Atangana-Baleanu Fractional Derivative Research Articles

MHD and porous effects on free convection flow of viscous fluid between vertical parallel plates: advance thermal analysis

We develop a non-local mathematical model in the current study that uses the time-fractional Fourier’s law to describe the thermal phenomena. The time-fractional generalized Atangana–Baleanu fractional derivative is used to define the new time-fractional constitutive equations. We establish analytical solutions for generalized convection flows of viscous fluid between two parallel plates with general boundary conditions by combining the Laplace transform with the finite sine-Fourier transform. The impacts of fractional and physical factors on the thermal and velocity fields were examined using numerical calculations and graphical representations prepared with the Mathcad program. From the generalized Atangana–Baleanu fractional derivative, we can recover the Atangana–Baleanu fractional derivative, Caputo–Fabrizio fractional derivative, and Caputo fractional derivative as a special case.

Modeling and analysis of Buck-Boost converter with non-singular fractional derivatives

Many electrical systems can be characterized more authentically by fractional order dynamic systems. The Caputo–Fabrizio and the Atangana–Baleanu fractional derivatives have solved the singularity problem in the Caputo derivative. This work uses Caputo–Fabrizio and Atangana–Baleanu fractional derivatives to model the fractional order Buck-Boost converter in the time domain. On this basis, the mean values of output voltage and inductor current are calculated. The characteristics of Buck-Boost with different orders in different fractional derivatives are analyzed. The results indicate that the Caputo–Fabrizio and Atangana–Baleanu fractional derivatives can be applied to the Buck-Boost converter to increase the design degree of freedom, which provides more choices for describing the nonlinear characteristics of the system.

New modied Atangana-Baleanu fractional derivative applied to solve nonlinear fractional dierential equations

The main goal of this work is to present a new modified version of the Atangana-Baleanu fractional derivative with Mittag-Leffler non-singular kernel and strong memory. This proposal presents important advantages when specific initial conditions are impossed. The new modified version of the Atangana-Baleanu fractional derivative with Mittag-Leffler non-singular kernel has been constructed considering the fulfillment of the initial conditions with special interest because they are decisive in the obtaintion of analytical and numerical solutions of the fractional differential equations. The advantage of this new fractional derivative in the fulfilling of initial conditions plays a central role for the implementation of different perturbative analytical methods, such as the homotopy perturbation method and the modified homotopy perturbation method. These methods will be applied to solve nonlinear fractional differential equations. This novel modified derivative can be applied in the future in different mathematical modeling areas where satisfy the initial conditions is of great relevance to get more accurate description of real-world problems.

SEMI-ANALYTICAL VIEW OF TIME-FRACTIONAL PDES WITH PROPORTIONAL DELAYS PERTAINING TO INDEX AND MITTAG-LEFFLER MEMORY INTERACTING WITH HYBRID TRANSFORMS

This paper focuses on the theoretical and computational investigation of the innovative nonlinear time-fractional PDEs incorporating the Caputo and Atangana–Baleanu fractional derivatives in the Caputo context using the [Formula: see text]-homotopy analysis transform method (HATM). The expected strategy employs a combination of [Formula: see text]-HATM and the Jafari transform with the assistance of Caputo and Atangana–Baleanu fractional derivative operators to obtain the methodology permits of PDEs with proportional delay. The fractional operators are employed in this research to demonstrate how crucial they are in generalizing frames involving singular and nonsingular kernels. The proposed series of solutions are closely in agreement with an exact solution. Several important challenges can be addressed to illustrate the validity of the proposed method. The outcomes of the proposed framework are displayed and assessed using numerical and graphical outputs. Furthermore, the results of our suggested strategy were compared to earlier outcomes. The proposed method requires less computation and has significantly better performance. Finally, the analysis shows that the enhanced technique is both reliable and meticulous when evaluating the impact of nonlinearities in science and technology.

Numerical study for a class of time fractional diffusion equations using operational matrices based on Hosoya polynomial

<abstract><p>In this paper, we develop a numerical method by using operational matrices based on Hosoya polynomials of simple paths to find the approximate solution of diffusion equations of fractional order with respect to time. This method is applied to certain diffusion equations like time fractional advection-diffusion equations and time fractional Kolmogorov equations. Here we use the Atangana-Baleanu fractional derivative. With the help of this approach we convert these equations to a set of algebraic equations, which is easier to be solved. Also, the error bound is provided. The obtained numerical solutions using the presented method are compared with the exact solutions. The numerical results show that the suggested method is convenient and accurate.</p></abstract>

Enhancing public health strategies for tungiasis: A mathematical approach with fractional derivative

<abstract> <p>In this study, we formulate a mathematical model in the framework of the Atangana-Baleanu fractional derivative in Caputo sense to study the transmission of tungiasis. In this formulation, interactions between the human host and the sand fleas are taken into consideration, including factors like infestation rate, incubation duration, and recovery rate. We calculate the basic reproduction parameter for the system, symbolized by $\mathcal{R}_0$ with the help of the next-generation matrix technique. A novel numerical scheme for encapsulating the non-local and memory-dependent aspects of the system is conceptualized via the Atangana-Baleanu fractional derivative. We prove the existence and uniqueness of the solution of the model of the infection and establish stability of the steady-states of the model. In addition to this, numerical simulations are carried out to evaluate the efficiency of interventions like campaigns for better sanitation and treatment, and to investigate the influence of various management techniques on the prevalence of tungiasis. The outcomes of the numerical simulations give us information about the possible efficacy of different control strategies in lowering the incidence of tungiasis. This research gives quantitative tools to enhance decision-making processes in public health treatments and advances our understanding of the dynamics of the tungiasis.</p> </abstract>

Computational modeling of financial crime population dynamics under different fractional operators

<abstract><p>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.</p></abstract>

Existence and data dependence results for neutral fractional order integro-differential equations

<abstract><p>We assess the multi-derivative nonlinear neutral fractional order integro-differential equations with Atangana-Baleanu fractional derivative of the Riemann-Liouville sense. We discuss results about the existence and difference solution on some data, based on the Prabhakar fractional integral operator $ \varepsilon^{\alpha}_{\delta, \eta, \mathcal{V}; c+} $ with generalized Mittag-Leffler function. The results are obtained by using Krasnoselskii's fixed point theorem and the Gronwall-Bellman inequality.</p></abstract>

Nonlocal impulsive differential equations and inclusions involving Atangana-Baleanu fractional derivative in infinite dimensional spaces

<abstract><p>The aim of this paper is to derive conditions under which the solution set of a non-local impulsive differential inclusions involving Atangana-Baleanu fractional derivative is a nonempty compact set in an infinite dimensional Banach spaces. Existence results for solutions in the presence of instantaneous or non-instantaneous impulsive effect are given. We considered the case where the right hand side is either a single valued function, or a multifunction. This generalizes recent results to the case when there are impulses, the right hand side is a multifunction, and where the dimension of the space is infinite. Examples are given to illustrate the effectiveness of the established results.</p></abstract>

THE IMPACT OF NONSINGULAR MEMORY ON THE MATHEMATICAL MODEL OF HEPATITIS C VIRUS

In this research, we examine the nonsingular memory effect when implementing the Atangana–Baleanu (AB) fractional derivative in the Caputo sense to the Hepatitis C virus (HCV) model. For this purpose, first, the sufficient conditions for the existence and uniqueness of the solutions under the AB fractional derivative of the model are expressed and proved. Therewithal, in order to show the response of the AB derivative on the system dynamics practically, a new version of the predictor–corrector method is used with the newly estimated model parameters from the literature. Finally, we provide some simulations of the results.

Assessment of the performance of the hyperbolic-NILT method to solve fractional differential equations

The performance of the hyperbolic–numerical inverse Laplace transform (hyperbolic-NILT) method is evaluated when it is used to solve time-fractional ordinary and partial differential equations. With this purpose, the formalistic fractionalization approach of Gompertz and diffusion equations are used as model problems, i.e., in the Gompertz and diffusion equations the integer-order time derivative is replaced by the Caputo or Atangana–Baleanu fractional derivative or the Caputo–Fabrizio non-integer order operator. The accuracy, stability and convergence of the numerical solutions are analyzed by comparing the numerical and exact solutions. From our analysis, we obtain an independent formula of the fractional order, which together with the initial condition is used to optimize the parameter of the hyperbolic-NILT method. This expression can be implemented in linear fractional differential equations with non-homogeneous initial condition. Finally, we show that the value of the parameter is transferred throughout the time domain with the certainty that the accuracy of the inverted solution remains between certain orders of magnitude. In fact, everything indicates that this conclusion fits well with model problems that are similar (fractional linear differential equations) to those studied here and for which we highly recommended the hyperbolic-NILT method to solve them.

On the chaotic nature of the Rabinovich system through Caputo and Atangana–Baleanu–Caputo fractional derivatives

The Rabinovich system can describe different physical interactions, including waves in plasmas, a convective fluid flow inside a rotating ellipsoid, and Kolmogorov’s flow interactions. This study considers the Rabinovich system through Caputo and Atangana–Baleanu fractional derivatives to detect its chaotic nature. First, the existence and uniqueness of the solutions of the fractional-order systems are proved using the combination of the Picard–Lindelöf theorem and the Banach contraction principle. Then, a numerical approximation of the fractional systems is developed. The fractional Rabinovich system is found to exhibit a chaotic behavior verified via Lyapunov exponents. However, the fractional-order models do not enter into chaotic behavior at the same fractional-derivative order. Bifurcation diagrams referring to variation of the fractional-order derivatives are provided. Chaotic attractors for both cases of the fractional-derivative representation of the system are depicted. The two fractional-order models of the system show sensitivity to initial conditions. A master–response synchronization was developed in the context of the Atangana–Baleanu fractional derivative. The master and the response systems showed a strong correlation, proving the system’s applicability in solving real problems, including secure communications.

Fractional Analysis of Nonlinear Boussinesq Equation under Atangana–Baleanu–Caputo Operator

This article proposed two novel techniques for solving the fractional-order Boussinesq equation. Several new approximate analytical solutions of the second- and fourth-order time-fractional Boussinesq equation are derived using the Laplace transform and the Atangana–Baleanu fractional derivative operator. We give some graphical and tabular representations of the exact and proposed method results, which strongly agree with each other, to demonstrate the trustworthiness of the suggested methods. In addition, the solutions we obtain by applying the proposed approaches at different fractional orders are compared, confirming that as the value trends from the fractional order to the integer order, the result gets closer to the exact solution. The current technique is interesting, and the basic methodology suggests that it might be used to solve various fractional-order nonlinear partial differential equations.

Numerical simulation of SIR childhood diseases model with fractional Adams–Bashforth method

This work investigates a fractional Susceptible Infected Recovered (SIR) model to study childhood disease. We analyzed the proposed model by applying Caputo, Caputo–Fabrizio, and Atangana–Baleanu fractional derivatives. Here, we use the fractional Adams–Bashforth method to solve the childhood disease model with nonlocal operator. The proposed numerical technique is developed by combining the fundamental theorem of integral calculus with Lagrange's interpolation. This numerical approach is more efficient than the other existing numerical techniques and is easily computed using Matlab as a programming language. The existence and uniqueness of this fractional model are studied with the fixed‐point theorem. These fractional derivatives show different asymptotic behaviors for the distinct values of the fractional order. The numerical results are presented graphically as well as in tabulated form for some value of fractional order.

Thermo-diffusion effect on magnetohydrodynamics flow of fractional Casson fluid with heat generation and first order chemical reaction over a vertical plate

Analytical solution of thermo diffusion effect on magnetohydrodynamics flow of fractionalized Cassonfluid over a vertical plate immersed in a porous media is obtained. Moreover, in the model of the problem, additional effects, like a chemical reaction, heat source/sink, and thermal radiation are also considered.The model is solved by three approaches, namely, Atangana-Baleanu, Caputo-Fabrizio, and Caputo fractionalderivative of non-integer order γ. The governing dimensionless equations for temperatures, concentrations,and velocities are solved using Laplace transform method and compared graphically. The effects of different parameters like fractional parameter γ, Thermo diffusion Sr, and magnetic parameter M are discussedthrough numerous graphs. Furthermore, comparisons among ordinary and fractionalized velocity fields arealso drawn. It is found that the velocity obtained with Atangana-Baleanu fractional derivative is less than thatobtained by Caputo, Caputo-Fabrizio, or ordinary derivatives.

QUALITATIVE ANALYSIS OF IMPLICIT DELAY MITTAG-LEFFLER-TYPE FRACTIONAL DIFFERENTIAL EQUATIONS

This research work is devoted to endeavor some results for a delay implicit impulsive type problem under Atangana–Baleanu fractional derivative. The concerned derivative utilizes a nonlocal and non-singular kernel. We build some hypotheses to prove our results. We use Banach and Krasnoselskii fixed point theorems to derive the required results. We consider the following problem involving nonlocal and non-singular fractional derivative with delay term: [Formula: see text] here [Formula: see text], represent the order of the derivative [Formula: see text] is bounded linear operator and [Formula: see text] shows a nonlinear continuous function. Stability theory of Ulam–Hyers is used to established the stability results. We provide some examples to demonstrate our theoretical findings.

An Efficient Technique for Time-Fractional Water Dynamics Arising in Physical Systems Pertaining to Generalized Fractional Derivative Operators

This study proposed the q ˜ -homotopy analysis transform method ( q ˜ -HATM) as a revolutionary mathematical method for addressing nonlinear time-fractional Boussinesq and approximate long wave dynamics models with the Caputo and Atangana–Baleanu fractional derivatives in the Caputo sense. Through a specific velocity distribution, these models play an essential role in explaining the physics of wave propagation. The q ˜ -HATM is a new improvement to the Elzaki transform (ET) that simplifies the computations. The presented scheme addresses computational complexity by avoiding the use of Adomian and He’s polynomials, which is a distinguishing feature of this innovative methodology over decomposition and the homotopy perturbation transform method. The convergence analysis and error analyses are carried out in the current investigation for the upcoming strategy. We provide illustrations to exemplify the suggested system’s strength and efficacy, and the error estimates are described to ensure reliability. The analytical and graphic illustrations show that the projected methodology is numerically very precise and pragmatic to analyze the solution of fractional associated dynamics that arise in physics and engineering.

Stability Analysis of an Extended SEIR COVID-19 Fractional Model with Vaccination Efficiency.

This work is aimed at presenting a new numerical scheme for COVID-19 epidemic model based on Atangana-Baleanu fractional order derivative in Caputo sense (ABC) to investigate the vaccine efficiency. Our construction of the model is based on the classical SEIR, four compartmental models with an additional compartment V of vaccinated people extending it SEIRV model, for the transmission as well as an effort to cure this infectious disease. The point of disease-free equilibrium is calculated, and the stability analysis of the equilibrium point using the reproduction number is performed. The endemic equilibrium's existence and uniqueness are investigated. For the solution of the nonlinear system presented in the model at different fractional orders, a new numerical scheme based on modified Simpson's 1/3 method is developed. Convergence and stability of the numerical scheme are thoroughly analyzed. We attempted to develop an epidemiological model presenting the COVID-19 dynamics in Italy. The proposed model's dynamics are graphically interpreted to observe the effect of vaccination by altering the vaccination rate.

Univariate and Multivariate Ostrowski-Type Inequalities Using Atangana–Baleanu Caputo Fractional Derivative

In this paper, we obtain some univariate and multivariate Ostrowski-type inequalities using the Atangana–Baleanu fractional derivative in the sense of Liouville–Caputo (ABC). The results obtained for both left and right ABC fractional derivatives can be applied to study further fractional inequalities and estimate various non-local function problems since the operator consists of a non-singular kernel. The obtained results are more generalized in nature.

Analytical Investigation of Nonlinear Fractional Harry Dym and Rosenau-Hyman Equation via a Novel Transform

We use a new integral transform approach to solve the fractional Harry Dym equation and fractional Rosenau-Hyman equation in this work. The Elzaki transform and the integral transformation are combined in the suggested method (ET). To handle two nonlinear problems, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD). The ultimate purpose of this study is to find an error analysis that demonstrates that our final result converges to the exact and approximate result. The convergent series form solution demonstrates the method’s efficiency in resolving several types of fractional differential equations. Furthermore, the solutions obtained in this study agree well with the exact solutions; thus, this strategy is powerful and efficient as an alternate way for obtaining approximate solutions to both linear and nonlinear fractional differential equations.