Atangana-Baleanu Fractional Derivative Research Articles

Existence analysis on multi-derivative nonlinear fractional neutral impulsive integro-differential equations

This article utilizes the Atangana-Baleanu (AB) fractional derivative to examine the behavior of multi-derivative fractional neutral impulsive integro-differential equations under non-local conditions. To demonstrate the existence, uniqueness, and controllability of the solutions, we utilize fixed point theory as our primary analytical tool. Furthermore, we include a detailed example to illustrate and validate the theoretical results obtained from our study.

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.

Fuzzy fractional delay integro-differential equation with the generalized Atangana-Baleanu fractional derivative

Abstract In this work, we consider a class of fuzzy fractional delay integro-differential equations with the generalized Caputo-type Atangana-Baleanu (ABC) fractional derivative. By using the monotone iterative method, we not only obtain the existence and uniqueness of the solution for the given problem with the initial condition but also give the monotone iteration sequence converging to the unique solution of the problem. Furthermore, we also give the continuous dependence of the unique solution on initial value. Finally, an example is presented to illustrate the main results obtained. The results presented in this study are new and open a new avenue of research for fuzzy fractional delay integro-differential equations with the generalized ABC fractional derivative.

An efficient numerical method for two-dimensional fractional integro-differential equations with modified Atangana–Baleanu fractional derivative using operational matrix approach

In the present paper, a numerical method is developed for solving non-homogeneous two-dimensional fractional integro-differential equations with a modified Atangana–Baleanu fractional derivative. The approach is based on the operational matrix method, involving the derivation of operational matrices for the terms of the equation to generate an algebraic system that is solved using mathematical software. First, basic definitions of fractional calculus and block pulse functions are presented. Then, the operational matrices and the method of solution are derived. The existence and uniqueness of the solution of the proposed equation are proved. Examples to test the numerical method are provided, demonstrating the efficiency of the proposed method.

INVESTIGATING VIRUS SPREAD ANALYSIS IN COMPUTER NETWORKS WITH ATANGANA–BALEANU FRACTIONAL DERIVATIVE MODELS

This paper proposes a mathematical model to investigate the dynamic behaviors of a modified computer virus model using the Atangana–Baleanu fractional derivative in the Caputo sense, aiming to elucidate the connection between its parameters and network attributes. By introducing a relatively new numerical method, we address the memory-dependent and nonlocal features of the system. The existence and uniqueness of the model’s solution are confirmed. To explore solution trajectories and assess the impact of various input factors on computer virus dynamics, we employ an efficient numerical technique. Our simulations provide insights into the consequences of fractional order, anti-virus measures, asymptotic fraction, damage rate, and removal rate in the system. These findings illuminate the relationships between model parameters, facilitating the design of networks that minimize the risk of virus outbreaks and prevent future cyber threats. By identifying critical factors involved in the progression of viruses, these results enable the development of more effective defense mechanisms.

Solvability of fractional differential equations with applications of Morgan Voyce polynomials

In this research, a new computational approach is presented to address multi-order fractional differential equations, including the renowned Bagley-Torvik and Painlevé equations. These equations are pivotal in scientific and engineering realms, like modelling the movement of a submerged plate restricted in a Newtonian fluid and gas in a fluid, along with simulating the coupled oscillations. We utilise the collocation approach employing a novel operational matrix derived for Morgan-Voyce polynomials via the Atangana-Baleanu fractional derivative. Initially, we introduce the fractional differential matrix to convert the problem and its constraints into a system of algebraic equations with unknown coefficients. These coefficients aid in finding numerical solutions for the given equations. To assess the efficiency of proposed method, various examples are simulated utilising the proposed approach and the outcomes are compared with existing results for different derivatives.

Solving Biological Population Model by Using FADM within Atangana-Baleanu fractional derivative

Solving Biological Population Model by Using FADM within Atangana-Baleanu fractional derivative

Comparative study of integer-order and fractional-order artificial neural networks: Application for mathematical function generation

This paper investigates the impact of fractional derivatives on the activation functions of an artificial neural network (ANN). Based on the results and analysis, a three-layer backpropagation neural network model with fractional and integer derivatives employed in the activation function and fractional gradient descent method backpropagation learning algorithm has been proposed. Specifically, three perceptrons have been proposed based on fractional and integer derivatives applied to activation functions and learning algorithms. They are fractional derivative activation function (FDAF) perceptron, integer derivative activation function (IDAF)-fractional derivative learning algorithm (FDLA) perceptron, and fractional derivative learning algorithm (FDLA) perceptron. The Riemann-Liouville (RL) fractional derivative, Grunwald-Letnikov (GL) derivative, Caputo-Fabrizio (CF) fractional derivative, Caputo (C) fractional derivative, and Atangana-Baleanu (AB) fractional derivative have been employed. The impact of these derivatives’ fractional-order (FO) is investigated in a wide range from 0.1−0.9 on testing mean square error (MSE) and time required to train the perceptron. FO-based and IO-based perceptrons are compared with the help of performance metrics such as testing MSE and the time required to train the models. The training and testing simulation results illustrate that Caputo-Fabrizio derivative-based perceptrons outperform other fractional derivatives. Also, the performance of the FO-based perceptron is better in terms of the least MSE.

An efficient computational scheme for solving coupled time-fractional Schrödinger equation via cubic B-spline functions.

The time fractional Schrödinger equation contributes to our understanding of complex quantum systems, anomalous diffusion processes, and the application of fractional calculus in physics and cubic B-spline is a versatile tool in numerical analysis and computer graphics. This paper introduces a numerical method for solving the time fractional Schrödinger equation using B-spline functions and the Atangana-Baleanu fractional derivative. The proposed method employs a finite difference scheme to discretize the fractional derivative in time, while a θ-weighted scheme is used to discretize the space directions. The efficiency of the method is demonstrated through numerical results, and error norms are examined at various values of the non-integer parameter, temporal directions, and spatial directions.

Fractional dynamics and sensitivity analysis of measles epidemic model through vaccination

Measles is a highly contagious disease that mainly affects children worldwide. Even though a reliable and effective vaccination is available, there were 140,000 measles deaths worldwide in 2018, and most of them were children under the age five years. In this paper, we comprehensively investigate a novel fractional SVEIR (Susceptible-Vaccinated-Exposed-Infected-Recovered) model of the measles epidemic powered by nonlinear fractional differential equations to understand the epidemic’s dynamical behaviour. We use a non-singular Atangana-Baleanu fractional derivative to analyze the proposed model, taking advantage of non-locality. The existence, uniqueness, positivity and boundedness of the solutions are shown via concepts of fixed point theory, and we also perform the Ulam-Hyers stability of the considered model. The parameter sensitivity is discussed in the context of the variance with each parameter using 3-D graphics based on the basic reproduction number. Moreover, with the Atangana-Toufik numerical scheme, numerical findings are depicted for different fractional-order values. The presented approach produce results that are efficiently consistent and in excellent agreement with the theoretical results.

Fractional-order brucellosis transmission model between interspecies with a saturated incidence rate

In this study, brucellosis dynamics between interspecies are discussed with the Atangana-Baleanu fractional derivative to examine the transmission of brucellosis by its behavior. The recovered compartment, recruitment, and natural death rate for humans are considered for the fractional order model to analyze the transmission dynamics in more detail from an epidemiological point of view. Additionally, the saturated incidence rate is suggested for brucellosis as indirectly transmitted to individuals from the environment. By fixed point theory, it is verified that developed fractional transmission dynamics have a unique solution. The model under consideration employs the Adams-type predictor-corrector method for numerical solution. All comparative results are plotted by MATLAB.

Coupled responses of thermomechanical waves in functionally graded viscoelastic nanobeams via thermoelastic heat conduction model including Atangana–Baleanu fractional derivative

Accurately characterizing the thermomechanical parameters of nanoscale systems is essential for understanding their performance and building innovative nanoscale technologies due to their distinct behaviours. Fractional thermal transport models are commonly utilized to correctly depict the heat transfer that occurs in these nanoscale systems. The current study presents a novel mathematical thermoelastic model that incorporates a new fractional differential constitutive equation for heat conduction. This heat equation is useful for understanding the effects of thermal memory. An application of a fractional-time Atangana–Baleanu (AB) derivative with a local and non-singular kernel was utilized in the process of developing the mathematical model that was suggested. To deal with effects that depend on size, nonlocal constitutive relations are introduced. Furthermore, in order to take into consideration, the viscoelastic behaviour of the material at the nanoscale, the fractional Kelvin–Voigt model is utilized. The proposed model is highly effective in properly depicting the unusual thermal conductivity phenomena often found in nanoscale devices. The study also considered the mechanical deformation, temperature variations, and viscoelastic characteristics of the functionally graded (FG) nanostructured beams. The consideration was made that the material characteristics exhibit heterogeneity and continuous variation across the thickness of the beam as the nanobeam transitions from a ceramic composition in the lower region to a metallic composition in the upper region. The complicated thermomechanical features of simply supported viscoelastic nanobeams that were exposed to harmonic heat flow were determined by the application of the model that was constructed. Heterogeneity, nonlocality, and fractional operators are some of the important variables that contribute to its success, and this article provides a full study and illustration of the significance of these characteristics. The results that were obtained have the potential to play a significant role in pushing forward the design and development of tools, materials, and nanostructures that have viscoelastic mechanical characteristics and graded functions.

Computational analysis of rabies and its solution by applying fractional operator

Numerous novel concepts in fractional mathematics have been created to provide numerical models for a variety of real-world, engineering, and scientific challenges because of the kernel's memory and non-local effects. In this post, we have looked at a deadly illness known as rabies. For our analysis, we employed the Atangana–Baleanu fractional derivative in Caputo sense. Additionally, the mathematical answer was obtained by applying the Laplace transform. Our approach is distinct, and we illustrated the vital role immunizations play in limiting the spread of the illness using graphical data. Furthermore, in this article, we have shown that fractional order systems are preferable to integer order systems.

Secure Multiple-Image Transfer by Hybrid Chaos System: Encryption and Visually Meaningful Images

The secure transmission of information is one of the most important topics in the field of information technology. Considering that images contain important visual information, it is crucial to create a safe platform for image transfer. One commonly employed tool to enhance the complexity and randomness in image encryption methods is the chaos system. The logistic and sine maps are utilized in encryption algorithms but these systems have some weaknesses, notably chaotic behavior in a confined area. In this study, to address these weaknesses, a hybrid system based on the Atangana–Baleanu fractional derivative is proposed. The various tests employed to evaluate the behavior of the new system, including the NIST test, histogram analysis, Lyapunov exponent calculation, and bifurcation diagram, demonstrate the efficiency of the proposed system. Furthermore, in comparison to the logistic and sine maps, the proposed hybrid exhibits chaotic behavior over a broader range. This system is utilized to establish a secure environment for the transmission of multiple images within an encryption algorithm, subsequently concealing them within a meaningful image. Various tools employed to assess the security of the proposed algorithm, including histogram analysis, NPCR, UACI, and correlation values, indicate that the proposed hybrid system has application value in encryption.

Application on Atangana-Baleanu fractional derivative via (τ, a)-weak contractive mappings

Application on Atangana-Baleanu fractional derivative via (τ, a)-weak contractive mappings

Application of the Mittag-Leffler kernel in stochastic differential systems for approximating the controllability of nonlocal fractional derivatives

This research focuses on utilizing the Mittag-Leffler kernel within stochastic differential systems to estimate the controllability of nonlocal Atangana–Baleanu fractional derivatives. By assuming the automatic control of the corresponding linear system, a novel set of necessary and sufficient conditions for the approximate controllability of the fractional stochastic differential inclusions of Atangana–Baleanu is established. Furthermore, the study explores the approximate controllability of the proposed system with infinite delay. The investigation relies on the fixed-point theorem for multivalued operators and fractional calculus to derive these outcomes. Lastly, an illustrative example is provided to highlight the practical implications of the research findings.

Analysis of the Spread of Covid-19 via Atangana-Baleanu Fractional Derivatives

We study the spread of the epidemic via Atangana-Baleanu Fractional Derivatives. We present the mathematical analysis and formulation of a fractional model for the epidemic. The existence and uniqueness of the solution for the proposed model are proved. The study also investigates the existence of disease-free equilibrium and analyzes its stability properties. To validate the theoretical results, we provide a numerical scheme for the fractional model and present various simulation results. These results can serve as a valuable resource in developing strategies to mitigate the spread of the epidemic.

Modeling and investigating the spread of COVID-19 dynamics with Atangana-Baleanu fractional derivative: a numerical prospective

Fractional-order models have been used in the study of COVID-19 to incorporate memory and hereditary properties into the systems Moira and Xu (2003) Respirology 8 S9â14. These models have been applied to analyze the dynamics and behavior of the novel coronavirus. Various fractional-order models have been proposed, including the SIR and SEIR models, with the addition of compartments such as asymptomatic classes and virus repositories. Overall, fractional-order models have proven to be effective in studying the dynamics and spread of COVID-19. In this paper, we propose a nonlinear fractional model employing the Atangana-Baleanu derivative to describe COVID-19. To offer a clearer perspective, our investigation incorporates two distinct quarantine stages within the population. The first quarantine class consists of individuals who have not yet contracted the virus but have chosen to self-isolate at home. The second quarantine class encompasses individuals who are infected and are undergoing quarantine in hospitals. Additionally, we introduce a vaccination class, consisting of the portion of the population that has received the COVID-19 vaccination and is now at a reduced risk of infection. Fixed point theorems are employed to prove the existence and uniqueness of the solutions. The model’s threshold parameter R0 is calculated to investigate the pandemic’s future dynamics. The Toufik-Atangana scheme is applied to obtain numerical solutions for the fractional model. Further, we analyzed the data to assess how vaccination impacts the spread of the virus.The model includes two novel parameters to determine the speed and effectiveness of vaccination measures. To evaluate the accuracy of our model, we simulate the graphical results of this stage and compared them with integer-order derivative. The mathematical model for COVID-19 shows that both equilibrium points are locally stable. Moreover, to gain a deeper understanding of the disease, we conduct a sensitivity analysis to examine the effect of parameters on R0 . The study recommends continuing hospitalization and home isolation for infected individuals until virus transmission reduces sufficiently after vaccination. The given model provides useful insights into the pandemic’s dynamics and suggests measures for controlling its spread.

Non-Standard Finite Difference and Vieta-Lucas Orthogonal Polynomials for the Multi-Space Fractional-Order Coupled Korteweg-de Vries Equation

This paper focuses on examining numerical solutions for fractional-order models within the context of the coupled multi-space Korteweg-de Vries problem (CMSKDV). Different types of kernels, including Liouville-Caputo fractional derivative, as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, are utilized in the examination. For this purpose, the nonstandard finite difference method and spectral collocation method with the properties of the Shifted Vieta-Lucas orthogonal polynomials are employed for converting these models into a system of algebraic equations. The Newton-Raphson technique is then applied to solve these algebraic equations. Since there is no exact solution for non-integer order, we use the absolute two-step error to verify the accuracy of the proposed numerical results.

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.