Mittag-Leffler Kernel Research Articles

Fractional order age dependent Covid-19 model: An equilibria and quantitative analysis with modeling

The presence of different age groups in the populations being studied requires us to develop models that account for varying susceptibilities based on age. This complexity adds a layer of difficulty to predicting outcomes accurately. Essentially, there are three main age categories: 0−19 years, 20−64 years, and >64 years. However, in this article, we only focus on two age groups (20−64 years and >64 years) because the age category 0−19 years is generally perceived as having a lower susceptibility to the virus due to its consistently low infection rate during the pandemic period of this research, particularly in the countries being examined. In this paper, we presented an age-dependent epidemic model for the COVID-19 Outbreak in Kuwait, France, and Cameroon in the fractal-fractional (FF) sense of derivative with the Mittag–Leffler kernel. The study includes positivity, stability, existence results, uniqueness, stability, and numerical simulations. Globally, the age-dependent COVID-19 fractal fractional model is examined using the first and second derivatives of Lyapunov. Fixed point theory is used to derive the existence and uniqueness of the fractional-order model. The numerical scheme of this paper is based on the Newton polynomial and is tested for a particular case with numerical values from Kuwait, France, and Cameroon. In our analysis, we explore the significance of these distinct parameters incorporated into the model, focusing particularly on the impact of vaccination and fractional order on the progression of the epidemic. The results are getting closer to the classical case for the orders reaching 1 while all other solutions are different with the same behavior. Consequently, the fractal fractional order model provides more substantial insights into the epidemic disease. We open a novel viewpoint on enhancing an age-dependent model and applying it to real-world data and parameters. Such a study will help determine the behavior of the virus and disease control methods for a population.

On configuring new choatic behaviours for a variable fractional-order memristor-based circuit in terms of Mittag-Leffler kernel

Memristors, which are inevitably dynamical and memory components, could replace capacitors in the next-generation dynamical supercomputing converters that can exhibit behavioural characteristics, such as chaos. The nonlocal and nonsingular kernel Mittag-Leffler function (MLF) in the context of Caputo is researched in a new memristor framework via variable order. The existence and uniqueness of the fractional depiction of the specific process are confirmed using the Banach fixed-point approach for the contraction criterion. A newly developed numerical scheme for fractional-order systems introduced by Toufik and Atangana is utilized to obtain the phase portraits of the suggested system for various variable-order values of the system. The memristor-based model and many chaotic patterns of the Newton–Leipnik, Rabinovich–Fabrikant, Dadras, Aizawa, Thomas’ and four-wing types were the core issues of our study. The sensitivity of the chaotic systems of the considered model will be focused with precision using the bifurcation diagrams and the Lyapunov exponent. The stability of the equilibrium points of the commensurable fractional-order chaotic system will be addressed in the context of fractional calculus. In other words, we will use the standard Matignon criterion to address the problem of stability. The main attraction and novelty of this paper will be the use of the Lyapunov exponent (LE˜)s. It is demonstrated that the attempted framework is chaotic as well as extremely responsive to modifications in the parameters and the small initial conditions.

Mathematical study of fractal-fractional leptospirosis disease in human and rodent populations dynamical transmission

In both industrialized and developing nations, leptospirosis is one of the most underdiagnosed and under-reported diseases. It is known that people are more likely to contract a disease depending on their employment habits and the environment they live in, which varies from community to community. The absence of global data for morbidity and mortality has contributed to leptospirosis' neglected disease status even though it is a life-threatening illness and is widely acknowledged as a significant cause of pulmonary hemorrhage syndrome. This study aims to examine the impact of rodent-borne leptospirosis on the human population by constructing and evaluating a compartmental mathematical model using fractional-order differential equations. The model considers both the presence of disease-causing agents in the environment and the rate of human infection resulting from interactions with infected rodents and the environment. Through this approach, the research investigates the dynamics and implications of leptospirosis transmission in the context of human-rodent interactions and environmental factors. We create a fractal-fractional model using the mittag-leffler kernel. The positivity and boundedness of solutions are first discussed. The model equilibria and fundamental reproduction number are then presented. With the use of the Lyapunov function method, the solutions are subjected to global stability analysis. The fixed-point theory is used to derive the fractional-order model's existence and uniqueness. Solutions are produced using a two-step Lagrange polynomial in the generalized form of the Mittag-Leffler kernel to explore the effect of the fractional operator with numerical simulations, which shows the influence of the sickness due to the effect of different parameters involved. Such a study will aid in the development of control strategies to combat the disease in the community and an understanding of the behavior of the Leptospira virus.

A fractal–fractional model of Ebola with reinfection

In this article, we have studied the dynamics of the Ebola virus disease by means of fractional differentiation combined with a fractal dimension. It has been shown explicitly that the Ebola model is positively invariant. The fixed point theorem procedures check the solution’s existence and uniqueness to the model using the Mittag-Leffler kernel. The stability of the Ebola fractal–fractional model is studied using the Hyers–Ulam stable analysis. The numerical simulation for the trajectories of the proposed model is obtained using Lagrangian interpolation. Also, the numerical sensitivity analysis of the model is presented. For instance, a reduction in the rate of human-to-human contact results in a decline in the rate of infection, implying that the disease at a point will die out. Again, an increase in the loss of immunity rate to unity leads to a massive spread of the Ebola disease in the population. The findings in our work depict that we can achieve an Ebola-free state should the health sector give more education on maintaining a strong immune system and reducing human-to-human contact, especially during outbreaks of the Ebola disease, using measures like isolation and quarantine. Finally, using the fractal–fractional operator, we observed that any amount of memory variation and repetition in the outbreak of the disease influences the spread of the Ebola disease.

Analysis of a fractional order Bovine Brucellosis disease model with discrete generalized Mittag–Leffler kernels

Bovine Brucellosis, a zoonotic disease, can infect cattle in tropical and subtropical areas. It remains a critical issue for both human and animal health in many parts of the world, especially those where livestock is an important source of food and income. An efficient method for monitoring the illness’s increasing prevalence and developing low-cost prevention strategies for both its effects and recurrence is brucellosis disease modeling. We create a fractional-order model of Bovine Brucellosis using a discrete modified Atangana–Baleanu fractional difference operator of the Liouville–Caputo type. An analysis of the suggested system’s well-posedness and a qualitative investigation are both conducted. The examination of the Volterra-type Lyapunov function for global stability is supported by the first and derivative tests. The Lipschitz condition is also used for the model in order to meet the criterion of the uniqueness of the exact solution. We created an endemic and disease-free equilibrium. Solutions are built in the discrete generalized form of the Mittag-Leffler kernel in order to analyze the effect of the fractional operator with numerical simulations and emphasize the effects of the sickness due to the many factors involved. The capacity of the suggested model to forecast an infectious disease like brucellosis can help researchers and decision-makers take preventive actions.

Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations.

Fractional diffusion and Fokker-Planck equationsare widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous-time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integrodifferential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We study here the dynamics of the associated generalized Fokker-Planck equationsfrom the perspective of the moments, the time-averaged mean-squared displacements, and the autocovariance functions. We also study generalized Langevin equationsbased on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integrodifferential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integrodifferential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equationsfor the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equationsare unveiled to share similar properties with scaled and fractional Brownian motion, respectively.

On Bounds of k-Fractional Integral Operators with Mittag-Leffler Kernels for Several Types of Exponentially Convexities

This paper aims to study the bounds of k-integral operators with the Mittag-Leffler kernel in a unified form. To achieve these bounds, the definition of exponentially (α,h−m)−p-convexity is utilized frequently. In addition, a fractional Hadamard type inequality which shows the upper and lower bounds of k-integral operators simultaneously is presented. The results are directly linked with the results of many published articles.

Analytic solution of a fractional order mathematical model for tumour with polyclonality and cell mutation

It is well established that gliomas are heterogeneous (polyclonal), that the degree of heterogeneity always rises with grade. It is believed that the more cancerous cells have a greater propensity to mutate, increasing heterogeneity. Consequently, it is anticipated that a tumour will have a variety of cell types. In this work, a fractional diffusion model for tumour growth where two cell populations are assumed, which could have different diffusivity and proliferation rates, is studied and analysed. The coupled system is solved analytically via the zeroth order finite Hankel transform employing the fractional derivatives with exponential and Mittag-Leffler kernels, respectively and the obtained solutions simulated using MATLAB. Important highlights of the simulations include: (i.) using the Caputo fractional derivative, and considering the scenario when the rate of loss of cell population u(r,t) is α=0.5, keeping the other values of the parameters ν=1.2,β=0.1, it is observed that, near the centre of the tumour where biopsy is to be carried out, the tumour cell concentrations u(r,t) dominates v(r,t) when t=0.5 while tumour cell population v(r,t) dominates u(r,t) at time t=2.0; (ii.) with the Atangana–Baleanu derivative, and considering the same scenario it is observed that, near the centre of the tumour, the cell concentration v(r,t) dominates u(r,t) when t=0.5 while the dominance is much higher at time t=2.0. Thus, it is concluded that, the nature of kernel in the fractional operator could indeed alter the dominance between the tumour cell concentrations.

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

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

New Solutions of the Fractional Differential Equations With Modified Mittag-Leffler Kernel

Abstract This paper is concerned with some features of the modified Caputo-type Mittag-Leffler fractional derivative operator and its associated fractional integral operator. Mainly, new types of solutions for fractional differential equations with Mittag-Leffler kernel are generated based on a numerical algorithm developed in this paper. The suggested algorithm is used to describe the solution behavior of models involving modified Caputo-type Mittag-Leffler fractional derivatives. The results described in this paper are expected to be effectively employed in the area of simulating related fractional models.

Exploring the role of fractal-fractional operators in mathematical modelling of corruption

In the proposed manuscript, we present a novel mathematical model for analysing the persistence of corruption in human communities, based on fractal-fractional concepts and the Mittag-Leffler kernel law. Corruption is considered analogous to a disease that can spread and influence others who are free from corruption. Our model evaluates the equilibrium points of corruption and tests their stability using the corruption reproduction number. We also apply the fixed point theory concept to check for the existence and uniqueness of a solution, in the context of a fractional fractal operator. Solution stability is verified using the perturbed Ulam Hyers technique, and an approximate solution is obtained through the use of Lagrangian polynomials. To test the validity of our model, we simulate all compartments at different fractional orders and time durations, providing additional insights into the dynamics of corruption beyond natural orders.

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.

Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators

Abstract Nonlocal operators with different kernels were used here to obtain more general harmonic oscillator models. Power law, exponential decay, and the generalized Mittag-Leffler kernels with Delta-Dirac property have been utilized in this process. The aim of this study was to introduce into the damped harmonic oscillator model nonlocalities associated with these mentioned kernels and see the effect of each one of them when computing the Bode diagram obtained from the Laplace and the Sumudu transform. For each case, we applied both the Laplace and the Sumudu transform to obtain a solution in a complex space. For each case, we obtained the Bode diagram and the phase diagram for different values of fractional orders. We presented a detailed analysis of uniqueness and an exact solution and used numerical approximation to obtain a numerical solution.

Fractional simulations for slip flow of Casson CMC-CNTs hybrid nanofluid with Mittag-Leffler kernel and Prabhakar fractional simulations

This work aims to investigate the mixed convection heat transfer in hybrid nanofluids across a vertical pours plate under the influence of Newtonian heating. It is also taken into account walls slipping slide and the applied magnetic field is angled. By using an effective mathematical fractional scheme known as the Prabhakar time-fractional derivative, the non-dimensional leading equations are converted into the fractional model. Water and carboxymethyl cellulose (CMC) are the foundation fluids for carbon nanotubes (CNTs), which are considered as base nanoparticles. The Laplace scheme is used to solve the momentum and energy equations, and several numerical techniques are taken into consideration for the Laplace inverse. The effects of various restrictions are visualized and quantitatively examined for both leading equations. It is concluded the consideration of CMC-based hybrid nanofluid has present peak thermal impact as compared to other hybrid nanofluid models. The increasing applications of nanoparticles volume fraction shows increasing trends for thermal phenomenon for both SWCNTs and MWCNTs.

On stability analysis of a fractional volterra integro delay differential equation in the context of Mittag–Leffler kernel

This paper deals with a nonlinear multi-derivative fractional Volterra integro-differential equation with finite delay involving a non-singular Mittag–Leffler kernel. Krasnoselskii's fixed-point theorem is used to obtain the current result. Also, we establish the generalization of Pachapate's inequality and provide its applications for the existence, uniqueness, stability, and data dependence of the suggested problem. All the results are justified by an example.

Controllability and analysis of sustainable approach for cancer treatment with chemotherapy by using the fractional operator

This research proposes a mathematical model of cancer treatment with chemotherapy using a fractal fractional Mittag-Leffler operator with non-integer order. The model is analyzed both qualitatively and quantitatively. The control of cancer treatment with chemotherapy effects is established using a fractal fractional operator with a Mittag-Leffler kernel and control theory. The effect of global derivative, existence of unique solutions, and boundedness of proposed system is verified. Solutions are produced using a two-step Lagrange polynomial, and numerical simulations are carried out to illustrate the theoretical results. Cancer treatment with chemotherapy effects are verified from our justified results and predictions are made by simulation using MATLAB. Controllability and observability of the proposed system is also treated for linear control system to observe the close loop design with chemotherapy as an input and treated cells as the output.

Correction to: A Fractional Order Covid-19 Epidemic Model with Mittag-Leffler Kernel

Correction to: A Fractional Order Covid-19 Epidemic Model with Mittag-Leffler Kernel

On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) \left({\tilde{h}}_{1},{\tilde{h}}_{2}) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel

Abstract Local fractional integral inequalities of Hermite-Hadamard type involving local fractional integral operators with Mittag-Leffler kernel have been previously studied for generalized convexities and preinvexities. In this article, we analyze Hermite-Hadamard-type local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) \left({\tilde{h}}_{1},{\tilde{h}}_{2}) -preinvex function comprising local fractional integral operators and Mittag-Leffler kernel. In addition, two examples are discussed to ensure that the derived consequences are correct. As an application, we construct an inequality to establish central moments of a random variable.

New Fractional Modelling and Simulations of Prey–Predator System with Mittag–Leffler Kernel

New Fractional Modelling and Simulations of Prey–Predator System with Mittag–Leffler Kernel

A Fractional Order Covid-19 Epidemic Model with Mittag–Leffler Kernel

We consider a nonlinear fractional-order Covid-19 model in a sense of the Atagana–Baleanu fractional derivative used for the analytic and computational studies. The model consists of six classes of persons, including susceptible, protected susceptible, asymptomatic infected, symptomatic infected, quarantined, and recovered individuals. The model is studied for the existence of solution with the help of a successive iterative technique with limit point as the solution of the model. The Hyers–Ulam stability is also studied. A numerical scheme is proposed and tested on the basis of the available literature. The graphical results predict the curtail of spread within the next 5000 days. Moreover, there is a gradual increase in the population of protected susceptible individuals.