Atangana-Baleanu-Caputo Research Articles

Computational analysis for the modeling of fractional time-dependent Maxwell fluid

Through the use of some extended fractional operators, this study models and analyzes time-dependent fractional Maxwell's fluid. The effect of buoyancy is taken into account in the presence of a magnetic field, to make the model as realistic as possible. The nanofluid model is based on the uniform dispersing of nano-size solid particles of silver in an engine taken as base fluid. Atangana Baleanu Caputo (ABC), Caputo Fabrizio (CF), and Caputo (C) fractional derivatives are used in the model formulation (FD). To find the behavior of the modeled problem, a finite difference scheme is suggested. To determine whether the fractional derivative operator was more accurate than the others, a simulation was run. Additionally, graphs are drawn to display the distribution of velocity and temperature as they change the physical parameters. A comparative analysis is also provided to demonstrate that the devised scheme is an effective tool for the topic under consideration and that it may be expanded to address a variety of physical challenges. Qualitative behavior was observed to be almost similar for C, CF and ABC fractional derivatives; however, ABC fractional derivative gives more appropriate and significant results quantitively.

Solving fractional variable-order differential equations of the non-singular derivative using Jacobi operational matrix

This research derives the shifted Jacobi operational matrix (JOM) with respect to fractional derivatives, implemented with the spectral tau method for the numerical solution of the Atangana-Baleanu Caputo (ABC) derivative. The major aspect of this method is that it considerably simplifies problems by reducing them to ones that can be solved by solving a set of algebraic equations. The main advantage of this method is its high robustness and accuracy gained by a small number of Jacobi functions. The suggested approaches are applied in solving non-linear and linear ABC problems according to initial conditions, and the efficiency and applicability of the proposed method are proved by several test examples. A lot of focus is placed on contrasting the numerical outcomes discovered by the new algorithm together with those discovered by previously well-known methods.

Approximate controllability of Atangana-Baleanu fractional neutral delay integrodifferential stochastic systems with nonlocal conditions☆

The study of approximate controllability results ensures the essential conditions required for a solution. Keeping the importance of the study, we initiate the existence and approximate controllability an Atangana-Baleanu-Caputo (ABC)-fractional order neutral delay integrodifferential stochastic systems with nonlocal conditions. For this purpose, the suggested ABC-fractional order neutral integrodifferential stochastic system is transferred into an equivalent fixed point problem via an integral operator. The operator is then analyzed for boundedness, continuity and equicontinuity. Then Arzela-Ascolli theorem ensures the compact multivalued function of the operator and Martelli’s fixed point theorem is utilized for the existence of the mild solution. Based on the fixed point theorem and Dunford-Pettis theorem joined with ζ -resolvent operators, we develop the approximate controllability results. Finally, an example is given to justify the theoretical results. The achieved results reveal that the proposed method is systematic and suitable for dealing with the stochastic fractional problems arising in physics, technology, and engineering in terms of the ABC fractional derivative.

Comparison of third-order fractional partial differential equation based on the fractional operators using the explicit finite difference method

In this research paper, the third-order fractional partial differential equation (FPDE) in the sense of the Caputo fractional derivative and the Atangana-Baleanu Caputo (ABC) fractional derivative is investigated for the first time. The importance of the proposed problem is more general than the third-order linear time-varying systems model. This fractional partial differential equation is converted to the abstract form of the ordinary differential equation at β = 1. A first-order finite difference scheme is created for the converted problem. The stability inequality of the (FDS) is demonstrated for this abstract form problem using Cardano's relation method in Hilbert space. The explicit finite difference method (EFDM) is used to obtain an approximate solution of the third-order fractional partial differential equation (FPDE) based on the Caputo fractional derivative and the (ABC) fractional derivative of order β∈(0,1]. Concerning these equations, the obtained approximate solutions using the proposed method are contrasted with the exact solution. Finally, the simulation for the third-order (FPDE) based on the fractional derivatives of Caputo and (ABC) for both the exact and approximate solutions is given.

Atangana–Baleanu–Caputo differential equations with mixed delay terms and integral boundary conditions

In this research work, we investigate a class of Atangana–Baleanu–Caputo (ABC) differential equations (DEs) having proportional and discrete delay terms accompanied by integral boundary conditions. We study the two important aspects, that is, the existence of the solution and stability analysis for the problem mentioned above. We confirm the existence of at least one solution via Schaefer's fixed‐point theorem and its uniqueness by Banach's fixed‐point theorem. Moreover, for stability analysis, we use the concept of Hyers–Ulam stability. For application purposes, we apply the main results to a numerical problem for specific values of the parameters involved.

Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator

PurposeThis paper aims to study qualitative properties and approximate solutions of a thermostat dynamics system with three-point boundary value conditions involving a nonsingular kernel operator which is called Atangana-Baleanu-Caputo (ABC) derivative for the first time. The results of the existence and uniqueness of the solution for such a system are investigated with minimum hypotheses by employing Banach and Schauder's fixed point theorems. Furthermore, Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias UHR stability and their generalizations are discussed by using some topics concerning the nonlinear functional analysis. An efficiency of Adomian decomposition method (ADM) is established in order to estimate approximate solutions of our problem and convergence theorem is proved. Finally, four examples are exhibited to illustrate the validity of the theoretical and numerical results.Design/methodology/approachThis paper considered theoretical and numerical methodologies.FindingsThis paper contains the following findings: (1) Thermostat fractional dynamics system is studied under ABC operator. (2) Qualitative properties such as existence, uniqueness and Ulam–Hyers–Rassias stability are established by fixed point theorems and nonlinear analysis topics. (3) Approximate solution of the problem is investigated by Adomain decomposition method. (4) Convergence analysis of ADM is proved. (5) Examples are provided to illustrate theoretical and numerical results. (6) Numerical results are compared with exact solution in tables and figures.Originality/valueThe novelty and contributions of this paper is to use a nonsingular kernel operator for the first time in order to study the qualitative properties and approximate solution of a thermostat dynamics system.

Finite-time boundedness and finite-time stabilization boundedness for nonlinear interconnected systems with Atangana–Baleanu–Caputo fractional derivative

In this paper, we consider the finite-time boundedness (FTB) and finite-time stabilization boundedness (FTSB) problems for a class of nonlinear interconnected systems with Atangana–Baleanu–Caputo (ABC) fractional derivative. By employing the FTB theory and some properties of ABC fractional derivative, and some basic mathematical transformations, we derive new conditions for the FTB and FTSB problems of ABC fractional-order nonlinear interconnected systems. These conditions are formed in linear matrix inequalities (LMIs), which can be effectively solved using the MATLAB LMI Control Toolbox. Finally, two numerical examples with simulations are provided to illustrate the efficiency of the proposed method.

Solutions of the mobile–immobile advection–dispersion model based on the fractional operators using the Crank–Nicholson difference scheme

In this manuscript, based on the Atangana–Baleanu Caputo (ABC) fractional derivative and the Caputo fractional derivative for the mobile–immobile advection–dispersion model are considered. For the proposed model, the Crank–Nicholson difference method (C-NDM) scheme is created. Utilizing the Von-Neumann analysis technique, the stability estimate for this difference scheme is demonstrated. The numerical results were obtained for the proposed model using the (C-NDM). The numerical results were produced using the MATLAB program. Then, the obtained approximate solutions for the proposed model through these two fractional derivative operators using the (C-NDM) were compared with the exact solution. The obtained results demonstrate the viability and utility of the proposed technique, and these two fractional derivatives yield a more comparable result. This study differs from previous studies by comparing this model which is based on the Caputo and the Atangana–Baleanu Caputo fractional derivatives using the Crank–Nicholson difference technique.

A New Approach for Solving a New Class of Nonlinear Optimal Control Problems Generated by Atangana-Baleanu- Caputo Variable Order Fractional Derivative and Fractional Volterra-Fredholm Integro-Differential Equations

In the sequel, the numerical solution of a new class of nonlinear optimal control problems (OCPs) generated by Atangana-Baleanu-Caputo (ABC) variable order (V-O) fractional derivative (FD) and fractional Volterra-Fredholm integro-differential equations (FVFIDEs) is found by Bezier curve method (BCM). The main idea behind this work is the use of the BCM. In this technique, the solution is found in the form of a rapid convergent series. Using this method, it is possible to obtain BCM solution of the general form of multipoint boundary value problems. To shown the efficiency of the developed method, numerical results are stated as the main results in this study.

Generalized Defensive Modeling of Malware Propagation in WSNs Using Atangana–Baleanu– Caputo (ABC) Fractional Derivative

The malware spreading in Wireless Sensor Network (WSN) has lately attracted the attention of many researchers as a hot problem in nonlinear systems. WSN is a collection of sensor nodes that communicate with each other wirelessly. These nodes are linked in a decentralised and distributed structure, allowing for efficient data collection and communication. Due to their decentralised architecture and limited resources, WSN is vulnerable to security risks, including malware attacks. Malware can attack sensor nodes, causing them to malfunction and consume more energy. These attacks can spread from one infected node to others in the network, making it essential to protect WSN against malware attacks. In this paper, we focus on the analysis of a novel fractional epidemiology model, specifically the fractional order SEIVR epidemic model in the sense of Caputo’s fractional derivative of order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0{ &lt; {\alpha }{\leq }}1$ </tex-math></inline-formula> with the goal of examining the efficacy of vaccination strategies and the heterogeneity of a scale-free network on epidemic spreading. First, using the next-generation technique and obtain the basic reproduction number of the proposed epidemic model, which is essential for determining both the locally asymptotically stable equilibrium point of the worm-free system and the unique existence of the endemic equilibrium point. To numerically solve the model, the Adam-Bashforth-Moulton predictor-corrector (ABM) method is applied. The fractional calculus enables us to deal directly with the “memory effect” of numerous phenomena, taking into account the system’s dependence on previous stages. This method provides the results of a complex system. Additionally, research demonstrates that vaccine treatments are quite effective at preventing the spread of malware. The outcome of the study reveals that the applied ABM predictor-corrector method is computationally strong and effective to analyse fractional order dynamical systems in the SEIVR epidemic model for malware propagation in WSN. The results show that the order of the fractional derivative has a significant effect on the dynamic process. Also, from the result, it is obvious that the memory effect is zero for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula> = 1. When the fractional order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula> is decreased from 1, the memory effect appears, and its dynamics vary according to the time. This memory effect points out the difference between derivatives of fractional and integer orders. The theorems and their proofs are presented to validate the validity of the proposed model. To validate the proposed model, extensive theoretical study and computational analysis have also been applied.

FRACTIONAL-ORDER SINE-GORDON EQUATION INVOLVING NONSINGULAR DERIVATIVE

The sine-Gordon equation has received attention since 1970s due to the existence of soliton solutions. The aforesaid equation has significant applications in the quantum field theory. The aforementioned problem has been treated by using various numerical and analytical techniques under the ordinary as well as fractional-order derivatives. The mentioned equation has been investigated under the usual Caputo fractional-order derivative. Since in some cases the nonsingular-type derivatives produce more significant results in the mathematical modelings of real-world nonlinear problems, therefore, the proposed problem is considered in this paper under the fractional-order case in the context of Atangana–Baleanu–Caputo (ABC) derivative for the analytical and approximate results. This fractional derivative has some useful properties involving Mittag-Leffler-type kernel that is nonlocal and nonsingular. Furthermore, Modified Homotopy Perturbation Method (MHPM) is utilized for the required approximate solution. We give appropriate examples depicting the sine-Gordon model. Also, we present our results for the approximate solution graphically to support all the results.

EXISTENCE AND STABILITY OF SOLUTION IN BANACH SPACE FOR AN IMPULSIVE SYSTEM INVOLVING ATANGANA–BALEANU AND CAPUTO–FABRIZIO DERIVATIVES

This paper investigates the necessary conditions relating to the existence and uniqueness of solution to impulsive system fractional differential equation with a nonlinear p-Laplacian operator. Our problem is based on two kinds of fractional order derivatives. That is, Atangana–Baleanu–Caputo (ABC) fractional derivative and the Caputo–Fabrizio derivative. To achieve our main aims, we will first convert the proposed impulse system into an integral equation form. Next, we prove the existence and uniqueness of solutions with the help of Leray–Schauder’s theory and the Banach contraction principle. We analyze the operator for continuity, boundedness, and equicontinuity. Further, we investigate the stability solution to the proposed impulsive system by using stability techniques. In the last part, we demonstrate the results via an illustrative example for the application of the results.

ITERATIVE INVESTIGATION OF KORTEWEG–DE VRIES EQUATION USING AB DERIVATIVE IN CAPUTO SENSE

This paper is devoted to investigate Korteweg–De Vries equations (KDVEs) with third order under fractional Atangana–Baleanu–Caputo (ABC) derivative. We use an analytical method due to the Laplace transform and the Adomian decomposition technique. We establish the convergence analysis results for the considered problem by using nonlinear functional analysis. First of all, a general algorithm is developed for the computation of the required analytical approximate solution by using the mentioned method. By applying the developed algorithm, we then investigate the approximate solutions for some examples. The computed solution has also been presented graphically using Matlab. Laplace Adomian decomposition method (LADM) is a powerful full technique that is easy to implement. In addition, the said method has also a fast convergent quality.

Computational modeling of financial crime population dynamics under different fractional operators

&lt;abstract&gt;&lt;p&gt;This paper presents an analysis and numerical simulation of financial crime population dynamics using fractional order calculus and Newton's polynomial. The dynamics of financial crimes are modeled as a fractional-order system, which is then solved using numerical methods based on Newton's polynomial. The results of the simulation provide insights into the behavior of financial crime populations over time, including the stability and convergence of the systems. The study provides a new approach to understanding financial crime populations and has potential applications in developing effective strategies for combating financial crimes. Fractional derivatives are commonly applied in many interdisciplinary fields of science because of its effectiveness in understanding and analyzing complicated phenomena. In this work, a mathematical model for the population dynamics of financial crime with fractional derivatives is reformulated and analyzed. A fractional-order financial crime model using the new Atangana-Baleanu-Caputo (ABC) derivative is introduced. The reproduction number for financial crime is calculated. In addition, the relative significance of model parameters is also determined by sensitivity analysis. The existence and uniqueness of the solution in consideration of the ABC derivative are discussed. A number of conditions are established for the existence and Ulam-Hyers stability of financial crime equilibria. A numerical scheme is presented for the proposed model, starting with the Caputo-Fabrizio fractional derivative, followed by the Caputo and Atangana-Baleanu fractional derivatives. Finally, we solve the models with fractal-fractional derivatives.&lt;/p&gt;&lt;/abstract&gt;

A Detailed Mathematical Analysis of the Vaccination Model for COVID-19

This study aims to structure and evaluate a new COVID-19 model which predicts vaccination effect in the Kingdom of Saudi Arabia (KSA) under Atangana-Baleanu-Caputo (ABC) fractional derivatives. On the statistical aspect, we analyze the collected statistical data of fully vaccinated people from June 01, 2021, to February 15, 2022. Then we apply the Eviews program to find the best model for predicting the vaccination against this pandemic, based on daily series data from February 16, 2022, to April 15, 2022. The results of data analysis show that the appropriate model is autoregressive integrated moving average ARIMA (1, 1, 2), and hence, a forecast about the evolution of theCOVID-19 vaccination in 60 days is presented. The theoretical aspect provides equilibrium points, reproduction number , and biologically feasible region of the proposed model. Also, we obtain the existence and uniqueness results by using the Picard-Lindel method and the iterative scheme with the Laplace transform. On the numerical aspect, we apply the generalized scheme of the Adams-Bashforth technique in order to simulate the fractional model. Moreover, numerical simulations are performed dependent on real data of COVID-19 in KSA to show the plots of the effects of the fractional-order operator with the anticipation that the suggested model approximation will be better than that of the established traditional model. Finally, the concerned numerical simulations are compared with the exact real available date given in the statistical aspect.

Numerical solution of Atangana–Baleanu–Caputo time-space fractional diffusion equation

In this article, the time-space fractional diffusion equation is solved by using the fractional operator in Atangana–Baleanu–Caputo (ABC) sense based on the Mittag-Leffler function involving non-singular and non-local kernels. This study mainly develop the numerical method for time-space linear and non-linear ABC fractional diffusion equation by utilising centred difference approximation with second-order accuracy of Riesz fractional derivative in space and trapezoidal formula for fractional integral approximation for a system of ABC integral equations. Also, we examine the stability and show that the proposed scheme converges at the rate of having space mesh size h and time step size t. Numerical simulations are presented based on the presented method. We finished the article with a conclusion.

Investigation of Novel Piecewise Fractional Mathematical Model for COVID-19

The outbreak of coronavirus (COVID-19) began in Wuhan, China, and spread all around the globe. For analysis of the said outbreak, mathematical formulations are important techniques that are used for the stability and predictions of infectious diseases. In the given article, a novel mathematical system of differential equations is considered under the piecewise fractional operator of Caputo and Atangana–Baleanu. The system is composed of six ordinary differential equations (ODEs) for different agents. The given model investigated the transferring chain by taking non-constant rates of transmission to satisfy the feasibility assumption of the biological environment. There are many mathematical models proposed by many scientists. The existence of a solution along with the uniqueness of a solution in the format of a piecewise Caputo operator is also developed. The numerical technique of the Newton interpolation method is developed for the piecewise subinterval approximate solution for each quantity in the sense of Caputo and Atangana-Baleanu-Caputo (ABC) fractional derivatives. The numerical simulation is drawn against the available data of Pakistan on three different time intervals, and fractional orders converge to the classical integer orders, which again converge to their equilibrium points. The piecewise fractional format in the form of a mathematical model is investigated for the novel COVID-19 model, showing the crossover dynamics. Stability and convergence are achieved on small fractional orders in less time as compared to classical orders.

Theoretical and numerical analysis of COVID-19 pandemic model with non-local and non-singular kernels

The global consequences of Coronavirus (COVID-19) have been evident by several hundreds of demises of human beings; hence such plagues are significantly imperative to predict. For this purpose, the mathematical formulation has been proved to be one of the best tools for the assessment of present circumstances and future predictions. In this article, we propose a fractional epidemic model of coronavirus (COVID-19) with vaccination effects. An arbitrary order model of COVID-19 is analyzed through three different fractional operators namely, Caputo, Atangana-Baleanu-Caputo (ABC), and Caputo-Fabrizio (CF), respectively. The fractional dynamics are composed of the interaction among the human population and the external environmental factors of infected peoples. It gives an extra description of the situation of the epidemic. Both the classical and modern approaches have been tested for the proposed model. The qualitative analysis has been checked through the Banach fixed point theory in the sense of a fractional operator. The stability concept of Hyers-Ulam idea is derived. The Newton interpolation scheme is applied for numerical solutions and by assigning values to different parameters. The numerical works in this research verified the analytical results. Finally, some important conclusions are drawn that might provide further basis for in-depth studies of such epidemics.

Stability analysis for Nabla discrete fractional-order of Glucose–Insulin Regulatory System on diabetes mellitus with Mittag-Leffler kernel

In this paper, we study an existence theory as well as stability results to the new fractional Nabla difference biological model of glucose–insulin regulatory system (GIRS) on diabetes mellitus involving the Atangana Baleanu–Caputo (ABC) derivative. We consider the proposed model under ABC derivative, and it has the nonsingular Mittag-Leffler function as its kernel. We utilize the fixed point technique for the existence and uniqueness analysis. The stability of the concerned solution in Hyers–Ulam sense is also investigated. Further to derive the approximate solution in the form of series to the considered model, we use iterative technique method. Numerical simulations are given to support the theoretical results. The results show that order of the fractional derivative has a significant effect on the dynamic process. Many informations on the dynamics of GIRS in diabetes mellitus were obtained using this model. It is recognized as a deterministic fractional Nabla difference model for diabetes mellitus that provides a better control approach at fractional values for the development of an artificial pancreas.

A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation

Abstract In this article, we present a novel numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ (BBMB) equation using Atangana Baleanu Caputo (ABC) derivative. First, we apply a linearization technique to deal with the generalized non-linear expression, and then the Crank–Nicolson finite difference formula is used in the temporal direction. A reliable numerical technique is applied to discretize the time-fractional ABC derivative, and the central difference formulae are used to approximate the derivatives in the spatial direction. The method is shown unconditionally stable and second-order convergent in both directions through the Fourier analysis. The numerical results of two test problems are analyzed to validate the theoretical results.