Fractional Adams-Bashforth Method Research Articles

Numerical and graphical simulation of the non-linear fractional dynamical system of bone mineralization.

The objective of the present study was to improve our understanding of the complex biological process of bone mineralization by performing mathematical modeling with the Caputo-Fabrizio fractional operator. To obtain a better understanding of Komarova's bone mineralization process, we have thoroughly examined the boundedness, existence, and uniqueness of solutions and stability analysis within this framework. To determine how model parameters affect the behavior of the system, sensitivity analysis was carried out. Furthermore, the fractional Adams-Bashforth method has been used to carry out numerical and graphical simulations. Our work is significant owing to its comparison of fractional- and integer-order models, which provides novel insight into the effectiveness of fractional operators in representing the complex dynamics of bone mineralization.

ADAMS–BASHFORTH NUMERICAL METHOD-BASED SOLUTION OF FRACTIONAL ORDER FINANCIAL CHAOTIC MODEL

A new definition of fractional differentiation of nonlocal and non-singular kernels has recently been developed to overcome the shortcomings of the traditional Riemann–Liouville and Caputo fractional derivatives. In this study, the dynamic behaviors of the fractional financial chaotic model have been investigated. Singular and non-singular kernel fractional derivatives are used to examine the proposed model. To solve the financial chaotic model with nonlocal operators, the fractional Adams–Bashforth method (ABM) is applied based on Lagrange polynomial interpolation (LPI). The existence and uniqueness of the solution of the model can be demonstrated using fixed point theory and nonlinear analysis. Further, the error analysis of the present method and Ulam–Hyers stability of the considered model have also been included. Obtained numerical simulations reveal that the model based on three different fractional derivatives shows various chaotic behaviors that may be useful in a practical sense which may not be observed in the integer case.

APPLICATION OF FIXED POINT THEOREM IN THE SOLUTION OF INTEGRO-DIFFERENTIAL EQUATIONS: A COMPLEX VALUED APPROACH

It's remarkable to note that Complex valued Integro-differential and integral type equations are currently intensifying the attention of appreciable researchers due to their comprehensive applications. Thus, this study is fully devoted to the application part of the complex valued controlled, double controlled metric ∂c. We introduce an extended version of the Fisher and Banach type contraction theorem and present some examples to sustain our results. As part of the main theorem's application, we address a common solution with uncertainty in two different folds as follows: [I] Applying the fractional Adams-Bashforth method to the (1.1) FVIdE. [II] Applying it to the integral type equation (1.2) in the setting of the Extended complex valued metric space.

A Fractional Order SITR Model for Forecasting of Transmission of COVID-19: Sensitivity Statistical Analysis

In this work, we investigate the effects of the contact rate between people on the covid-19 virus transmission through a susceptible-infected-treatment-recovered (SITR) fractional mathematical model. Several strategies are introduced, and the development methodology is constructed up in various cases based on the rate of individual contact, due to confinement and social distancing rules, which can be useful in reducing infection. The existence and uniqueness of the proposed model solution are established, as well as the basic reproduction number. The basic reproduction number has been used to control the dynamics of the fractional SITR model completely, which determines whether or not the infection is extinguished. The global stability of the infection-free balance and endemic equilibrium point of the proposed model has been fully established using the Lyapunov-LaSalle type theorem. Furthermore, a sensitivity analysis is carried out to find out which parameter is the most dominant to affect the disease's endemicity and to see how changes in parameters affect Covid-19's beginning disease transmission. The fractional Adams-Bashforth method is used to compute an iterative solution to the model. Finally, using the model parameter values to explain the importance of the arbitrary fractional-order derivative, the numerical results using MATLAB are presented.

Piecewise Fractional Analysis of Omicron Type Covid-19 Infection

In this article, we investigate the new type of COVID-19 caused by the Omicron virus in the sense of piece-wise fractional derivative. The total interval is divided into two sub-intervals under fractional Caputo and Atangana Baleanu operators respectively. The whole model is divided into six compartments in which the agent of infection from Omicron is included. The proposed piecewise fractional model is tested for fixed points using the different theorems of fixed point theory. The approximate solution is carried out by the technique of the piecewise fractional Adams-Bashforth method. All the agents of the considered problem are tested for graphical representation. The first sub-interval is graphed for the Caputo derivative while the second interval dynamics are checked by Atangana Baleanu fractional operator. The numerical simulation results are established at different fractional orders along with the comparison of integer orders. This consideration will also show the cross-over behavior of the Omicron dynamics in human life and will be essential for its controlling and future prediction on various sub-intervals. The sensitivity of different parameters is also checked graphically. Keywords: Piecewise Fractional Derivative, Fractional mathematical model Omicron; Qualitative analysis; Caputo derivative; Fractional Adams-Bashforth technique.

A fractional–order model with different strains of COVID-19

This study examines the dynamics of COVID-19 variants using a Caputo–Fabrizio fractional order model. The reproduction ratio R0 and equilibrium solutions are determined. The purpose of this article is to use a non-integer order derivative in order to present information about the model solutions, uniqueness, and existence using a fixed point theory. A detailed analysis of the existence and uniqueness of the model solution is conducted using fixed point theory. For the computation of the iterative solution of the model, the fractional Adams–Bashforth method is used. Using the estimated values of the model parameters, numerical results are used to support the significance of the fractional-order derivative. The graphs provide useful information about the complexity of the model, and provide reliable information about the model for any case, integer or non-integer. Also, we demonstrate that any variant with the largest basic reproduction ratio will automatically outperform the other variant.

Study of a Fractal-Fractional Smoking Models with Relapse and Harmonic Mean Type Incidence Rate

This manuscript investigates fractal-fractional order smoking models with relapse and harmonic mean type incidence rate under the Caputo derivative. We derive the existence and unique results about the solution for the considered model via fixed point theory. For the stability of the considered system, Ulam-Hyers (UH) approach is used. We compute the numerical solution by using fractional Adams-Bashforth method. For the simulation of the model, we consider different values of fractional order δ and fractal dimension θ by using some real values of the parameters. The proposed scheme is used to simulate the available data for some smoking community including potential, light, and quit smokers. Various graphical presentations are given to understand the dynamics of the model at various fractional orders.

Qualitative and Quantitative study of Zika virus epidemic model under Caputo’s fractional differential operator

The Zika virus is considered to be one of the dangerous virus, which spread in the community through small insect so-called mosquitoes. Some of the other will known methods by which the consider virus may transform are: pregnant mother can pass it to her baby during the pregnancy or birth time, it can also spread through sexual intercourse. Therefore, the world health organization (WHO) declares the Zika infection with clusters of microcephaly and other neurological disorders constitutes a public health emergency of international concern. To understand the dynamics of Zika infectious virus, we investigate the fractional order mathematical model for the transference of Zika infectious disease. The proposed model is divided into seven compartments. Particularly, the author’s used the results of fixed point theory to discuss the existence and uniqueness of the solution and stability analysis for the proposed model. Furthermore, we have calculated the equilibrium points, basic reproduction number and stability of the basic reproduction number. In addition, for the numerical solution of the said model, the author’s used two steps fractional Adam’s Bashforth method (FABm). Finally, numerical simulation and discussion of the graphical representation of the solutions obtained via FABm are given.

Fractional order of pneumococcal pneumonia infection model with Caputo Fabrizio operator

In this study, we present the Pneumococcal Pneumonia infection model using fractional order derivatives in the Caputo-Fabrizio sense. We use fixed-point theory to prove the existence of the solution and investigate the uniqueness of the model variables. The fractional Adams-Bashforth method is used to compute an iterative solution to the model. Finally, using the model parameter values to explain the importance of the arbitrary fractional order derivative, the numerical results are presented.

Fractal-fractional mathematical modeling and forecasting of new cases and deaths of COVID-19 epidemic outbreaks in India

Fractional-order derivative-based modeling is very significant to describe real-world problems with forecasting and analyze the realistic situation of the proposed model. The aim of this work is to predict future trends in the behavior of the COVID-19 epidemic of confirmed cases and deaths in India for October 2020, using the expert modeler model and statistical analysis programs (SPSS version 23 & Eviews version 9). We also generalize a mathematical model based on a fractal fractional operator to investigate the existing outbreak of this disease. Our model describes the diverse transmission passages in the infection dynamics and affirms the role of the environmental reservoir in the transmission and outbreak of this disease. We give an itemized analysis of the proposed model including, the equilibrium points analysis, reproductive number R0, and the positiveness of the model solutions. Besides, the existence, uniqueness, and Ulam-Hyers stability results are investigated of the suggested model via some fixed point technique. The fractional Adams Bashforth method is applied to solve the fractal fractional model. Finally, a brief discussion of the graphical results using the numerical simulation (Matlab version 16) is shown.

Analysis of the fractional tumour-immune-vitamins model with Mittag–Leffler kernel

Recently, Atangana-Baleanu fractional derivative has got much attention of the researchers due to its non-locality and non-singularity. This operator contains an accurate kernel that describes the better dynamics of systems with a memory effect. In this paper, we investigate the fractional-order tumour-immune-vitamin model (TIVM) under Mittag–Leffler derivative. The existence of at least one solution and a unique solution has discussed through fixed point results. We established the Hyres-Ulam stability of the proposed model under the Mittag–Leffler derivative. The fractional Adams–Bashforth method has used to achieve numerical results. Finally, we simulate the obtained numerical results for different fractional orders to show the effect of vitamin intervention on decreased tumour cell growth and cancer risk. At the end of the paper, the conclusion has provided.

A comparative study of spreading of novel corona virus disease by ussing fractional order modified SEIR model

In this research work, a non-linear dynamical modified SEIR model of the recent pandemic, due to Coronavirus-19 disease (COVID-19) for different countries like Malaysia and Pakistan, is considered under nonsingular fractional order derivative. For this model, some qualitative results, existence theory, and numerical solution are studied by using fixed point approach and fractional Adams-Bashforth method. The results are simulated corresponding to some real data of various fractional order by using Matlab. Hence, the suitability of the considered COVID-19 model for the current outbreak in two different countries Malaysia and Pakistan are shown by simulation.

On analysis of the fractional mathematical model of rotavirus epidemic with the effects of breastfeeding and vaccination under Atangana-Baleanu (AB) derivative

Abstract In this article, the fractional-order model of the rotavirus epidemic with the effect of breastfeeding and vaccination under the Atangana-Baleanu derivative is investigated. To show the existence and uniqueness of the model fixed point theorems of Krasnoselskii and Banach are used. The Ulam-Hyres stability is carried out for the proposed model by using the notion of non-linear functional analysis. The numerical scheme is determined for finding the approximate solution of the proposed model by using the fractional Adams-Bashforth method. Numerical simulations are provided for different fractional order to give the transmission dynamics of the rotavirus epidemic.

Existence theory and numerical analysis of three species prey\u2013predator model under Mittag-Leffler power law

In this manuscript, the fractional Atangana–Baleanu–Caputo model of prey and predator is studied theoretically and numerically. The existence and Ulam–Hyers stability results are obtained by applying fixed point theory and nonlinear analysis. The approximation solutions for the considered model are discussed via the fractional Adams Bashforth method. Moreover, the behavior of the solution to the given model is explained by graphical representations through the numerical simulations. The obtained results play an important role in developing the theory of fractional analytical dynamic of many biological systems.

On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative.

The major purpose of the presented study is to analyze and find the solution for the model of nonlinear fractional differential equations (FDEs) describing the deadly and most parlous virus so-called coronavirus (COVID-19). The mathematical model depending of fourteen nonlinear FDEs is presented and the corresponding numerical results are studied by applying the fractional Adams Bashforth (AB) method. Moreover, a recently introduced fractional nonlocal operator known as Atangana-Baleanu (AB) is applied in order to realize more effectively. For the current results, the fixed point theorems of Krasnoselskii and Banach are hired to present the existence, uniqueness as well as stability of the model. For numerical simulations, the behavior of the approximate solution is presented in terms of graphs through various fractional orders. Finally, a brief discussion on conclusion about the simulation is given to describe how the transmission dynamics of infection take place in society.

Modelling and simulation of a dynamical system with the Atangana-Baleanu fractional derivative

In this paper, we model an ecological system consisting of a predator and two preys with the newly derived two-step fractional Adams-Bashforth method via the Atangana-Baleanu derivative in the Caputo sense. We analyze the dynamical system for correct choice of parameter values that are biologically meaningful. The local analysis of the main model is based on the application of qualitative theory for ordinary differential equations. By using the fixed point theorem idea, we establish the existence and uniqueness of the solutions. Convergence results of the new scheme are verified in both space and time. Dynamical wave phenomena of solutions are verified via some numerical results obtained for different values of the fractional index, which have some interesting ecological implications.