved with the aid of Ulam–Hyers (UH) stability. In addition, the numerical computations are performed using Variational Iteration Method (VIM) and the visualized results are shown using MATLAB.

Synchronization of Multi-Term Fractional-Order Neural Networks with Switching Parameters via Hybrid Impulsive Control

This paper investigates the hybrid impulsive control synchronization problem of multi-term fractional-order neural networks (MFNNs) with switching parameters. Firstly, a novel MFNN with switching parameters model is introduced by incorporating multi-term Caputo fractional-order derivative to extend the existing framework for fractional-order cases. Then, the relationship between multi-term fractional-order derivative and distributed-order derivative is analyzed, and a synchronization criterion for a class of multi-term fractional-order impulsive switched systems is derived by utilizing the properties of the distributed-order derivative weight function. Furthermore, a hybrid impulsive controller is designed to obtain sufficient conditions for synchronization of MFNNs with switching parameters. To validate the effectiveness of the obtained conclusions, a numerical example is presented to demonstrate the validity of the proposed MFNN model and synchronization criterion.