Atangana-Baleanu Fractional Derivative Research Articles

Initial‐boundary value and interface problems on the real half line for the fractional advection–diffusion‐type equation

We use the unified transform method (UTM) to solve initial‐boundary value problems for the fractional advection–diffusion‐type equation (FADE) on the real half line. We generalize this equation using the modified definition of the Atangana–Baleanu fractional derivative of order in order to satisfy the initial condition. A solution methodology is proposed when the UTM is implemented in fractional differential equations with boundary conditions of Dirichlet and Robin type, in particular when using the modified definitions of fractional operators with non‐singular kernel. In addition, an interface problem is stated and solved in the adjacent domains and , where perfect contact continuity conditions are imposed. The exact solutions obtained include as a particular case, the diffusion and advection–diffusion equations with integer‐order derivatives. Finally, representative curves for the solution are shown by varying the fractional order.

Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions

Abstract The generalization of the classical FitzHugh–Nagumo model provides a more accurate description of the physical phenomena of neurons by incorporating both nonlinearity and fractional derivatives. In this article, we present a numerical method for solving the time-fractional FitzHugh–Nagumo equation (TFFNE) in the sense of the Atangana–Baleanu fractional derivative using B-spline functions. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while θ \theta -weighted scheme is used to discretize the space directions. The efficiency of the scheme is demonstrated through numerical results and rate of convergence. The convergence order and error norms are studied at different values of the noninteger parameter, temporal directions, and spatial directions. Finally, the effectiveness of the proposed methodology is examined through the analysis of three applications.

A projection method based on the piecewise Chebyshev cardinal functions for nonlinear stochastic ABC fractional integro‐differential equations

In this study, the Atangana–Baleanu fractional derivative in the Caputo type (as a kind of non‐local and non‐singular derivative) is used to define a new class of stochastic fractional integro‐differential equations. A projection method (more precisely, a Galerkin approach) based on the piecewise Chebyshev cardinal functions is developed to solve these stochastic fractional equations. To construct this method, the operational matrices of fractional and stochastic integrals of these basis functions are obtained and used in the established method. By approximating the solution of the problem with a finite expansion of the expressed basis functions (in which the expansion coefficients are unknown), a system of algebraic equations is obtained. By solving this system, the expansion coefficients and subsequently the solution of the original stochastic fractional problem are obtained. The convergence analysis of the proposed method is investigated, theoretically and numerically. The accuracy of the established procedure is illustrated by solving several numerical examples.

A new investigation on fractionalized modeling of human liver

This study focuses on improving the accuracy of assessing liver damage and early detection for improved treatment strategies. In this study, we examine the human liver using a modified Atangana-Baleanu fractional derivative based on the mathematical model to understand and predict the behavior of the human liver. The iteration method and fixed-point theory are used to investigate the presence of a unique solution in the new model. Furthermore, the homotopy analysis transform method, whose convergence is also examined, implements the mathematical model. Finally, numerical testing is performed to demonstrate the findings better. According to real clinical data comparison, the new fractional model outperforms the classical integer-order model with coherent temporal derivatives.

Delay differential equations with fractional differential operators: Existence, uniqueness and applications to chaos

<abstract><p>In this study, we consider a chaotic model in which fractional differential operators and the delay term are added. Using the Carathéodory existence-uniqueness theorem for this chaotic model modified with the Caputo fractional derivative, we show that the solution of the associated system exists and is unique. We consider the chaotic model with a delay term with Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives and present a numerical algorithm for these models. We then present the numerical solution of chaotic models with delay terms by using piecewise differential operators, where fractional, classical and stochastic processes can be used. We present the numerical solution of chaotic models with delay terms, as modified by using piecewise differential operators. The graphical representations of these models are simulated for different values of the fractional order.</p></abstract>

A fractional dual-phase-lag thermoelastic model for a solid half-space with changing thermophysical properties involving two-temperature and non-singular kernels

<abstract> <p>The thermal and mechanical properties of materials show differences depending on the temperature change, which necessitates consideration of the dependence of the properties of these materials on this change in the analysis of thermal stress and deformation of the material. As a result, in the present work, a mathematical framework for thermal conductivity was formulated to describe the behavior of non-simple elastic materials whose properties depend on temperature changes. This derived model includes generalized fractional differential operators with non-singular kernels and two-stage delay operators. The fractional derivative operators under consideration include both the Caputo-Fabrizio fractional derivative and the Atangana-Baleanu fractional derivative, in addition to the traditional fractional operator. Not only that, but the system of governing equations includes the concept of two temperatures. Based on the proposed model, the thermodynamic response of an unlimited, constrained thermoelastic medium subjected to laser pulses was considered. It was taken into account that the thermal elastic properties of the medium, such as the conductivity coefficient and specific heat, depend on the temperature. The governing equations of the problem were formulated and then solved using the Laplace transform method, followed by the numerical inverse. By presenting the numerical results in graphical form, a detailed analysis and discussion of the effects of fractional factors and the dependence of properties on temperature are presented. The results indicate that the fractional order coefficient, discrepancy index, and temperature-dependent properties significantly affect the behavior fluctuations of all physical domains under consideration.</p> </abstract>

Existence and uniqueness of nonlinear fractional differential equations with the Caputo and the Atangana-Baleanu derivatives: Maximal, minimal and Chaplygin approaches

<p>This work provided a detailed theoretical analysis of fractional ordinary differential equations with Caputo and the Atangana-Baleanu fractional derivative. The work started with an extension of Tychonoff's fixed point and the Perron principle to prove the global existence with extra conditions due to the properties of the fractional derivatives used. Then, a detailed analysis of the existence of maximal and minimal solutions was presented for both cases. Then, using Chaplygin's approach with extra conditions, we also established the existence and uniqueness of the solutions of these equations. The Abel and the Bernoulli equations were considered as illustrative examples and were solved using the fractional middle point method.</p>

Computational analysis of time fractional diffusion wave equation via novel B-splines-based technique

<abstract><p>In this paper, the authors propose and utilize a novel B-splines-based technique to solve time fractional diffusion wave equations numerically. The present approach employs the arbitrary-order fractional derivative in conjunction with the $ \theta $-weighted scheme. The integration of the Atangana-Baleanu fractional derivative into time fractional diffusion wave equations represents a novel advancement within B-spline methodologies. Furthermore, the stability and convergence of the presented schemes are given. The suggested approach not only exhibits unconditional stability, it also attains second-order convergence in both temporal and spatial dimensions. We conclude by applying our theoretical results to certain numerical examples to demonstrate their efficiency and accuracy.</p></abstract>

Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $ in infinite dimensional Banach spaces

<abstract><p>In this paper, we improved recent results on the existence of solutions for nonlinear fractional boundary value problems containing the Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $. We also derived the exact relations between these fractional boundary value problems and the corresponding fractional integral equations in infinite dimensional Banach spaces. We showed that the continuity assumption on the nonlinear term of these equations is insufficient, give the derived expression for the solution, and present two results about the existence and uniqueness of the solution. We examined the case of impulsive impact and provide some sufficiency conditions for the existence and uniqueness of the solution in these cases. We also demonstrated the existence and uniqueness of anti-periodic solution for the studied problems and considered the problem when the right-hand side was a multivalued function. Examples were given to illustrate the obtained results.</p></abstract>

Numerical Investigation of the Fractional Diffusion Wave Equation with the Mittag–Leffler Function

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a θ-weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order O(h2+(Δt)2). Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques.

Comparative Analysis and Definitions of Fractional Derivatives

Fractional Calculus (FC) has emerged as a valuable tool in various fields. This study explores the historical development of (FC) and examines prominent definitions regarding Fractional Derivatives (FD), such as the Riemann-Liouville, Grunwald-Letnikov, Caputo Fractional Derivative, Katugampula derivatives, Caputo Fractional Derivative, Caputo-Fabrizio Fractional Derivative and as well as Atangana-Baleanu Fractional Derivative. It critically evaluates their strengths, weaknesses and implications on (FD) equations. The findings contribute to establishing a clearer understanding of Fractional Derivatives (FD) and guiding their appropriate use in practical scenarios. It investigates the impact of different definitions on the properties and behaviors of (FD), providing valuable insights for researchers.

Computational results of convective heat transfer for fractionalized Brinkman type tri-hybrid nanofluid with ramped temperature and non-local kernel

Engineers have recently become attracted to electrically conducting nanofluids (NFs) due to various applications in several applied science and engineering disciplines. They have been employed in magnetic refrigeration, cancer treatment (hyperthermia), medicine, and magnetic resonance imaging, among other things. Considering the importance of electrically conducting NFs, in this article, we have proposed a fractionalized MHD (Magnetohydrodynamics) and thermal transmission of a Brinkman-type tri-hybrid nanofluid over an infinite plate saturated through a porous medium with generalized velocity and ramped conditions. For the solution of the governed fractional model, we have utilized a recent definition of fractional derivatives, known as Atangana-Baleanu (AB) fractional derivative and Laplace transformation. The computational results are exhibited for tri-hybrid (TiO2-Al2O3-CuO/H2O) NF and described through graphical diagrams and tables to discover the physics of numerous relevant flow parameters on temperature and velocity profiles. It is detected that both thermal transmission and momentum profile for tri-hybrid NF is a better technique as compared to hybrid NF and NF for both graphical and numerical comparisons.

Magnetite/engine oil Casson nanofluid flow over a Riga plate with Atangana–Baleanu fractional derivative

The Riga plate is useful to enhance the conductivity of working fluids in many critical technical applications like MHD generators, nuclear reactor design, flow meters, and thermal nuclear reactors. The present work is carried out to analyze the heat transfer of ferrofluid with Engine oil as base fluid, flowing parallel to a vertical Riga plate. Atangana–Baleanu fractional derivative is used to model the unsteady distributions of velocity, temperature, and concentration fields, and exact solutions are acquired by applying the Laplace transformation technique and are exhibited with the help of special functions such as the Lorenzo–Hartley’s G function and the Mittag-Leffler function. The influence of all physical and rheological characteristics on flow distribution is illustrated graphically. From the study, it is revealed that the effect of modified Hartmann number plays a significant role in the velocity field and also that heat transfer is increased with increasing volume fraction of the nanoparticles. The effect of significant parameters on Sherwood number, Nusselt number, and skin friction is scrutinized in the tables.

Mathematical modeling of a novel fractional-order monkeypox model using the Atangana–Baleanu derivative

In this research endeavor, we undertake a comprehensive analysis of a compartmental model for the monkeypox disease, leveraging the Atangana–Baleanu fractional derivative framework. Our primary objective is to investigate the effectiveness of a range of control strategies in containing the transmission of this infectious ailment. The parameterization of the model is executed meticulously via the application of the maximum likelihood estimation technique. Our study involves a rigorous mathematical analysis of the considered model, which encompasses an exploration of the existence and uniqueness of solutions, as well as the establishment of conditions ensuring the compactness and continuity of these solutions. Subsequently, we embark on an extensive stability analysis of the model, complemented by the computation of both the effective and basic reproduction numbers. These calculations are instrumental in illuminating the long-term behavior of the epidemic. Additionally, we perform a sensitivity analysis of the basic reproduction number to discern the influence of various factors on disease transmission dynamics. To derive our numerical results, we implement the Adams–Bashforth predictor–corrector algorithm tailored for the Atangana–Baleanu fractional derivatives. We employ this numerical technique to facilitate the simulation of the model under a spectrum of fractional-order values, offering a visual representation of our findings. Our study underscores the pivotal roles of infection awareness, vaccination campaigns, and effective treatment in significantly curtailing disease transmission, thus contributing valuable insight to the field of epidemiology.

A non-integer order model for Zika and Dengue co-dynamics with cross-enhancement

A novel fractional derivative model with nine compartments is formulated to investigate the transmission dynamics of zika and dengue co-infection. The Atangana–Baleanu fractional derivative in the Caputo sense was employed. The conditions for a unique solution are identified, and the solutions’ positivity and boundedness are demonstrated. The disease-free equilibrium point (DFE) and basic reproduction number, R0, were obtained. The DFE was shown to be locally asymptotically stable when the basic reproduction number is less than one. Zika-associated reproduction number, R0z, and dengue-associated reproduction number, R0d, were estimated to be 1.0144 and 1.1724, respectively. The system was shown to be generalized Ulam Hyers–Rassias stable, and the Adam–Bashforth method was used to provide its’ numerical solution. Sensitivity analysis using the Latin Hyper-cube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) (|PRCC|> 0.45) with 200 runs was carried out using various variables as response functions per time. The most significant parameters were found to be zika human-to-human transmission rate, βhz1, vector death rate, μv, zika recovery rate, γhz1 and dengue vector-to-human transmission rate, βhd. Real data from Espirito Santo in Brazil is used to validate the model and fit needed parameter values. Numerical simulations illustrated the impact of varying the fractional order derivative, recovery rates, transmission rates, and cross-enhancement parameters on the infected human compartments. The zika Human-to-human transmission rate, βhz1, was found to be a very significant parameter in the control of zika disease transmission. Increasing the vector death rate, μv, was more important in curbing dengue prevalence and incidence than the attainment of recovery from the dengue disease, and the absence of the zika Vector-to-human transmission rate, βhz3, was almost insignificant in the presence of the zika Human-to-human transmission rate, βhz1, for disease eradication. This study suggested control measures and strategies to decrease the dengue and zika human-to-human transmission rates.

Analytical Solution of Diffusion Thermo Effect on MHD Second Grade Fluid Flow with Heat Generation and Chemical Reaction through an Accelerated Vertical Plate

Abstract: The objective of this model is to examine the Dufour effect on unsteady free convection second-grade fluid flow past an accelerated moving plate subjected to the magnetic field through a porous medium. The thermal radiation and chemical reactions are also taken into account. The constitutive governing equations of the model with all levied initial and boundary conditions are written in non-dimensional form. The non-dimensional equations that govern the flow model are transformed into a time-fractional model using the Caputo, Caputo–Fabrizio, and Atangana–Baleanu time-fractional derivatives. The Laplace transform technique is applied to the differential equations of the flow model to obtain the exact solution for concentration, temperature, and velocity fields. The expression for the Sherwood number, the Nusselt number, and skin friction are also derived analytically. The effects of diffusion-thermo, chemical reactions, second-grade parameterfractional parameter (γ), porosity, magnetic parameter, heat absorption/generation, and thermal radiation on velocity profiles are studied through various figures. It is observed that the velocity profiles for Caputo–Fabrizio fractional derivatives are higher as compared to Caputo and Atangana–Baleanu fractional derivatives. It is also seen that for the value of fractional parameter γ→1, the velocity profiles obtained via Caputo, Caputo–Fabrizo, and Atangana–Baleanu derivatives are identical. Keywords: Second-grade fluid, Free convection, Chemical reaction, Diffusion thermo, Heat generation, Caputo, Caputo–Fabrizio, Atangana–Baleanu fractional derivative.

A rock creep model based on Atangana-Baleanu fractional derivative with coupled damage effects of pore water pressure, stress and time

Pore water pressure is an important factor affecting long-term rock failure and instability. On the basis of the assumptions that the rock microunit strength follows Weibull distribution and the rock failure compliances Drucker-Prager criterion, a pore water pressure-stress coupled damage (PSCD) variable is constructed in this paper. Considering the effect of pore water pressure, stress and time (PST), a pore water pressure-stress-time coupled damage (PSTCD) variable is put forward by introducing the damage variable considering time effect. Based on the characteristics and advantages of fractional differential theory in viscoelastic analysis of rock materials, one-dimensional and three-dimensional pore water pressure-stress-time coupled damage fractional (PSTCDF) creep models are proposed by Atangana-Baleanu (AB) fractional derivative. By fitting and analyzing the laboratory experimental data of rock creep under the combined action of PST, the relevant parameters for PSTCDF creep model can be deduced. The analysis of the goodness of fit and the generalization accuracy prove the rationality, validity and generalization of the proposed model. The sensitivity of the PSTCDF model is analyzed by different the stress level, pore water pressure, fractional derivative order and damage variable coefficient. The results show that the creep model changes flexibly when the model parameters change, and the specific creep state of rock can be reflected by reasonable selection of the model parameters.

Symmetry Analyses of Epidemiological Model for Monkeypox Virus with Atangana–Baleanu Fractional Derivative

The monkeypox virus causes a respiratory illness called monkeypox, which belongs to the Poxviridae virus family and the Orthopoxvirus genus. Although initially endemic in Africa, it has recently become a global threat with cases worldwide. Using the Antangana–Baleanu fractional order approach, this study aims to propose a new monkeypox transmission model that represents the interaction between the infected human and rodent populations. An iterative method and the fixed-point theorem are used to prove the existence and uniqueness of the symmetry model’s system of solutions. It shows that the symmetry model has equilibrium points when there are epidemics and no diseases. As well as the local asymptotic stability of the disease-free equilibrium point, conditions for the endemic equilibrium point’s existence have also been demonstrated. For this purpose, the existence of optimal control is first ensured. The aim of the proposed optimal control problem is to minimize both the treatment and prevention costs, and the number of infected individuals. Optimal conditions are acquired Pontryagin’s maximum principle is used. Then, the stability of the symmetry model is discussed at monkeypox-free and endemic equilibrium points with treatment strategies to control the spread of the disease. Numerical simulations clearly show how necessary and successful the proposed combined control strategy is in preventing the disease from becoming epidemic.

Numerous analytical wave solutions to the time-fractional unstable nonlinear Schrödinger equation with beta derivative

Fractional nonlinear evolution equations are mathematical representations used to explain a wide range of complex phenomena occurring in nature. By incorporating fractional order viscoelasticity, these equations can accurately depict the intricate behaviour of materials or mediums, requiring fewer parameters compared to classical models. Furthermore, fractional viscoelastic models align with molecular theories and thermodynamics, making them highly compatible. Consequently, the scientific community has shown significant interest in fractional nonlinear evolution equations and their soliton solutions. This study employs the extended Kudryashov method to derive soliton solutions for the time-fractional unstable nonlinear Schrödinger equation, utilizing the Atangana–Baleanu fractional derivative known as the beta derivative. The obtained solitons exhibit various shapes, including V-shaped, periodic, singular periodic, flat kink, and singular bell, under specific conditions. To better understand their physical characteristics, 3D and contour plots are presented by assigning parameter values to certain solutions. Additionally, 2D graphs are generated to observe how the fractional parameter affects the solutions. The Hamiltonian function is determined to further analyse the dynamics of the phase plane. The simulations were conducted using Mathematica and MATLAB software tools. The outcomes of this research contribute to a deeper understanding of the behaviour of fractional nonlinear evolution equations and their soliton solutions, offering insights into the complex dynamics of viscoelastic systems.

On the Analysis of a Fractional Tuberculosis Model with the Effect of an Imperfect Vaccine and Exogenous Factors under the Mittag–Leffler Kernel

This research study aims to investigate the effects of vaccination on reducing disease burden by analyzing a complex nonlinear ordinary differential equation system. The study focuses on five distinct sub-classes within the system to comprehensively explore the impact of vaccination. Specifically, the mathematical model employed in this investigation is a fractional representation of tuberculosis, utilizing the Atangana–Baleanu fractional derivative in the Caputo sense. The validity of the proposed model is established through a rigorous qualitative analysis. The existence and uniqueness of the solution are rigorously determined by applying the fundamental theorems of the fixed point approach. The stability analysis of the model is conducted using the Ulam–Hyers approach. Additionally, the study employs the widely recognized iterative Adams–Bashforth technique to obtain an approximate solution for the suggested model. The numerical simulation of the tuberculosis model is comprehensively discussed, with a particular focus on the assumptions made regarding vaccination. The model assumes that only a limited portion of the population is vaccinated at a steady rate, and the efficacy of the vaccine is a critical factor in reducing disease burden. The findings of the study indicate that the proposed model can effectively assess the impact of vaccination on mitigating the spread of tuberculosis. Furthermore, the numerical simulation underscores the significance of vaccination as an effective control measure against tuberculosis.