Atangana-Baleanu Fractional Operators Research Articles

Analysis of Cauchy reaction-diffusion equations involving Atangana-Baleanu fractional operator

This study investigates the Cauchy reaction-diffusion equation (CRDE) with the Atangana-Baleanu differential operator. The existence and uniqueness of solutions to fractional starting value issues are begun using the fixed-point theorem and contraction principle, respectively. The proposed study uses the natural variation iteration technique (NVIM) to get an approximate solution for nonlinear fractional reaction-diffusion equations. This study's approximate answers are compared to other solutions found using known methodologies, and the results are discussed. The devised technique has benefits in terms of accuracy and computational cost efficiency, which may be used to solve nonlinear fractional reaction-diffusion equations.

Gronwall-Type Inequalities and Qualitative Studies onHigher-Variable Orders of Atangana-Baleanu Fractional Operators via Increasing Functions

This paper introduces a novel extension of Caputo-Atangana-Baleanu and Riemann-Atangana-Baleanu fractional derivatives from constant to increasing variable order. We generalize the fractional order from a fixed value in (0, 1] to a time-dependent function in (k, k + 1], where k ≥ 0. The corresponding Atangana-Baleanu fractional integral is also extended. Key properties ofthese new definitions are explored, including a generalized Gronwall inequality. We then delve into the analysis of higher-variable initial fractional differential equations using the Caputo-Atangana-Baleanu operator with an increasing function, establishing existence and uniqueness results via Picard’s iterative method. The findings presented in this work are expected to stimulate further research on inequalities and fractional differential equations related to Atangana-Baleanu fractional calculus with respect to increasing functions. Concrete examples are provided to illustrate the practical applications of our results.

On Efficient Method For Fractional-Order Two-Dimensional Navier-Stokes Equations

For the solution of fractional-order two-dimensional Navier-Stokes equations (FOTDNSEs), the current work proposes a novel variant of the Laplace Adomian decomposition approach with the Atangana-Baleanu fractional operator in the Caputo sense (ABCFO). In this approach, the solution is considered as a Taylor series expansion that converges rapidly to the exact solution. Only two components are required for the new approximation analytical technique. When compared to other strategies, the current method is very simple, requires less calculation, and is extremely accurate.

Study of fractional order rabies transmission model via Atangana–Baleanu derivative

In this work, we aim at disentangling the theoretical contribution through mathematical modeling approach to advance the understanding of rabies dynamics and control in livestock population. A fractional order model of rabies, using Atangana–Baleanu fractional operator is created. The analysis of suggested system and its application are both conducted. The capacity of proposed model to forecast the disease can help researchers and livestock health care agencies to take preventive actions.

Existence and Uniqueness of Solution of Fractional Differential Equation Using Non-local Operator

Objectives : To investigate the existence and uniqueness of a solution to the non-local Cauchy problem of order using the Atangana Baleanu (AB) fractional derivative operator. Methods : Using Schauder's fixed point theorem and the Arzela-Ascoli theorem, the study proved the existence of a solution to the given problem. Further, it obtains results for the uniqueness of the solution. Findings : This study proves the existence of a solution to the non-local Cauchy problem under the given conditions. Results are also provided for the uniqueness of the solution. Novelty : A novel approach to fractional differential equations is represented by applying Atangana-Baleanu fractional derivative operator to the non-local Cauchy problem. Schauder's fixed point theorem and Arzela-Ascoli's theorem, are used to show existence and uniqueness. Further, one detailed example has been solved. Keywords: Existence, Uniqueness, Non-local operator, Schauder fixed point theorem, Arzela-Ascoli theorem, Cauchy problem

The Approximate Solutions of 2D- Burger’s Equations

This paper investigates the use of the fractional variational iteration method (FVIM) to obtain approximate analytical solutions to two dimensional Burger’s Equations with the Atangana-Baleanu fractional operator (ABFO). This study provides insight on the fractional variational iteration method's accuracy and reliability while approximating fractional differential equation solutions.

Magneto-thermoelastic responses in an unbounded porous body with a spherical cavity subjected to laser pulse heating via an Atangana-Baleanu fractional operator

Poro-thermoelasticity models are crucial for analyzing porous materials in geotechnical engineering, such as how soil and rock respond to thermal and mechanical loads, fluid flow, and heat transmission. This study introduces a fractional dual-phase lag (DPL) porous thermoelasticity model using the Atangana-Baleanu fractional derivative to analyze thermoelastic responses in a material with unlimited porosity containing a spherical cavity under a pulsed magnetic field. The governing equations were solved using Laplace transforms, and numerical solutions were obtained by inverting the transforms. Graphical representations of deformation, temperature variation, and thermal stresses were generated for analysis. The results reveal significant insights into the behavior of porous materials like concrete and masonry under thermal and mechanical stress, aiding the development of more robust and flexible structures. For example, the proposed model predicts a 20 % reduction in the amount of thermal stress compared to conventional models, highlighting the influence of phase lag and fractional derivatives in accurately simulating realistic conditions.

Analytical and fractional model for power transmission of lossy transmission line

ABSTRACT A lossy transmission line can draw current from DC source if DC voltage is applied to constant resistance. This manuscript proposes a fractional modeling of lossy transmission line based on the partial differential equations by employing Kirchoff’s current and voltage laws via Fourier analysis. The governing equation of lossy transmission line is fractionalized by means of modern fractional differential operators. The optimal solution of voltage is investigated by means of Fourier sine and Laplace transforms subject to the imposed conditions. The solutions of voltage over the transmission line have been investigated in terms of exponential and gamma functions. The comparative analysis of voltage over the transmission line through Caputo-Fabrizio and Atangana-Baleanu fractional operators has been presented for line losses on the basis of conductance, resistance and inductance for power transmission.

Inequalities for strongly s-convex functions via Atangana-Baleanu fractional integral operators

It is more convenient to use fractional derivatives and integrals to express and represent rapid changes than to use integer derivatives and integrals. For this reason, fractional analysis has been found worthy of study in many fields. In recent years, fractional derivatives and integrals have been discussed together with inequality theory and the studies have attracted attention. In this article, we discuss new Hermite-Hadamard type approximations for strongly convex functions with the help of Atangana-Baleanu fractional integral operators. Additionally, new upper bounds have been obtained using various auxiliary inequalities with the help of twice differentiable strongly convex functions.

Modelling the transmission behavior of measles disease considering contaminated environment through a fractal-fractional Mittag-Leffler kernel

This work utilises a fractal-fractional operator to examine the dynamics of transmission of measles disease. The existence and uniqueness of the measles model have been thoroughly examined in the context of the fixed point theorem, specifically utilising the Atangana-Baleanu fractal and fractional operators. The model has been demonstrated to possess both Hyers-Ulam stability and Hyers-Ulam Rassias stability. Furthermore, a qualitative analysis of the model was performed, including examination of key parameters such as the fundamental reproduction number, the measles-free and measles-present equilibria, and assessment of global stability. This research has shown that the transmission of measles disease is affected by natural phenomena, as changes in the fractal-fractional order lead to changes in the disease dynamics. Furthermore, environmental contamination has been shown to play a significant role in the transmission of the measles disease.

Biomechanical response of skin tissue under ramp-type heating by incorporating a modified bioheat transfer model and the Atangana–Baleanu fractional operator

Biomechanical response of skin tissue under ramp-type heating by incorporating a modified bioheat transfer model and the Atangana–Baleanu fractional operator

Modeling and Analysis of an Age-Structured Malaria Model in the Sense of Atangana–Baleanu Fractional Operators

In this paper, integer- and fractional-order models are discussed to investigate the dynamics of malaria in a human host with a varied age distribution. A system of differential equation model with five human state variables and two mosquito state variables was examined. Preschool-age (0–5) and young-age individuals make up our model’s division of the human population. We investigated the existence of an area in which the model is both mathematically and epidemiologically well posed. According to the findings of our mathematical research, the disease-free equilibrium exists whenever the fundamental reproduction number R0 is smaller than one and is asymptotically stable. The disease-free equilibrium point is unstable when R0>1. We showed that the endemic equilibrium point is unique for R0>1. Also, the most influential control parameters of the spread of malaria were identified. Numerical simulations of both classical and fractional order were conducted, and we used ODE (45) for classical part and numerical technique developed by Toufik and Atangana for fractional order. The infected population will grow because of the high biting frequency of the mosquito and the high likelihood of transmission from the infected mosquito to the susceptible human. R=1.622, which is more than one, indicating that the mosquito vector keeps on growing. This supports the stability of the endemic equilibrium point theorem, which states that the disease becomes endemic when R=1. The susceptible human population will decrease because of the presence of the infective mosquito, which has a high biting frequency for the first couple of days. Since the infective mosquito bit the susceptible human, the susceptible human became infected and went to the infected human compartments. Then, the susceptible population will decrease and the infested human population will increase. After a certain amount of time, it becomes zero due to the growth of protected classes. In this case, a disease-free equilibrium point exists and is stable. This condition exists because R0=2.827×10−5 is less than 1. This supports the theorem that the stability of the disease-free equilibrium point is obtained when R0<1. Depending on equation, we have shown that the possibility of some endemic equilibria exists when R0<1, that is, it undergoes backward bifurcation, even when the disease-free equilibrium is locally stable, and the result means that the society may misunderstand the level of malaria prevalence in the community.

Fractional integral inequalities for preinvex functions via Atangana-Baleanu integral operators

Different types of functions have been used in many various areas of mathematics until today, and today they are used to obtain new equality and inequalities in fractional analysis, which is one of the important areas of mathematics. In this study, we obtained integral inequalities for preinvex functions with the help of Atangana Baleanu fractional integral operator.

Fractional operators on the bounded symmetric domains of the Bergman spaces

<abstract><p>Mathematics has several uses for operators on bounded symmetric domains of Bergman spaces including complex geometry, functional analysis, harmonic analysis and operator theory. They offer instruments for examining the interaction between complex function theory and the underlying domain geometry. Here, we extend the Atangana-Baleanu fractional differential operator acting on a special type of class of analytic functions with the $ m $-fold symmetry characteristic in a bounded symmetric domain (we suggest the open unit disk). We explore the most significant geometric properties, including convexity and star-likeness. The boundedness in the weighted Bergman and the convex Bergman spaces associated with a bounded symmetric domain is investigated. A dual relations exist in these spaces. The subordination and superordination inequalities are presented. Our method is based on Young's convolution inequality.</p></abstract>

Analytical study of a Hepatitis B epidemic model using a discrete generalized nonsingular kernel

<abstract><p>Hepatitis B is a worldwide viral infection that causes cirrhosis, hepatocellular cancer, the need for liver transplantation, and death. This work proposed a mathematical representation of Hepatitis B Virus (HBV) transmission traits emphasizing the significance of applied mathematics in comprehending how the disease spreads. The work used an updated Atangana-Baleanu fractional difference operator to create a fractional-order model of HBV. The qualitative assessment and well-posedness of the mathematical framework were looked at, and the global stability of equilibrium states as measured by the Volterra-type Lyapunov function was summarized. The exact answer was guaranteed to be unique using the Lipschitz condition. Additionally, there were various analyses of this new type of operator to support the operator's efficacy. We observe that the explored discrete fractional operators will be $ \chi^2 $-increasing or decreasing in certain domains of the time scale $ \mathbb{N}_j: = {j, j + 1, ... } $ by looking at the fundamental characteristics of the proposed discrete fractional operators along with $ \chi $-monotonicity descriptions. For numerical simulations, solutions were constructed in the discrete generalized form of the Mittag-Leffler kernel, highlighting the impacts of the illness caused by numerous causes. The order of the fractional derivative had a significant influence on the dynamical process utilized to construct the HBV model. Researchers and policymakers can benefit from the suggested model's ability to forecast infectious diseases such as HBV and take preventive action.</p></abstract>

Computational analysis of multi-layered Navier–Stokes system by Atangana–Baleanu derivative

In our present paper, we have employed the Atangana–Baleanu derivative in the multi-layered Navier–Stokes condition to analyse the nature of the flow. Actually, the stream, the impacts and modifications caused by the operator and the mathematical technique approach are the main subjects of our attention. In this exploratory work, the Laplace transform strategy and the Atangana–Baleanu fractional operator were combined. Due to the Atangana–Baleanu operator's convergence in many engineering and technological issues, it is employed. We have also carefully evaluated the presence and uniqueness of the outcome. To visualize the variations in the flow, we have presented the graphs of the stream's velocity components in each direction during the study.

A study on fractional order infectious chronic wasting disease model in deers

In this paper, the fractional order analysis of the behaviour of the four-dimensional chronic wasting disease (CWD) communities model has been presented. The complexity and dynamical behaviour of the CWD model have been calculated using two fractional derivatives. CWD, a neurological disease that affects deer, has resulted in many deaths and infections among deer populations around the world. To better understand and tackle this eco-epidemiological issue, we used two numerical schemes utilizing the Caputo fractional operator and the Atangana-Baleanu (AB) fractional operator. We have investigated the stability of the eco-epidemiological CWD model. The fixed point theory is a volumetric role play in analyzing the existence and uniqueness of the solution. We use bifurcation diagrams, time series diagrams, and phase diagrams to analyze fractional-order eco-epedimological systems with derivative orders and parameters varying. We examine the approximate result of the eco-epidemiological CWD model with the Atangana-Baleanu (AB) operator and the Caputo operator, and we also compare both solutions. We do a brief analysis of the simulated results, which reveals that the suggested methodologies are novel, dependable, and remarkably easy to implement. In addition to determining the direction, stability, bifurcating, and numerical solutions, graphic depiction and graph bifurcation provide better information about the proposed model.

Monotonicity and positivity analyses for two discrete fractional-order operator types with exponential and Mittag–Leffler kernels

The discrete analysed fractional operator technique was employed to demonstrate positive findings concerning the Atangana-Baleanu and discrete Caputo-Fabrizo fractional operators. Our tests utilized discrete fractional operators with orders between 1<φ<2, as well as between 1<φ<32. We employed the initial values of Mittag–Leffler functions and applied the principle of mathematical induction to ensure the positivity of the discrete fractional operators at each time step. As a result, we observed a significant impact of the positivity of these operators on ∇Q(τ) within Np0+1 according to the Riemann–Liouville interpretation. Furthermore, we established a correlation between the discrete fractional operators based on the Liouville-Caputo and Riemann–Liouville definitions. In addition, we emphasized the positivity of ∇Q(τ) in the Liouville-Caputo sense by utilizing this relationship. Two examples are presented to validate the results.

Nonlocal Hybrid Integro-Differential Equations Involving Atangana–Baleanu Fractional Operators

In this study, we develop a theory for the nonlocal hybrid boundary value problem for the fractional integro-differential equations featuring Atangana–Baleanu derivatives. The corresponding hybrid fractional integral equation is presented. Then, we establish the existence results using Dhage’s hybrid fixed point theorem for a sum of three operators. We also offer additional exceptional cases and similar outcomes. In order to demonstrate and verify the results, we provide an example as an application.

Modified Atangana-Baleanu fractional operators involving generalized Mittag-Leffler function

In this paper, we are going to deal with fractional operators (FOs) with non-singular kernels which is not an easy task because of its restriction at the origin. In this work, we first show the boundedness of the extended form of the modified Atangana-Baleanu (A-B) Caputo fractional derivative operator. The generalized Laplace transform is evaluated for the introduced operator. By using the generalized Laplace transform, we solve some fractional differential equations. The corresponding form of the Atangana-Baleanu Caputo fractional integral operator is also established. This integral operator is proved bounded and obtained its Laplace transform. The existence and Hyers-Ulam stability is explored. In the last results, we studied the relation between our defined operators. The operators in the literature are obtained as special cases for these newly explored FOs.