Atangana-Baleanu Operator Research Articles

Numerical Solution of Fractional Model of HIV-1 Infection in Framework of Different Fractional Derivatives

In this paper, we have extended the model of HIV-1 infection to the fractional mathematical model using Caputo-Fabrizio and Atangana-Baleanu fractional derivative operators. A detailed proof for the existence and the uniqueness of the solution of fractional mathematical model of HIV-1 infection in Atangana-Baleanu sense is presented. Numerical approach is used to find and study the behavior of the solution of the stated model using different derivative operators, and the graphical comparison between the solutions obtained for the Caputo-Fabrizio and the Atangana-Baleanu operator is presented to see which fractional derivative operator is more efficient.

An efficient approach for fractional nonlinear chaotic model with Mittag-Leffler law

In this work, we exemplify the behaviour of the nonlinear model of arbitrary order differential equations by adopting q-homotopy analysis transform method (q-HATM). In the present study, the illustrated scheme is a graceful amalgamation of Laplace transform with q-homotopy analysis algorithm and we considered arbitrary order derivative using Atangana-Baleanu (AB) operator. The suggested nonlinear system exhibits chaotic behaviour in nature with respect to considered initial conditions. Fixed point hypothesis heard present the existence and uniqueness for the attained solution. We exemplified suggested arbitrary order system with to illustrate and confirm the efficiency of the projected solution procedure. Further, the numerical simulation is illustrated and also the chaotic behaviour of the obtained result captured with respect to arbitrary order in terms of plots. The obtained results confirm the projected scheme is highly methodical, easy to implement and very powerful to exemplify the nature of the dynamical system of arbitrary order.

Study of HIV Disease and Its Association with Immune Cells under Nonsingular and Nonlocal Fractal‐Fractional Operator

HIV, like many other infections, is a severe and lethal infection. Fractal‐fractional operators are frequently used in modeling numerous physical processes in the current decade. These operators provide better dynamics of a mathematical model because these are the generalization of integer and fractional‐order operators. This paper aims to study the dynamics of the HIV model during primary infection by fractal‐fractional Atangana–Baleanu (AB) operators. The sufficient conditions for the existence and uniqueness of the solution of the proposed model under the AB operator are derived via fixed point theory. The numerical scheme is presented by using the Adams–Bashforth method. Numerical results are demonstrated for different fractal and fractional orders to see the effect of fractional order and fractal dimension on the dynamics of HIV and CD4+ T‐cells during primary infection.

Analysis and Dynamics of Fractional Order Mathematical Model of COVID-19 in Nigeria Using Atangana-Baleanu Operator

We propose a mathematical model of the coronavirus disease 2019 (COVID-19) to investigate the transmission and control mechanism of the disease in the community of Nigeria. Using stability theory of differential equations, the qualitative behavior of model is studied. The pandemic indicator represented by basic reproductive number R0 is obtained from the largest eigenvalue of the next-generation matrix. Local as well as global asymptotic stability conditions for the disease-free and pandemic equilibrium are obtained which determines the conditions to stabilize the exponential spread of the disease. Further, we examined this model by using Atangana–Baleanu fractional derivative operator and existence criteria of solution for the operator is established. We consider the data of reported infection cases from April 1, 2020, till April 30, 2020, and parameterized the model. We have used one of the reliable and efficient method known as iterative Laplace transform to obtain numerical simulations. The impacts of various biological parameters on transmission dynamics of COVID-19 is examined. These results are based on different values of the fractional parameter and serve as a control parameter to identify the significant strategies for the control of the disease. In the end, the obtained results are demonstrated graphically to justify our theoretical findings.

Mathematical modeling for the outbreak of the coronavirus (COVID-19) under fractional nonlocal operator

A mathematical model for the spread of the COVID-19 disease based on a fractional Atangana–Baleanu operator is studied. Some fixed point theorems and generalized Gronwall inequality through the AB fractional integral are applied to obtain the existence and stability results. The fractional Adams–Bashforth is used to discuss the corresponding numerical results. A numerical simulation is presented to show the behavior of the approximate solution in terms of graphs of the spread of COVID-19 in the Chinese city of Wuhan. We simulate our table for the data of Wuhan from February 15, 2020 to April 25, 2020 for 70 days. Finally, we present a debate about the followed simulation in characterizing how the transmission dynamics of infection can take place in society.

Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

The dynamics of dengue infection through fractal-fractional operator with real statistical data

The purpose of this work is to analyze the dynamics of dengue fever in newly introduced operator known as fractal-fractional Atangana-Baleanu. Initially, we formulate a new dengue model with hospitalization class of notified infected cases and present the model basic results. Stability results for the model are shown when R0<1. We estimate the parameters for the model considering the infected cases occur in East Java Indonesia for the year 2018. We estimated that the basic reproduction number for this model is approximately equal to R0≈1.8463. We show the application of the fractal-fractional Atangana-Baleanu operator to the dengue model and present its numerical solution with a new method. We show the fractal-fractional model versus model fitting for different values of fractal and fractional orders and suggest that the fractal-fractional model provides better fitting to the infected cases of dengue infection. Some numerical results for different orders of fractal and fractional are shown for the validity and the applicability of the newly introduced operator.

A reliable analytical technique for fractional Caudrey-Dodd-Gibbon equation with Mittag-Leffler kernel

AbstractThe pivotal aim of the present work is to find the solution for fractional Caudrey-Dodd-Gibbon (CDG) equation using q-homotopy analysis transform method (q-HATM). The considered technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the projected fractional-order model. In order to illustrate and validate the efficiency of the future technique, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of q-HATM solutions have been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The obtained results elucidate that, the considered algorithm is easy to implement, highly methodical as well as accurate and very effective to examine the nature of nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering.

New approach for the model describing the deathly disease in pregnant women using Mittag-Leffler function

Abstract In this paper, numerical solution of the mathematical model describing the deathly disease in pregnant women with fractional order is investigated with the help of q-homotopy analysis transform method (q-HATM). This sophisticated and important model is consisted of a system of four equations, which illustrate a deathly disease spreading pregnant women called Lassa hemorrhagic fever disease. The fixed point theorem is considered so as to demonstrate the existence and uniqueness of the obtained numerical solution for the governing fractional model. The proposed method is also included the Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. In order to illustrate and validate the efficiency of the future technique, the projected model in the sense of fractional order is also considered. Moreover, the physical behaviors of the obtained numerical results are presented in terms of simulations for diverse fractional order.

A novel approach for nonlinear equations occurs in ion acoustic waves in plasma with Mittag-Leffler law

Purpose The purpose of this paper is to find the solution for special cases of regular-long wave equations with fractional order using q-homotopy analysis transform method (q-HATM). Design/methodology/approach The proposed technique (q-HATM) is the graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme and fractional derivative defined with Atangana-Baleanu (AB) operator. Findings The fixed point hypothesis considered to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional-order model. To illustrate and validate the efficiency of the future technique, the authors analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order. Originality/value To illustrate and validate the efficiency of the future technique, we analysed the projected nonlinear equations in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order. The obtained results elucidate that, the proposed algorithm is easy to implement, highly methodical, as well as accurate and very effective to analyse the behaviour of nonlinear differential equations of fractional order arisen in the connected areas of science and engineering.

Non integer order, state space model of heat transfer process using Atangana-Baleanu operator

Non integer order, state space model of heat transfer process using Atangana-Baleanu operator

Solution for fractional generalized Zakharov equations with Mittag-Leffler function

Abstract The pivotal aim of the present work is to find the solution for fractional generalized Zakharov (FGZ) equations using q -homotopy analysis transform method ( q -HATM). The proposed technique is graceful amalgamations of Laplace transform technique with q -homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to illustrate and validate the efficiency of the proposed technique, we analysed the projected model in terms of fractional order. Moreover, the physical behaviour of the obtained solution has been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The obtained results elucidate that, the considered algorithm is easy to implement, highly methodical as well as accurate and very effective to analyse the behaviour of coupled nonlinear differential equations of arbitrary order arisen in the connected areas of science and engineering.

Fractional-order dynamics of Rift Valley fever in ruminant host with vaccination

Rift Valley fever is the most terrifying animal disease around the globe, which transfers through mosquitoes and caused by a virus. This infection is life-threatening and heavily affects the economic sectors. Therefore, it is valuable to conceptualize the dynamics of Rift Valley fever to understand its transmission pathway to provide better control policies. Here, we construct an epidemic model for Rift Valley fever with vaccination through fractional derivatives. Firstly, we present the proposed Rift Valley fever dynamics in the Caputo framework. The basic knowledge of fractional calculus is used to determine the rudimentary properties of the proposed fractional model, which include positivity, uniqueness, and boundedness of the solutions. We investigate our constructed model of Rift Valley fever for equilibria and determined the basic reproduction number of the system through next-generation technique, indicated by R0. The stability results are established for the infection-fee steady-state of the system. Numerical simulations are conducted and sensitivity analysis of R0 through partial rank correlation coefficient (PRCC) method is carried out to show the importance of different parameters in R0. Then the Rift Valley fever model is analyzed in the Atangana-Baleanu framework, furthermore, we present a numerical scheme for the proposed fractional model to illustrate the solution pathway of the model. We notice that the fractional-order dynamics can explain the complex system of Rift Valley fever infection more precisely and accurately rather than the integer-order dynamics. It is also observed that the Atangana-Baleanu operator provides more accurate results than the Caputo fractional derivative.

A mathematical model of tuberculosis (TB) transmission with children and adults groups: A fractional model

We consider a novel fractional model to investigates the (tuberculosis) TB model dynamics with two age groups of human, that is, the children and the adults. First, we formulate the model and present the basic results associated to the model. Then, using the fractional operators, Caputo and the Atangana-Baleanu and obtain a generalized model. Further, we give a novel numerical approach for the solution of the fractional model and obtain their approximate solution. We show graphical results with various values of the fractional order. A comparison of the two operators are shown graphically. The results obtained through Atangana-Baleanu operator is flexible than that of Caputo derivative. The infection in tuberculosis (TB) infected people decreases fast when decreasing the fractional order.

Fractional order SIR model with generalized incidence rate

The purpose of this work is to present the dynamics of a fractional SIR model with generalized incidence rate using two differential derivatives, that are the Caputo and the AtanganaBaleanu. Firstly, we formulate the proposed model in Caputo sense and carried out the basic mathematical analysis such as positivity, basic reproduction number, local and global dynamics of the disease free and endemic case. The disease free equilibrium is locally and globally asymptotically stable when the basic reproduction number is less than 1. We also show that the SIR fractional model is locally and globally asymptotically stable at endemic state when the basic reproduction number greater than unity. Further, to analyze the dynamics of Caputo model we provide some numerical results by considering various incidence rates and with different fractional order parameters. Then, we reformulate the same model using the Atangana-Baleanu operator having nonsingular and non-local kernel. We prove the existence and uniqueness of the Atangana-Baleanu SIR model using PicardLindelof approach. Furthermore, we present an iterative scheme of the model and provide graphical results for various values of the fractional order α. From the graphical interpretation we conclude that the Atangana-Baleanu derivative is more prominent and provides biologically more feasible results than Caputo operator.

Fractional Approach for Equation Describing the Water Transport in Unsaturated Porous Media With Mittag-Leffler Kernel

In this paper, we find the solution for fractional Richards equation describing the water transport in unsaturated porous media using q-homotopy analysis transform method (q-HATM).The proposed technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to validate and illustrate the efficiency of the future technique, we analysed the projected model in terms of fractional order. Meanwhile, the physical behaviour of the q-HATM solutions have been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The achieved results illuminate that, the future algorithm is easy to implement, highly methodical as well as effective and very accurate to analyse the behaviour of nonlinear differential equations of fractional order arisen in the connected areas of science and engineering.

A powerful approach for fractional Drinfeld–Sokolov–Wilson equation with Mittag-Leffler law

The pivotal aim of the present work is to find the solution for fractional Drinfeld–Sokolov–Wilson equation using q-homotopy analysis transform method (q-HATM). The proposed technique is graceful amalgamations of Laplace transform technique with q-homotopy analysis scheme, and fractional derivative defined with Atangana-Baleanu (AB) operator. The fixed point hypothesis considered in order to demonstrate the existence and uniqueness of the obtained solution for the proposed fractional order model. In order to validate and illustrate the efficiency of the future technique, we analysed the projected model in terms of fractional order. Meanwhile, the physical behaviour of the q-HATM solutions have been captured in terms of plots for diverse fractional order and the numerical simulation is also demonstrated. The achieved results illuminate that, the future algorithm is easy to implement, highly methodical as well as effective and very accurate to analyse the behaviour of coupled nonlinear differential equations of fractional order arisen in the connected areas of science and engineering.

Accuracy estimation of the approximated Atangana-Baleanu operator

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Mittag–Leffler Memory Kernel in Lévy Flights

In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag–Leffler kernel in the context of Atangana–Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana–Baleanu in time derivative, i.e., a A B D t α ρ ( x , t ) = K α , μ ∂ x μ ρ ( x , t ) . We present the exact solution to model and discuss how the Mittag–Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems.

A comparison study of bank data in fractional calculus

Abstract The present paper investigate the dynamics of the bank data through a competition model with real field data for the year 2004–2014. Initially, we formulate a competition model for the bank data and then use different fractional approaches to simulate the model with the real data for many fractional order parameters α. Then, we present a novel approach for each fractional model and provide a graphical illustration with real data. We show that all these fractional approaches have good resemblance to each other and can be used to model such real data case. We prove in general that the results of the fractional approaches utilized here are good for modeling purpose but also we prove the results of the fractional Atangana–Baleanu operator is more accurate and flexible and can be used confidently to modeled such real case problem.