Fractional-order Derivative Operator 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.

Nonlinear fractional-order differential equations: New closed-form traveling-wave solutions

Abstract The fractional-order differential equations (FO-DEs) faithfully capture both physical and biological phenomena making them useful for describing nature. This work presents the stable and more effective closed-form traveling-wave solutions for the well-known nonlinear space–time fractional-order Burgers equation and Lonngren-wave equation with additional terms using the exp ( − Φ ( ξ ) ) (-\Phi (\xi )) expansion method. The main advantage of this method over other methods is that it provides more accuracy of the FO-DEs with less computational work. The fractional-order derivative operator is the Caputo sense. The transformation is used to reduce the space–time fractional differential equations (FDEs) into a standard ordinary differential equation. By putting the suggested strategy into practice, the new closed-form traveling-wave solutions for various values of parameters were obtained. The generated 3D graphical soliton wave solutions demonstrate the superiority and simplicity of the suggested method for the nonlinear space–time FDEs.

Numerical solution of fractional PDEs through wavelet approach

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1]$$\\end{document} is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof.

Numerical approximation of fractional SEIR epidemic model of measles and smoking model by using Fibonacci wavelets operational matrix approach

In the present article, we have considered two essential models (The epidemic model of measles and the smoking model). Across the globe, the primary cause of health problems is smoking. Measles can be controlled in infectious populations using the mathematical model representing the direct transmission of infectious diseases. The Caputo fractional derivative operator of order α∈[0,1] is used to determine the solution of the system of fractional differential equations in both models. We employed the Fibonacci wavelets operational matrix of integration and developed the Fibonacci wavelet collocation method (FWCM) to solve a nonlinear coupled system of fractional differential equations (NCSFDEs). The proposed approach transforms the given model into a system of algebraic equations. The Newton-Raphson method is considered to extract the unknown coefficient values. Numerical outcomes are obtained to illustrate the simplicity and effectiveness of the proposed method. Graphs and tables show how consistently and effectively the developed strategy works. Mathematical software called Mathematica has been used to perform all calculations. Theorems explain the convergence of this approach.

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.

Quasi-Synchronization of Fractional Multiweighted Coupled Neural Networks via Aperiodic Intermittent Control.

This article investigates the quasi-synchronization for fractional multiweighted coupled neural networks (FMCNNs) with discontinuous activation functions and mismatched parameters. First, under the generalized Caputo fractional-order derivative operator, a novel piecewise fractional differential inequality is established to study the convergence of fractional systems, which significantly extends some related published results. Subsequently, by exploiting the new inequality and Lyapunov stability theory, some sufficient quasi-synchronization conditions of FMCNNs are presented by aperiodic intermittent control. Meanwhile, the exponential convergence rate and synchronization error's bound are given explicitly. Finally, the validity of theoretical analysis is confirmed by numerical examples and simulations.

Transmission dynamics of breast cancer through Caputo Fabrizio fractional derivative operator with real data

<abstract><p>In this paper, we studied the dynamical behavior of various phases of breast cancer using the Caputo Fabrizio (CF) fractional order derivative operator. The Picard-Lindelof (PL) method was used to investigate the existence and uniqueness of the proposed system. Moreover, we investigated the stability of the system in the sense of Ulam Hyers (UH) criteria. In addition, the two-step Adams-Bashforth (AB) technique was employed to simulate our methodology. The fractional model was then simulated using real data, which includes reported breast cancer incidences among females of Saudi Arabia from 2004 to 2016. The real data was used to determine the values of the parameters that were fitted using the least squares method. Also, residuals were computed for the integer as well as fractional-order models. Based on the results obtained, the CF model's efficacy rates were greater than those of the existing classical model. Graphical representations were used to illustrate numerical results by examining different choices of fractional order parameters, then the dynamical behavior of several phases of breast cancer was quantified to show how fractional order affects breast cancer behavior and how chemotherapy rate affects breast cancer behavior. We provided graphical results for a breast cancer model with effective parameters, resulting in fewer future incidences in the population of phases Ⅲ and Ⅳ as well as the disease-free state. Chemotherapy often raises the risk of cardiotoxicity, and our proposed model output reflected this. The goal of this study was 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 of patients and provide information of treatment strategies for breast cancer patients.</p></abstract>

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.

Caputo fractional differential variational–hemivariational inequalities involving history-dependent operators: Global error bounds and convergence

In this paper, we introduce and investigate a new differential variational–hemivariational inequality (for brevity, FDVHI) involving history-dependent operators and Caputo fractional order derivative operator. We first propose the new concept of gap functions for the history-dependent variational control system of FDVHI. Furthermore, using the estimate technologies involving Caputo fractional derivatives and history-dependent operators, we establish a RG-function of the Fukushima type (in which RG is a brevity of “regularized gap”) and provide some nice properties of the RG-functions. Then, global error bounds for the FDVHI controlled by the Fukushima RG-function are derived under some suitable assumptions. Finally, by applying penalty methods, we formulate FDVHI to a family penalized problems that are fractional differential variational–hemivariational inequalities without constraints, and establish the convergence result that the solution to the original problem FDVHI can be approached by the solutions of the corresponding penalized problems, as the penalty parameter converges to zero.

MHD Free Convection Flows for Maxwell Fluids over a Porous Plate via Novel Approach of Caputo Fractional Model

The ultimate goal of the article is the analysis of free convective flow of an MHD Maxwell fluid over a porous plate. The study focuses on understanding the dynamics of fluid flow over a moving plate in the presence of a magnetic field, where the magnetic lines of force can either be stationary or in motion along the plate. Further, we will investigate the heat and mass transfer characteristics of the system under specific conditions: constant species and thermal conductivity as functions of time. The study involves a symmetric temperature distribution that provides heat on both sides of the plane. Our analysis includes the study of the model for different instances of plate motion and variations in temperature. The fluid dynamics of the system are mathematically described using a system of fractional-order partial differential equations. To make the model independent of geometric units, dimensionless variables are introduced. Moreover, we employ the concept of fractional-order derivative operators in the sense of Caputo, which introduces a fractional dimension to the equations. Additionally, the integral Laplace transform and numerical algorithms are utilized to solve the problem. Finally, by using graphical analysis the contribution of physical parameters on the fluid dynamics is discussed and valuable findings are documented.

Analyzing the convergence of a semi-numerical-analytical scheme for non-linear fractional PDEs

The purpose of this work is to develop a semi-analytical numerical scheme for solving fractional order non-linear partial differential equations (FOPDEs), particularly inhomogeneous FOPDEs, expressed in terms of the Caputo-Fabrizio fractional order derivative operator. To achieve this goal, we examine several fractional versions of nonlinear model equations from the literature. We then present the proposed scheme, discussing its stability and convergence properties. We show that the proposed scheme is efficient and accurate, and we provide numerical examples to illustrate its performance. Our findings demonstrate that the scheme has significant potential for solving a wide range of complex FOPDEs. Overall, this work contributes to the advancement of numerical techniques for solving fractional order non-linear partial differential equations and lays a foundation for further research in this area.

A fractional-order nonlinear model for a within-host chikungunya virus dynamics with adaptive immunity using Caputo derivative operator

The study of within-host dynamics of viral infection is a vital process of understanding how healthy cells in the body are affected by viral foreign bodies and how body immunity responds to such changes. This paper presents and analyses the dynamics of a within-host Chikungunya virus transmission with adaptive immune responses using a fractional-order derivative operator of the Caputo type. The formulated model describes the interaction of uninfected cells, infected cells and viral particles in the presence of Cytotoxic T-Lymphocytes and antibody immune responses. Several analytical methods are employed to analyse the fractional-order within-host model. Positivity and boundedness theory are used to investigate the properties of solutions of the model via the generalized mean value theorem approach. The existence and uniqueness of solutions are examined using the Banach fixed point method. The normalized forward sensitivity method determines how different model parameters affect the fractional-order system. It was shown that transmission rate and parameters representing adaptive immune responses played a significant role in the Chikungunya virus model’s dynamics. Further, the time-dependent optimal control fractional-order model is analysed to optimize the performance of adaptive immune responses using the optimal control theory approach made popular by Pontryagin’s maximum principle. Simulations are performed to visualize the theoretical results and show the influence of memory on the trajectories of concentrations of uninfected cells, infected cells, viral particles, Cytotoxic T-Lymphocytes and antibodies.

Analysis, modeling and simulation of a fractional-order influenza model

The primary goal of this study is to provide a novel mathematical model for Influenza using the Atangana–Baleanu Caputo fractional-order derivative operator (ABC-Operator) in place of the standard operator. There will be an examination of how the influenza-positive solutions reacts to real-world data. The fractional Euler Method will be utilized to reveal the dynamics of the influenza mathematical model. Both the stability of the disease-free equilibrium and the endemic equilibrium points, two symmetrical extrema of the proposed dynamical model, are examined. It will be shown, using numerical comparisons, that the findings obtained by employing the fractional-order model are considerably more similar to certain actual data than the integer-order model's results. These should shed light on the significance of fractional calculus when confronting epidemic risks.

Dynamics analysis of a predator–prey fractional-order model incorporating predator cannibalism and refuge

In this article, we consider a predator–prey interaction incorporating cannibalism, refuge, and memory effect. To involve the memory effect, we apply Caputo fractional-order derivative operator. We verify the non-negativity, existence, uniqueness, and boundedness of the model solution. We then analyze the local and global stability of the equilibrium points. We also investigate the existence of Hopf bifurcation. The model has four equilibrium points, i.e., the origin point, prey extinction point, predator extinction point, and coexistence point. The origin point is always unstable, while the other equilibrium points are conditionally locally asymptotically stable. The stability of the coexistence point depends on the order of the Caputo derivative, α. The prey extinction point, predator extinction point, and coexistence point are conditionally globally and asymptotically stable. There exists Hopf bifurcation of coexistence point with parameter α. The analytic results of stability properties and Hopf bifurcations are confirmed by numerical simulations.

A novel fractional-order hyperchaotic complex system and its synchronization

A novel fractional-order hyperchaotic complex system is proposed by introducing the Caputo fractional-order derivative operator and a constant term to the complex simplified Lorenz system. The proposed system has different numbers of equilibria for different ranges of parameters. The dynamics of the proposed system is investigated by means of phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The results show abundant dynamical characteristics. Particularly, the phenomena of extreme multistability as well as hidden attractors are discovered. In addition, the complex generalized projective synchronization is implemented between two fractional-order hyperchaotic complex systems with different fractional orders. Based on the fractional Lyapunov stability theorem, the synchronization controllers are designed, and the theoretical results are verified and demonstrated by numerical simulations. It lays the foundation for practical applications of the proposed system.

A Hybrid Approximation Method for Integer-Order Approximate Realization of Fractional-Order Derivative Operators

The use of fractional-order (FO) calculus for the solution of different problems in many fields has increased recently. However, the usage of FO system models in practice brings some difficulties. The FO operator, fractance device, is usually realized via several integer-order approximation methods, which have pros and cons in the aspect of operation frequency, time response and stability region. These methods may not meet all performance expectations. In this regard, author proposes an efficient hybrid integer-order approximation method for FO derivative operator without causing any additional difficulty in realization. The proposed method combines Matsuda and modified stability boundary locus (M-SBL) approximation methods. The advantage of each method is combined in a single hybrid function by considering root mean square error (RMSE) rates for step response. The performance of hybrid transfer function is analyzed in comparison with Matsuda, Oustaloup, continued fraction expansion (CFE) and M-SBL transfer functions for both frequency and time response. Analog realization of the proposed model is performed experimentally via partial fraction expansion method. Analog design is verified via both Multisim simulations and experimental results. The improvements due to the hybrid behavior and the consistency of experimental results with theoretical and simulation results demonstrate the practicality and usefulness of the hybrid model.

A Numerical Confirmation of a Fractional-Order COVID-19 Model’s Efficiency

In the past few years, the world has suffered from an untreated infectious epidemic disease (COVID-19), caused by the so-called coronavirus, which was regarded as one of the most dangerous and viral infections. From this point of view, the major objective of this intended paper is to propose a new mathematical model for the coronavirus pandemic (COVID-19) outbreak by operating the Caputo fractional-order derivative operator instead of the traditional operator. The behavior of the positive solution of COVID-19 with the initial condition will be investigated, and some new studies on the spread of infection from one individual to another will be discussed as well. This would surely deduce some important conclusions in preventing major outbreaks of such disease. The dynamics of the fractional-order COVID-19 mathematical model will be shown graphically using the fractional Euler Method. The results will be compared with some other concluded results obtained by exploring the conventional model and then shedding light on understanding its trends. The symmetrical aspects of the proposed dynamical model are analyzed, such as the disease-free equilibrium point and the endemic equilibrium point coupled with their stabilities. Through performing some numerical comparisons, it will be proved that the results generated from using the fractional-order model are significantly closer to some real data than those of the integer-order model. This would undoubtedly clarify the role of fractional calculus in facing epidemiological hazards.

On a Weak Solution of a Fractional-order Temporal Equation

Several real-world phenomena emerging in engineering and science fields can be described successfully by developing certain models using fractional-order partial differential equations. The exact, analytical, semi-analytical or even numerical solutions for these models should be examined and investigated by distinguishing between their solvablities and non-solvabilities. In this paper, we aim to establish some sufficient conditions for exploring the existence and uniqueness of solution for a class of initial-boundary value problems with Dirichlet condition. The gained results from this research paper are established for the class of fractional-order partial differential equations by a method based on Lax Milgram theorem, which relies in its construction on properties of the symmetric part of the bilinear form. Lax Milgram theorem is deemed as a mathematical scheme that can be used to examine the existence and uniqueness of weak solutions for fractional-order partial differential equations. These equations are formulated here in view of the Caputo fractional-order derivative operator, which its inverse operator is the Riemann-Louville fractional-order integral one. The results of this paper will be supportive for mathematical analyzers and researchers when a fractional-order partial differential equation is handled in terms of finding its exact, analytical, semi-analytical or numerical solution.

A generalized dusty Brinkman type fluid of MHD free convection two phase flow between parallel plates

Fractional order derivative operators are regarded to be very complex mathematical tools for implementing more realistic solutions in a wide range of physics and engineering problems. Due to significance applications of fractional derivatives, the aim of this article is analyzed the combinational effect of the magnetic field and heat transfer on Magneto hydrodynamic, free convective, two-phase flow of generalized Brinkman type dusty fluid among two parallel plates. The buoyant force causes the flow, which is aided by natural convection to transfer heat. Furthermore, the left plate travels at a constant speed and the right plate remains fixed, and all spherically dust particles are homogeneously dispersed among the fluid. The problem is modeled mathematically via set of partial differential equations. The Caputo-Fabrizio fractional derivative is employed to generalize the derived PDE's. A combination of Laplace and finite sine Fourier transformations is utilized to get the closed form solution of the problem. The impacts of many factors on temperature, Brinkman fluid, and dust particle velocity have also been examined, including a Grashof number, magnetic parameter, Reynold number, Peclet number and Dusty fluid parameter. Mathcad-15 is used to plot the graphical findings for the dusty fluid, Brinkman fluid, and temperature profiles. For various embedded parameters, the Brinkman fluid and dusty fluid behave similarly. It is concluded that the fractional Brinkman type dusty fluid is more realistic feature as compared to the classical one. The rate of heat transfer can be enhance with the passage of time.

A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay

Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor’s series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.