Mittag-Leffler Kernel Research Articles

On a Hybrid Class of $p$-Laplacian Initial Value Problems with Modified Mittag-Leffler Kernel

In this work, we establish key results on the existence theory for a category of initial value problems (IVPs) involving hybrid fractional integro-differential equations (HFIDEs) with a $p$-Laplacian operator, utilizing the modified Mittag-Leffler kernel. By employing Krasnoselskii and Banach fixed point theorems (FPTs), we determine the conditions required for the existence of solutions. Additionally, we examine the Hyers-Ulam (H-U) stability of the problem. Lastly, we present an example to confirm our theoretical results.

Chlamydia infection with vaccination asymptotic for qualitative and chaotic analysis using the generalized fractal fractional operator

In this work, we solve a system of fractional differential equations utilizing a Mittag-Leffler type kernel through a fractal fractional operator with two fractal and fractional orders. A six-chamber model with a single source of chlamydia is studied using the concept of fractal fractional derivatives with nonsingular and nonlocal fading memory. The fractal fractional model of the Chlamydia system can be solved by using the characteristics of a non-decreasing and compact mapping. A suggested model with the Lipschitz criteria and linear growth is studied both qualitatively and quantitatively, taking into account boundedness, uniqueness, and positive solutions at equilibrium points with Leray-Schauder results under time scale concepts. We examined the framework of local and global stability and insight into Lyapunov function properties for the infectious disease model. Chaos Control will employ the regulate for linear responses approach to stabilize the system following its equilibrium points. This will take into consideration a fractional order framework with a managed design, where solutions are bounded in the feasible domain and have a greater impact at the lower minimum infectious rate. To illustrate the implications of fractional and fractal dimensions with varying interest rate values through simulations with Newton’s polynomial method under the Mittag-Lefller kernel. Additionally, a comparative analysis of results is also derived by employing power and exponential decay kernels at various fractional orders.

Chaos and feasibility analysis of a healthy lifestyle’s sustainability using fractional-order polycystic ovarian syndrome’s effect on infertility

In this study, we developed a fractional-order model for polycystic ovarian syndrome issue transmission in women through four compartments with the Mittag-Leffler kernel. For biological feasibility, we discussed the positivity and boundedness of solutions with fixed point mappings in Banach spaces. Sensitivity analysis is performed using different variables as response functions each time, employing Latin Hyper-cube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) with 200 iterations. Further, the reproductive number is determined for the model’s biological feasibility with an analysis of parameter impact through Matlab software. The solutions are examined for global stability using the Lyapunov function with a first derivative. Chaos Control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium. Numerical solutions are derived by using a Newton polynomial in the generalized form of the Mittag-Leffler kernel, which illustrates the effect of the fractional operator and parameters involved in the system. Results are also analyzed with different assumptions of parameter values and initial conditions to observe the different impacts of treatment to overcome the risk of infertility in women. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies to overcome the risk of problems and best management care pre-detection of the issue and for treatment.

Modeling and analysis using piecewise hybrid fractional operator in time scale measure for ebola virus epidemics under Mittag–Leffler kernel

This emerging infectious disease poses one of the most severe threats to public health in these locations, but there are not many reliable therapies yet. In this work, we developed the Ebola virus dynamics and control factors epidemic model with a piecewise hybrid fractional Operator in time scale measure insight of Mittag–Leffler kernel. Patterns and structures that repeat at various scales are the focus of fractal analysis, which has applications in complex systems such as biological ones. Both qualitatively and statistically, a proposed model with the Lipschitz criteria and linear growth is examined, considering positive solutions, boundedness, and uniqueness at equilibrium points with Leray-Schauder results under time scale ideas. The regulation for linear responses approach will be used by Chaos Control to stabilize the system after its equilibrium points. A fractional-order framework with a controlled design will be considered, where solutions are bounded in the feasible domain of relations of different compartments. Ulam–Hyers stability results in the solution are treated when function (constant or rising) for the component of qualitative inquiry in generalized form. The dynamical behaviors of the suggested model are discussed with the Newton polynomial approach used to implement on model in the sense of classical piecewise and Mittag–Leffler kernel at different fractional order values. The model shows that solutions are stable and confined within a feasible range, ensuring reliability. Through detailed simulations, it effectively captures how different interventions and infection rates influence Ebola spread. This fractional-order model enhances understanding of Ebola transmission, providing a strong basis for predicting outbreaks and planning effective control measures, with practical applications for analyzing real-world data.

Mathematical Analysis of a Navier–Stokes Model with a Mittag–Leffler Kernel

In this paper, we establish the existence and uniqueness results of the fractional Navier–Stokes (N-S) evolution equation using the Banach fixed-point theorem, where the fractional order β is in the form of the Atangana–Baleanu–Caputo fractional order. The iterative method combined with the Laplace transform and Sumudu transform is employed to find the exact and approximate solutions of the fractional Navier–Stokes equation of a one-dimensional problem of unsteady flow of a viscous fluid in a tube. In the domains of science and engineering, these methods work well for solving a wide range of linear and nonlinear fractional partial differential equations and provide numerical solutions in terms of power series, with terms that are simple to compute and that quickly converge to the exact solution. After obtaining the solutions using these methods, we use Mathematica software Version 13.0.1.0 to present them graphically. We create two- and three-dimensional plots of the obtained solutions at various values of β and manipulate other variables to visualize and model relationships between the variables. We observe that as the fractional order β becomes closer to the integer order 1, the solutions approach the exact solution. Lastly, we plot a 2D graph of the first-, second-, third-, and fourth-term approximations of the series solution and observe from the graph that as the number of iterations increases, the approximate solutions become close to the series solution of the fourth-term approximation.

Existence and Uniqueness Solution for a Mathematical Model with Mittag-Leffler Kernel

In this work, we analyse the fractional order West Nile Virus model involving the Atangana-Baleanu derivatives. Existence and uniqueness solutions were obtained by the fixed-point theorem. Another impressive aspect of the work is illustrated by simulations of different fractional orders by calculating the numerical solutions of the mathematical model.

The Peano–Sard theorem for fractional operators with Mittag-Leffler kernel and application in classical numerical approximation

We investigate fractional Peano kernels for continuous linear functionals, in the context of differintegral operators with Mittag-Leffler kernel. New bounds for polynomial interpolation are obtained and numerical computations are shown, indicating improvements.

Exact and fractional solution of MHD generalized Couette hybrid nanofluid flow with Mittag–Leffler and power law kernel

This study investigates the complex behavior of Jeffrey nanofluid flow in a porous oscillating microchannel under the influence of magnetohydrodynamic (MHD) effects. The research explores how magnetic field distortions lead to diverse accumulation patterns of nanofluid particles, a phenomenon attributed to homogeneous magnetization in fluid dynamics. Nanoparticles ranging from 0.25 % to 0.5 % (low and inexpensive concentrations) are remarkably consistent for best results in most machining procedures. Different concentrations of various nanomaterial's is utilized (φ1 = φ2 = 0.01, 0.02, 0.03, 0.04) to make the simple nanfluid and hybrid nanofluid suspensions. By employing fractal-fractional derivatives governed by power law, a mathematical model developed to describe the time-varying, compressible MHD flow of Jeffrey nanofluid. The model incorporates the effects of heat transfer, pressure, and magnetic fields on the fluid dynamics. A novel fractional approach utilizing the Laplace transform is applied to solve the fractal MHD hybrid-fluid model integrated into a porous medium. The study reveals velocity flow decrease with increasing Reynolds numbers but increase with channel inclination. Additionally, both the Darcy number and magnetic field orientation enhance heat transfer rates. In addition, the velocity profile enhanced by the hybrid nanofluid suspension as compared to simple nanofluid flow. The research validates its findings by demonstrating the convergence of fractional and numerical solution methods. Furthermore, the study compares the performance of different hybrid nanofluids, concluding that water-based (H2O + GO + MoS2) hybrid fluids exhibit slightly superior characteristics compared to (CMC + GO + MoS2) hybrid nanofluids.

A nonlinear fractional fishery resource system model with Crowley–Martin functional response under Mittag-Leffler kernel

This article studies the action of predators and the predator-dependent functional response in the fishery resource model. We employ the Atangana–Baleanu–Caputo fractional derivative to study the proposed fractional fishery resource model in the presence of predators with Crowley–Martin functional response. We present a theoretical and numerical analysis of the governing nonlinear differential equations of the model consisting of the biomass density of the fish population inside the unrestricted fishing zone, the biomass density of the fish population inside the reserve or restricted fishing zone, and the predator population. Using the fixed point theory and nonlinear analysis, we establish the existence and uniqueness results of the proposed ABC fractional fishery resource model. We establish stability analysis of the fishery model using the Ulam–Hyers stability approach. The numerical scheme of the fractional Adams–Bashforth method is provided and the approximate solutions for the model under consideration are given and discussed. We observe that an increase in the fish capturing rates increases the size of the predator population and reduces the fish subpopulations. To maintain a high number of fish species, we recommend a control measure to reduce the fish capturing rate by the predators.

Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability

Abstract Spacecraft models that mimic the Planck satellite’s behaviour have produced information on cosmic microwave background radiation, assisting physicists in their understanding of the composition and expansion of the universe. For achieving the intended formation, a framework for a discrete fractional difference spacecraft model is constructed by the use of a discrete nabla operator of variable order containing the Mittag–Leffler kernel. The efficacy of the suggested framework is evaluated employing a numerical simulation of the concerning dynamic systems of motion while taking into account multiple considerations such as exterior disruptions, parameterized variations, time-varying feedback delays, and actuator defects. The implementation of the Banach fixed-point approach provides sufficient requirements for the presence of the solution as well as a distinctive feature for such mechanisms Furthermore, the consistent stability is examined. With the aid of discrete nabla operators, we monitor the qualitative behavioural patterns of spacecraft systems to provide justification for structure’s chaos. We acquire the fixed points of the proposed trajectory. At each fixed point, we calculate the eigenvalue of the spacecraft system’s Jacobian matrix and check for zones of instability. The outcomes exhibit a wide range of multifaceted behaviours resulting from the interaction with various fractional orders in the offered system. To maintain stability and synchronize the system, nonlinear controllers are additionally provided. The study highlights the technique’s vulnerability to fractional-order factors, resulting in exclusive, changing trends and equilibrium frameworks. Because of its diverse and convoluted behaviour, the spacecraft chaotic model is an intriguing and crucial subject for research.

New computations of the fractional worms transmission model in wireless sensor network in view of new integral transform with statistical analysis; an analysis of information and communication technologies

Wireless sensor networks (WSNs) have attracted a lot of interest due to their enormous potential for both military and civilian uses. Worm attacks can quickly target WSNs because of the network's weak security. The worm can spread throughout the network by interacting with a single unsafe node. Moreover, the analysis of worm spread in WSNs can benefit from the use of mathematical epidemic models. We suggest a five-compartment model to characterize the mechanisms of worm proliferation with respect to time in WSN. Taking into account the ZZ transform convoluted with the Atangana-Baleanu-Caputo (ABC) fractional derivative operator, we employ it to analyze the characteristics and applications of the ZZ transformation using the Mittag-Leffler kernel. Moreover, we construct a new algorithm for the homotopy perturbation method (HPM) in conjunction with the ZZ transform technique to generate analytical solutions for the worm transmission model. Also, we address the qualitative aspects such as positivity, boundness, worm-free state, endemic state, basic reproduction number (R0) and worm-free equilibrium stability. Furthermore, we prove that the virus rate in sensor nodes is extinct if R0<1 and the virus persists if R0>1. In addition, we develop analytical findings to evaluate the series of solutions. Furthermore, a detailed statistical analysis is conducted to verify the nonlinear dynamics of the system by verifying the 0−1 test to determine whether uncertainty exists using approximation entropy and the C0 data. An extensive analysis of the vaccination class with respect to the transmitting class as well as the susceptible class is being used to investigate the effects of stepping up precautions on WP in WSN. Moreover, the modeling of the WSN revealed that reducing the fractional-order from 1 requires that the recommended approach be implemented at the highest rate so that there is no long-lasting immunization; instead, nodes remain briefly defensive before becoming vulnerable to future worm attacks.

Mathematical modeling of neuron model through fractal-fractional differentiation based on maxwell electromagnetic induction: application to neurodynamics

The electrical activities of the reliable neuron models have different responses within intrinsic biophysical effects and can functionalize for asymmetric coexisting electrical activities under anti-monotonicity phenomenon. This manuscript presents mathematical analysis of neuron model based on Maxwell electromagnetic induction through newly proposed fractal-fractional differential and integral operators. The neuron model based on Maxwell electromagnetic induction changes with time along a fractal dimension that describes the cumulative chaotic phenomenon. The cumulative chaotic phenomenon of neuron model is mathematically modeled via exponential and Mittag–Leffler kernels with variable and fixed fractal and fractional orders. In order to exhibit fractal properties and memory effects, the neuron model is discretized by means of Adams–Bashforth-Moulton method that allows explicitly to compute the approximate solution of neuron model. The comparison of neuron model based on memory effect and fractal dimension have distinguished the evolution of neuron model at (i) variability of fractal order with fixed fractional order, (ii) variability of fractional order with fixed fractal order, and (iii) variability of fractal order as well fractional order.

Dynamical and numerical analysis of the hepatitis B virus treatment model through fractal–fractional derivative

Infection with the hepatitis B virus (HBV) is a global health problem and may be controlled via appropriate treatment. We use fractional models to understand infectious diseases because fractional models help us understand treatments' effects on hepatitis B better than integer‐order models. In this article, we introduce a new mathematical model for HBV based on the fractal–fractional derivative with a generalized Mittag–Leffler kernel. Firstly, we discuss the fundamental properties, like the equilibria of the model and the primary reproduction number. Then, we investigate the existence of a unique solution to our model. We use an iterative method to solve the proposed model. Further, we discuss the stability of the model through stability theory. Finally, we offer some graphical illustrations for various values of the parameters.

Robustness and exploration between the interplay of the nonlinear co-dynamics HIV/AIDS and pneumonia model via fractional differential operators and a probabilistic approach

In this article, we considered a nonlinear compartmental mathematical model that assesses the effect of treatment on the dynamics of HIV/AIDS and pneumonia (H/A-P) co-infection in a human population at different infection stages. Understanding the complexities of co-dynamics is now critically necessary as a consequence. The aim of this research is to construct a co-infection model of H/A-P in the context of fractional calculus operators, white noise and probability density functions, employing a rigorous biological investigation. By exhibiting that the system possesses non-negative and bounded global outcomes, it is shown that the approach is both mathematically and biologically practicable. The required conditions are derived, guaranteeing the eradication of the infection. Furthermore, adequate prerequisites are established, and the configuration is tested for the existence of an ergodic stationary distribution. For discovering the system’s long-term behavior, a deterministic-probabilistic technique for modeling is designed and operated in MATLAB. By employing an extensive review, we hope that the previously mentioned approach improves and leads to mitigating the two diseases and their co-infections by examining a variety of behavioral trends, such as transitions to unpredictable procedures. In addition, the piecewise differential strategies are being outlined as having promising potential for scholars in a range of contexts because they empower them to include particular characteristics across multiple time frame phases. Such formulas can be strengthened via classical techniques, power law, exponential decay, generalized Mittag-Leffler kernels, probability density functions and random procedures. Furthermore, we get an accurate description of the probability density function encircling a quasi-equilibrium point if the effect of H/A-P minimizes the propagation of the co-dynamics. Consequently, scholars can obtain better outcomes when analyzing facts using random perturbations by implementing these strategies for challenging issues. Random perturbations in H/A-P co-infection are crucial in controlling the spread of an epidemic whenever the suggested circulation is steady and the amount of infection eliminated is closely correlated with the random perturbation level.

Numerical study and dynamics analysis of diabetes mellitus with co-infection of COVID-19 virus by using fractal fractional operator

COVID-19 is linked to diabetes, increasing the likelihood and severity of outcomes due to hyperglycemia, immune system impairment, vascular problems, and comorbidities like hypertension, obesity, and cardiovascular disease, which can lead to catastrophic outcomes. The study presents a novel COVID-19 management approach for diabetic patients using a fractal fractional operator and Mittag-Leffler kernel. It uses the Lipschitz criterion and linear growth to identify the solution singularity and analyzes the global derivative impact, confirming unique solutions and demonstrating the bounded nature of the proposed system. The study examines the impact of COVID-19 on individuals with diabetes, using global stability analysis and quantitative examination of equilibrium states. Sensitivity analysis is conducted using reproductive numbers to determine the disease’s status in society and the impact of control strategies, highlighting the importance of understanding epidemic problems and their properties. This study uses two-step Lagrange polynomial to analyze the impact of the fractional operator on a proposed model. Numerical simulations using MATLAB validate the effects of COVID-19 on diabetic patients and allow predictions based on the established theoretical framework, supporting the theoretical findings. This study will help to observe and understand how COVID-19 affects people with diabetes. This will help with control plans in the future to lessen the effects of COVID-19.

Fractional order cancer model infection in human with CD8+ T cells and anti-PD-L1 therapy: simulations and control strategy

In order to comprehend the dynamics of disease propagation within a society, mathematical formulations are essential. The purpose of this work is to investigate the diagnosis and treatment of lung cancer in persons with weakened immune systems by introducing cytokines (IL2&IL12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ IL_{2} \\& IL_{12}$$\\end{document}) and anti-PD-L1 inhibitors. To find the stable position of a recently built system TCDIL2IL12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$IL_{2} IL_{12}$$\\end{document}Z, a qualitative and quantitative analysis are taken under sensitive parameters. Reliable bounded findings are ensured by examining the generated system’s boundedness, positivity, uniqueness, and local stability analysis, which are the crucial characteristics of epidemic models. The positive solutions with linear growth are shown to be verified by the global derivative, and the rate of impact across every sub-compartment is determined using Lipschitz criteria. Using Lyapunov functions with first derivative, the system’s global stability is examined in order to evaluate the combined effects of cytokines and anti-PD-L1 inhibitors on people with weakened immune systems. Reliability is achieved by employing the Mittag-Leffler kernel in conjunction with a fractal-fractional operator because FFO provide continuous monitoring of lung cancer in multidimensional way. The symptomatic and asymptomatic effects of lung cancer sickness are investigated using simulations in order to validate the relationship between anti-PD-L1 inhibitors, cytokines, and the immune system. Also, identify the actual state of lung cancer control with early diagnosis and therapy by introducing cytokines and anti-PD-L1 inhibitors, which aid in the patients’ production of anti-cancer cells. Investigating the transmission of illness and creating control methods based on our validated results will both benefit from this kind of research.

Global stability and modeling with a non-singular kernel for fractional order heroin epidemic model: Insights from different population studies

There has been a worldwide epidemic of heroin that has affected people, families, societies, and cultures across the world. Now, the heroin epidemic has transitioned from heroin abuse to the overuse of synthetic narcotics, which are widely accessible and inexpensively produced. In this work, a novel mathematical approach is applied to investigate the dynamics of the heroin epidemic model and its harmful effect on society with different population data. A heroin model has been constructed with the importance of a non-singular kernel in the sense of a generalized Mittag-Leffler kernel. The well-posedness of the proposed model is proven via fixed-point theory. To examine the heroin model, two equilibrium states have been determined. These equilibrium states are proven to be locally and globally asymptotically stable. To analyze the behavior of heroin, a basic reproduction number and sensitivity analysis are used to determine the impact of different parameters mathematically as well as through simulations. To find the approximate solution, we implement the Toufik–Atangana numerical method at different fractional order values. The sensitivity of the heroin model is carried out, and 3-D graphs show the significance of the parameter involved in the model. Finally, the numerical outcomes are presented with different values of fractional parameters.

THE Q-ANALOGUES OF NONSINGULAR FRACTIONAL OPERATORS WITH MITTAG-LEFFLER AND EXPONENTIAL KERNELS

This paper is devoted to introducing a new [Formula: see text]-fractional calculus in the framework of Atangana–Baleanu ([Formula: see text]) and Caputo–Fabrizio ([Formula: see text]) operators. First, an appropriate [Formula: see text]-Mittag-Leffler function is defined, and then [Formula: see text]-analogues of fractional derivatives of Atangana–Baleanu–Riemann ([Formula: see text]) and Atangana–Baleanu–Caputo ([Formula: see text]) are derived. Next, the [Formula: see text]-analogues of proper fractional integrals in the [Formula: see text] sense are proved. Several important properties of these definitions are investigated by using the [Formula: see text]-Laplace transform. Additionally, a suitable [Formula: see text]-exponential function is defined, and the [Formula: see text]-analogues of [Formula: see text] fractional derivatives with their inverse operators are introduced. The higher-order extension of the [Formula: see text]-analogues of [Formula: see text] and [Formula: see text] fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of [Formula: see text]-[Formula: see text] calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus.

Power and Mittag–Leffler laws for examining the dynamics of fractional unemployment model: A comparative analysis

Unemployment is a major problem worldwide and is one of the key factors determining a nation’s economic status. The issue of unemployment is made more difficult globally by the ongoing rise in labor force participation and the scarcity of job positions. In this work, we study the unemployment model with two distinct fractional-order derivatives: the Caputo operator and the Atangana–Baleanu operator in the sense of Caputo (ABC). These derivatives under consideration are the operators widely utilized in modeling real-world phenomena in fractional dynamics. The existence and uniqueness of the solutions to the fractional model under consideration are ascertained using the fixed-point theory. The Hyers-Ulam analysis is employed to determine stability. For the numerical results, we present an Adams-type predictor–corrector (PC) technique for Caputo derivative and an extended Adams Bashforth (ABM) method for Atangana–Baleanu derivative. The outcomes achieved with the Atangana–Baleanu–Caputo and Caputo derivatives are identical to those of the regular case when fractional order ν=1.00. However, the results obtained change slightly as fractional order assumes values smaller than one, and this variation becomes most noticeable when the fractional order ν<0.72. This is because of the fractional derivative definitions’ underlying kernel. It is shown that the Mittag–Leffler kernel derivative provides better results for smaller fractional orders.

On a fractional derivative operator with a singular kernel: definition, properties and numerical simulation

This paper is concerned with proposing a novel nonlocal fractional derivative operator with a singular kernel. We considered a fractional integral operator as a single integral of convolution type combined with a Mittag-Leffler kernel of Prabhakar type. The proposed singular fractional derivative operator is formulated as a proper inverse of the considered integral operator. We provided some useful features and relationships of the proposed derivative and introduced comparisons with the Caputo derivative which can be utilized for potential applications. Next, we presented numerical solutions for some nonlinear fractional order models incorporating the proposed derivative using a numerical algorithm developed in this paper. As a case study, we discussed the dynamic behavior of a fractional logistic model with the proposed derivative.