Atangana-Baleanu Fractional Derivative Research Articles

A fractional order model for Dual Variants of COVID-19 and HIV co-infection via Atangana-Baleanu derivative

In this paper, a new mathematical model for dual variants of COVID-19 and HIV co-infection is presented and analyzed. The existence and uniqueness of the solution of the proposed model have been established using the well known Banach fixed point theorem. The model is solved semi-analytically using the Laplace Adomian decomposition Method. The impact of the Atangana-Baleanu fractional derivative on the dynamics of the proposed model is studied. The work also highlights the impact of COVID-19 vaccination on the dynamics of the co-infection of both diseases. The model is fitted to real COVID-19 data from Botswana. The impact of COVID-19 variants on HIV prevalence using simulations is also assessed. Simulation for the class of individuals co-infected with HIV and the wild or Delta COVID-19 variant reveals a significant decrease, as vaccination rate is increased. The impact of fractional order on different epidemiological classes is also studied. Drawing the plot of total infected population with the wild and Delta COVID-19 variants, at different vaccination rates, it is concluded that, as vaccination rate is increased, there is a significant reduction in population infected with the wild and delta COVID-19 variants. The plot of class of individuals co-infected with HIV and the wild or Delta COVID-19 variant is more interesting; as vaccination rate is increased, the co-infected populations experience a significant decrease. Thus, stepping up vaccination against the different variants of COVID-19 could reduce co-infection cases largely, among people already infected with HIV.

Study of Fuzzy Fractional Third-Order Dispersive KdV Equation in a Plasma under Atangana-Baleanu Derivative

Motivated by the wide-spread of both integer and fractional third-order dispersive Korteweg-de Vries (KdV) equations in explaining many nonlinear phenomena in a plasma and many other fluid models, thus, in this article, we constructed a system for calculating an analytical solution to a fractional fuzzy third-order dispersive KdV problems. We implemented the Shehu transformation and the iterative transformation technique under the Atangana-Baleanu fractional derivative. The achieved series result was contacted and determined the analytic value of the suggested models. For the confirmation of our system, three various problems have been represented, and the fuzzy type solution was determined. The fuzzy results of upper and lower section of all three problems are simulate applying two different fractional orders among zero and one. Because it globalises the dynamic properties of the specified equation, it delivers all forms of fuzzy solutions occurring at any fractional order among zero and one. The present results can help many researchers to explain the nonlinear phenomena that can create and propagate in several plasma models.

Numerical and theoretical analysis of an awareness COVID-19 epidemic model via generalized Atangana-Baleanu fractional derivative

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

On Analytical Solution of Time-Fractional Biological Population Model by means of Generalized Integral Transform with Their Uniqueness and Convergence Analysis

This research utilizes the generalized integral transform and the Adomian decomposition method to derive a fascinating explicit pattern for outcomes of the biological population model (BPM). It assists us in comprehending the dynamical technique of demographic variations in BPMs and generates significant projections. Besides that, generalized integral transforms are the unification of other existing transforms. To investigate the closed form solutions, we employed a fractional complex transform to deal with a partial differential equation of fractional order and a generalized decomposition method was applied to analyze the nonlinear equation. Several aspects of the Caputo and Atangana–Baleanu fractional derivative operators are discussed with the aid of a generalized integral transform. In mathematical terms, the variety of equations and their solutions have been discovered and identified with various novel features of the projected model. To provide additional context for these ideas, numerous sorts of illustrations and tabulations are presented. The precision and efficacy of the proposed technique suggest that it can be used for a variety of nonlinear evolutionary problems.

A Comparative Analysis of the Fractional-Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law

This article applies efficient methods, namely, modified decomposition method and new iterative transformation method, to analyze a nonlinear system of Korteweg–de Vries equations with the Atangana–Baleanu fractional derivative. The nonlinear fractional coupled systems investigated in this current analysis are the system of Korteweg–de Vries and the modified system of Korteweg–de Vries equations applied as a model in nonlinear physical phenomena arising in chemistry, biology, physics, and applied sciences. Approximate analytical results are represented in the form of a series with straightforward components, and some aspects showed an appropriate dependence on the values of the fractional-order derivatives. The convergence and uniqueness analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. The series result achieved applying this technique is proved to be accurate and reliable with minimal calculations. The numerical simulations for obtained solutions are discussed for different values of the fractional order.

Numerical solutions of time fractional Burgers’ equation involving Atangana–Baleanu derivative via cubic B-spline functions

The current paper uses the cubic B-spline functions and θ-weighted scheme to achieve numerical solutions of the time fractional Burgers’ equation with Atangana–Baleanu derivative. A non-singular kernel is involved in the Atangana–Baleanu fractional derivative. For discretization along temporal and spatial grids, the proposed numerical technique employs the finite difference approach and cubic B-spline functions, respectively. This scheme is unconditionally stable and second order convergent in spatial and temporal directions. The presented approach is endorsed by some numerical examples, which show that it is applicable and accurate.

Fractional mathematical modeling of malaria disease with treatment & insecticides

Many fatal diseases spread through vertical transmission while some of them spread through horizontal transmission and others transmit through both modes of transmission. Horizontal transmission illnesses are usually carried by a vector, which might be an animal, a bird, or an insect. Plasmodium parasites that dwell in red blood cells produce malaria, an infectious illness. This parasite is mostly transmitted to humans via mosquitoes. The dynamics of Malaria illness among human persons and vectors are examined in this study. The impact of the vector (mosquito) on disease transmission is also taken into account. The problem is described using nonlinear ODEs that are then generalized using the Atangana–Baleanu fractional derivative. Some theoretical analyses such as existence and uniqueness and stability via Ulam–Hyres stability analysis and optimal control strategies have been done. The numerical solution has been achieved via a numerical technique by implementing MATLAB software. Results of fractional, as well as classical order, are portrayed through different graphs while some figures are displayed for the global asymptotical stability of the model. From the graphical results, it can be noticed that the control parameters drastically decrease the number of infected human and vector population which will off course minimize the spread of infection among the human population. In addition to that, from the graphical results, it also be noticed that our model is globally asymptomatically stable as the solution converges to its equilibrium. Moreover, the use of bednets and insecticides can reduce the spread of infection dramatically while the impact of medication and treatment on the control of infection is comparatively less.

Constructing analytical estimates of the fuzzy fractional-order Boussinesq model and their application in oceanography

The main idea of this article is the investigation of atmospheric internal waves, often known as gravity waves. This arises within the ocean rather than at the interface. A shallow fluid assumption is illustrated by a series of nonlinear partial differential equations in the framework. Because the waves are scattered over a wide geographical region, this system can precisely replicate atmospheric internal waves. In this research, the numerical solutions to the fuzzy fourth-order time-fractional Boussinesq equation (BSe) are determined for the case of the aquifer propagation of long waves having small amplitude on the surface of water from a channel. The novel scheme, namely the generalized integral transform (proposed by H. Jafari [35]) coupled with the Adomian decomposition method (GIADM), is used to extract the fuzzy fractional BSe in R,Rn and (2nth)-order including gH-differentiability. To have a clear understanding of the physical phenomena of the projected solutions, several algebraic aspects of the generalized integral transform in the fuzzy Caputo and Atangana-Baleanu fractional derivative operators are discussed. The confrontation between the findings by Caputo and ABC fractional derivatives under generalized Hukuhara differentiability are presented with appropriate values for the fractional order and uncertainty parameters ℘∈[0,1] were depicted in diagrams. According to proposed findings, hydraulic engineers, being analysts in drainage or in water management, might access adequate storage volume quantity with an uncertainty level.

New numerical dynamics of the fractional monkeypox virus model transmission pertaining to nonsingular kernels.

Monkeypox (MPX) is a zoonotic illness that is analogous to smallpox. Monkeypox infections have moved across the forests of Central Africa, where they were first discovered, to other parts of the world. It is transmitted by the monkeypox virus, which is a member of the Poxviridae species and belongs to the Orthopoxvirus genus. In this article, the monkeypox virus is investigated using a deterministic mathematical framework within the Atangana-Baleanu fractional derivative that depends on the generalized Mittag-Leffler (GML) kernel. The system's equilibrium conditions are investigated and examined for robustness. The global stability of the endemic equilibrium is addressed using Jacobian matrix techniques and the Routh-Hurwitz threshold. Furthermore, we also identify a criterion wherein the system's disease-free equilibrium is globally asymptotically stable. Also, we employ a new approach by combining the two-step Lagrange polynomial and the fundamental concept of fractional calculus. The numerical simulations for multiple fractional orders reveal that as the fractional order reduces from 1, the virus's transmission declines. The analysis results show that the proposed strategy is successful at reducing the number of occurrences in multiple groups. It is evident that the findings suggest that isolating affected people from the general community can assist in limiting the transmission of pathogens.

Numerical analysis of fractional-order Whitham-Broer-Kaup equations with non-singular kernel operators

<abstract><p>This paper solves a fractional system of non-linear Whitham-Broer-Kaup equations using a natural decomposition technique with two fractional derivatives. Caputo-Fabrizio and Atangana-Baleanu fractional derivatives were applied in a Caputo-manner. In addition, the results of the suggested method are compared to those of well-known analytical techniques such as the Adomian decomposition technique, the Variation iteration method, and the optimal homotopy asymptotic method. Two non-linear problems are utilized to demonstrate the validity and accuracy of the proposed methods. The analytical solution is then utilized to test the accuracy and precision of the proposed methodologies. The acquired findings suggest that the method used is very precise, easy to implement, and effective for analyzing the nature of complex non-linear applied sciences.</p></abstract>

The study of fractional-order convection-reaction-diffusion equation via an Elzake Atangana-Baleanu operator

<abstract><p>The major goal of this research is to use a new integral transform approach to obtain the exact solution to the time-fractional convection-reaction-diffusion equations (CRDEs). The proposed method is a combination of the Elzaki transform and the homotopy perturbation method. He's polynomial is used to tackle the nonlinearity which arise in our considered problems.Three test examples are considered to show the accuracy of the proposed scheme. In order to find satisfactory approximations to the offered problems, this work takes into account a sophisticated methodology and fractional operators in this context. In order to achieve better approximations after a limited number of iterations, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for CRDEs. It has been found that the proposed method's solution converges at the desired rate towards the accurate solution. We give some graphical representations of the accurate and analytical results, which are in excellent agreement with one another, to demonstrate the validity of the suggested methodology. For validity of the present technique, the convergence of the fractional solutions towards integer order solution is investigated. The proposed method is found to be very efficient, simple, and suitable to other nonlinear problem raised in science and engineering.</p></abstract>

Qualitative analysis of a fuzzy Volterra-Fredholm integrodifferential equation with an Atangana-Baleanu fractional derivative

<abstract><p>The point of this work was to analyze and investigate the sufficient conditions of the existence and uniqueness of solutions for the nonlinear fuzzy fractional Volterra Fredholm integro-differential equation in the frame of the Atangana-Baleanu-Caputo fractional derivative methodology. To begin with, we give the parametric interval form of the Atangana-Baleanu-Caputo fractional derivative on fuzzy set-valued functions. Then, by employing Schauder's and Banach's fixed point procedures, we examine the existence and uniqueness of solutions for fuzzy fractional Volterra Fredholm integro-differential equation with the Atangana-Baleanu-Caputo fractional operator. It turns out that the last interval model is a combined arrangement of nonlinear equations. In addition, we consider results by applying the Adams Bashforth fractional technique and present two examples that have been numerically solved using graphs.</p></abstract>

Strong interaction of Jafari decomposition method with nonlinear fractional-order partial differential equations arising in plasma via the singular and nonsingular kernels

<abstract><p>This research utilizes the Jafari transform and the Adomian decomposition method to derive a fascinating explicit pattern for the outcomes of the KdV, mKdV, K(2,2) and K(3,3) models that involve the Caputo fractional derivative operator and the Atangana-Baleanu fractional derivative operator in the Caputo sense. The novel exact-approximate solutions are derived from the formulation of trigonometric, hyperbolic, and exponential function forms. Laser and plasma sciences may benefit from these solutions. It is demonstrated that this approach produces a simple and effective mathematical framework for tackling nonlinear problems. To provide additional context for these ideas, simulations are performed, employing a computationally packaged program to assist in comprehending the implications of solutions.</p></abstract>

Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory

<abstract><p>In this research, the $ \bar{\mathbf{q}} $-homotopy analysis transform method ($ \bar{\mathbf{q}} $-HATM) is employed to identify fractional-order Whitham–Broer–Kaup equation (WBKE) solutions. The WBKE is extensively employed to examine tsunami waves. With the aid of Caputo and Atangana-Baleanu fractional derivative operators, to obtain the analytical findings of WBKE, the predicted algorithm employs a combination of $ \bar{\mathbf{q}} $-HAM and the Aboodh transform. The fractional operators are applied in this work to show how important they are in generalizing the frameworks connected with kernels of singularity and non-singularity. To demonstrate the applicability of the suggested methodology, various relevant problems are solved. Graphical and tabular results are used to display and assess the findings of the suggested approach. In addition, the findings of our recommended approach were analyzed in relation to existing methods. The projected approach has fewer processing requirements and a better accuracy rate. Ultimately, the obtained results reveal that the improved strategy is both trustworthy and meticulous when it comes to assessing the influence of nonlinear systems of both integer and fractional order.</p></abstract>

On new computations of the fractional epidemic childhood disease model pertaining to the generalized fractional derivative with nonsingular kernel

<abstract><p>The present research investigates the Susceptible-Infected-Recovered (SIR) epidemic model of childhood diseases and its complications with the Atangana-Baleanu fractional derivative operator in the Caputo sense (ABC). With the aid of the Elzaki Adomian decomposition method (EADM), the approximate solutions of the aforesaid model are discussed by exerting the Adomian decomposition method. By employing the fixed point postulates and the Picard–Lindelöf approach, the stability, existence, and uniqueness consequences of the model are demonstrated. Furthermore, we illustrate the essential hypothesis for disease control in order to find the role of unaware infectives in the spread of childhood diseases. Besides that, simulation results and graphical illustrations are presented for various fractional-orders. A comparison analysis is shown with the previous findings. It is hoped that ABC fractional derivative and the projected algorithm will provide new venues in futuristic studies to manipulate and analyze several epidemiological models.</p></abstract>

Results on impulsive fractional integro-differential equations involving Atangana-Baleanu derivative

In this paper, we consider the impulsive fractional integro-differential equations involving Atangana-Baleanu fractional derivative. The main tools consist a fractional integral operator contains generalized Mittag-Leffler function, Gronwall-Bellman inequality with continuous functions and the Krasnoselskii?s fixed point theorem.

Fuzzy fractional estimates of Swift-Hohenberg model obtained using the Atangana-Baleanu fractional derivative operator

<abstract><p>Swift-Hohenberg equations are frequently used to model the biological, physical and chemical processes that lead to pattern generation, and they can realistically represent the findings. This study evaluates the Elzaki Adomian decomposition method (EADM), which integrates a semi-analytical approach using a novel hybridized fuzzy integral transform and the Adomian decomposition method. Moreover, we employ this strategy to address the fractional-order Swift-Hohenberg model (SHM) assuming g$ {\bf H} $-differentiability by utilizing different initial requirements. The Elzaki transform is used to illustrate certain characteristics of the fuzzy Atangana-Baleanu operator in the Caputo framework. Furthermore, we determined the generic framework and analytical solutions by successfully testing cases in the series form of the systems under consideration. Using the synthesized strategy, we construct the approximate outcomes of the SHM with visualizations of the initial value issues by incorporating the fuzzy factor $ \varpi\in[0, 1] $ which encompasses the varying fractional values. Finally, the EADM is predicted to be effective and precise in generating the analytical results for dynamical fuzzy fractional partial differential equations that emerge in scientific disciplines.</p></abstract>

A study of behaviour for fractional order diabetes model via the nonsingular kernel

<abstract><p>A susceptible diabetes comorbidity model was used in the mathematical treatment to explain the predominance of mellitus. In the susceptible diabetes comorbidity model, diabetic patients were divided into three groups: susceptible diabetes, uncomplicated diabetics, and complicated diabetics. In this research, we investigate the susceptible diabetes comorbidity model and its intricacy via the Atangana-Baleanu fractional derivative operator in the Caputo sense (ABC). The analysis backs up the idea that the aforesaid fractional order technique plays an important role in predicting whether or not a person will develop diabetes after a substantial immunological assault. Using the fixed point postulates, several theoretic outcomes of existence and Ulam's stability are proposed for the susceptible diabetes comorbidity model. Meanwhile, a mathematical approach is provided for determining the numerical solution of the developed framework employing the Adams type predictor–corrector algorithm for the ABC-fractional integral operator. Numerous mathematical representations correlating to multiple fractional orders are shown. It brings up the prospect of employing this structure to generate framework regulators for glucose metabolism in type 2 diabetes mellitus patients.</p></abstract>

Investigation of the dynamics of COVID-19 with SEIHR nonsingular and nonlocal kernel fractional model

ABSTRACT We considered an SEIHR mathematical model to investigate the dynamics of COVID-19. Real data of COVID-19 confirmed cases are fitted to the model. The basic reproductive number is approximated to be . The most globally sensitive parameter of the basic reproductive number is the incubation period of coronavirus. The Atangana Baleanu fractional derivative in Caputo sense (ABC) representation of the model is formulated, and a numerical approximation developed by Toufik and Atangana is used. It is shown that decreasing the fractional-order leads to decreasing the infection of the virus in the infected compartments.

Comparison of Caputo and Atangana–Baleanu fractional derivatives for the pseudohyperbolic telegraph differential equations

Comparison of Caputo and Atangana–Baleanu fractional derivatives for the pseudohyperbolic telegraph differential equations