Mittag-Leffler Law Research Articles

Analyzing the dynamics of human papillomavirus transmission via fractal and fractional dimensions under Mittag-Leffler Law

Analyzing the dynamics of human papillomavirus transmission via fractal and fractional dimensions under Mittag-Leffler Law

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.

Stability and complex dynamical analysis of COVID-19 epidemic model with non-singular kernel of Mittag-Leffler law

Stability and complex dynamical analysis of COVID-19 epidemic model with non-singular kernel of Mittag-Leffler law

Ultrafast Diffusion Modeling via the Riemann–Liouville Nonlocal Structural Derivative and Its Application in Porous Media

Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of cement materials, the motion of particles on their surfaces satisfies the conditions for ultrafast diffusion. The investigation of the diffusion behavior in cementitious materials is crucial for predicting the mechanical properties of cement. In this study, we first attempted to investigate the dynamic of ultrafast diffusion in cementitious materials underlying the Riemann–Liouville nonlocal structural derivative. We constructed a Riemann–Liouville nonlocal structural derivative ultrafast diffusion model with an exponential function and then extended the modeling strategy using the Mittag–Leffler function. The mean square displacement is analogous to the integral of the corresponding structural derivative, providing a reference standard for the selection of structural functions in practical applications. Based on experimental data on cement mortar, the accuracy of the Riemann–Liouville nonlocal structural derivative ultrafast diffusion model was verified. Compared to the power law diffusion and the exponential law diffusion, the mean square displacement with respect to the Mittag–Leffler law is closely tied to the actual data. The modeling approach based on the Riemann–Liouville nonlocal structural derivative provides an efficient tool for depicting ultrafast diffusion in porous media.

A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law

In this paper, new generalized weighted fractional derivatives with respect to another function are derived in the sense of Caputo and Riemann–Liouville involving a new modified version of a generalized Mittag–Leffler function with three parameters, as well as their corresponding fractional integrals. In addition, several new and existing operators of nonsingular kernels are obtained as special cases of our operator. Many important properties related to our new operator are introduced, such as a series version involving Riemann–Liouville fractional integrals, weighted Laplace transforms with respect to another function, etc. Finally, an example is given to illustrate the effectiveness of the new results.

Analysis of age wise fractional order problems for the Covid-19 under non-singular kernel of Mittag-Leffler law

The developed article considers SIR problems for the recent COVID-19 pandemic, in which each component is divided into two subgroups: young and adults. These subgroups are distributed among two classes in each compartment, and the effect of COVID-19 is observed in each class. The fractional problem is investigated using the non-singular operator of Atangana Baleanu in the Caputo sense (ABC). The existence and uniqueness of the solution are calculated using the fundamental theorems of fixed point theory. The stability development is also determined using the Ulam-Hyers stability technique. The approximate solution is evaluated using the fractional Adams-Bashforth technique, providing a wide range of choices for selecting fractional order parameters. The simulation is plotted against available data to verify the obtained scheme. Different fractional-order approximations are compared to integer-order curves of various orders. Therefore, this analysis represents the recent COVID-19 pandemic, differentiated by age at different fractional orders. The analysis reveals the impact of COVID-19 on young and adult populations. Adults, who typically have weaker immune systems, are more susceptible to infection compared to young people. Similarly, recovery from infection is higher among young infected individuals compared to infected cases in adults.

Stability analysis of a non-singular fractional-order covid-19 model with nonlinear incidence and treatment rate

In this paper, a non-singular SIR model with the Mittag-Leffler law is proposed. The nonlinear Beddington-DeAngelis infection rate and Holling type II treatment rate are used. The qualitative properties of the SIR model are discussed in detail. The local and global stability of the model are analyzed. Moreover, some conditions are developed to guarantee local and global asymptotic stability. Finally, numerical simulations are provided to support the theoretical results and used to analyze the impact of face masks, social distancing, quarantine, lockdown, immigration, treatment rate of the disease, and limitation in treatment resources on COVID-19. The graphical results show that face masks, social distancing, quarantine, lockdown, immigration, and effective treatment rates significantly reduce the infected population over time. In contrast, limitation in the availability of treatment raises the infected population.

Piecewise Business Bubble System under Classical and Nonsingular Kernel of Mittag-Leffler Law.

This study aims to investigate the dynamics of three agents in the emerging business bubble model based on the Mittag-Leffler law pertaining to the piecewise classical derivative and non-singular kernel. By generalizing the business bubble dynamics in terms of fractional operators and the piecewise concept, this study presents a new perspective to the field. The entire set of intervals is partitioned into two piecewise intervals to analyse the classical order and conformable order derivatives of an Atangana-Baleanu operator. The subinterval analysis is critical for removing discontinuities in each sub-partition. The existence and uniqueness of the solution based on a piecewise global derivative are tested for the considered model. The approximate root of the system is determined using the piecewise numerically iterative technique of the Newton polynomial. Under the classical order and non-singular law, the approximate root scheme is applied to the piecewise derivative. The curve representation for the piece-wise globalised system is tested by applying the data for the classical and different conformable orders. This establishes the entire density of each compartment and shows a continuous spectrum instead of discrete dynamics. The concept of this study can also be applied to investigate crossover behaviours or abrupt changes in the dynamics of the values of each market.

An Analytical Study of Fractional Delay Impulsive Implicit Systems with Mittag-Leffler Law

An Analytical Study of Fractional Delay Impulsive Implicit Systems with Mittag-Leffler Law

Theoretical and numerical analysis of a prey–predator model (3-species) in the frame of generalized Mittag-Leffler law

Abstract In the present paper, a new fractional order predator–prey model is considered. The applied fractional operator is a generalized Atangana–Baleanu–Caputo (ABC) derivative, which does not require any restrictions on the initial conditions as in the case of classical ABC fractional derivatives. On the theoretical aspect, we prove the existence, uniqueness, and Ulam–Hyers stability results by using some fixed point theorems and nonlinear analysis techniques. The numerical aspect discusses the approximation solutions for the proposed model by applying the generalized scheme of the Adams–Bashforth technique. At the end, we explain the behavior of the solution to the studied model through graphical representations and numerical simulations.

A modeling approach to stress-dependent porosity and permeability decays of rocks

The rock porosity or permeability highly depends on stress, a crucial property in resource exploitation and geological storage engineering. However, due to factors such as rock types, loading paths, and loading ranges, the stress-porosity/permeability relationships are pretty different, exhibiting linear, nonlinear, or even heavy-tailed characteristics. The exponential and power-law models, two mainstream empirical relationships for describing the rock permeability and porosity decays, are used to fit the data with “heavy tail” characteristics but yield poor fitting or outrageous predictions for specific stress ranges. Based on the physical interpretation of the compaction-induced microstructural evolution inside the rock, this paper proposes fractional-order relaxation equations, which consider the memory effect of permeability/porosity variations with stress, leading to accurate descriptions of effective stress-porosity/permeability relationships by the Mittag-Leffler (ML) law. The fitting on low-permeability shales and relatively high-permeability sandstones shows that the ML law agrees better with the experimental data, especially with “heavy tail” characteristics than the two classical laws. Moreover, the numerical solutions for the proposed ML models are presented via the predictor-corrector algorithm. The relationship between the ML, exponential, and power laws is also discussed.

On Implicit Time–Fractal–Fractional Differential Equation

An implicit time–fractal–fractional differential equation involving the Atangana’s fractal–fractional derivative in the sense of Caputo with the Mittag–Leffler law type kernel is studied. Using the Banach fixed point theorem, the well-posedness of the solution is proved. We show that the solution exhibits an exponential growth bound, and, consequently, the long-time (asymptotic) property of the solution. We also give examples to illustrate our problem.

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.

On three dimensional fractal dynamics with fractional inputs and applications

&lt;abstract&gt;&lt;p&gt;The environment around us naturally represents number of its components in fractal structures. Some fractal patterns are also artificially simulated using real life mathematical systems. In this paper, we use the fractal operator combined to the fractional operator with both exponential and Mittag-leffler laws to analyze and solve generalized three-dimensional systems related to real life phenomena. Numerical solutions are provided in each case and applications to some related systems are given. Numerical simulations show the existence of the models' initial three-dimensional structure followed by its self- replication in fractal structure mathematically produced. The whole dynamics are also impacted by the fractional part of the operator as the derivative order changes.&lt;/p&gt;&lt;/abstract&gt;

Solution of Linear Fuzzy Fractional Differential Equations Using Fuzzy Natural Transform

The purpose of this paper is to apply the fuzzy natural transform (FNT) for solving linear fuzzy fractional ordinary differential equations (FFODEs) involving fuzzy Caputo’s H-difference with Mittag-Leffler laws. It is followed by proposing new results on the property of FNT for fuzzy Caputo’s H-difference. An algorithm was then applied to find the solutions of linear FFODEs as fuzzy real functions. More specifically, we first obtained four forms of solutions when the FFODEs is of order α∈(0,1], then eight systems of solutions when the FFODEs is of order α∈(1,2] and finally, all of these solutions are plotted using MATLAB. In fact, the proposed approach is an effective and practical to solve a wide range of fractional models.

Applications of Magnetohydrodynamic Couple Stress Fluid Flow between Two Parallel Plates with Three Different Kernels

In this paper, we investigate the implementations of newly introduced nonlocal differential operators as convolution of power law, exponential decay law, and the generalized Mittag-Leffler law with fractal derivative in fluid dynamics. The new operators are referred as fractal-fractional differential operators. The governing equations for the problem are constructed with the fractal-fractional differential operators. We present the stability analysis and the error analysis.

Local existence and uniqueness for a fractional SIRS model with Mittag–Leffler law

In this paper, we study an epidemic model with Atangana-Baleanu-Caputo fractional derivative. We obtain a special solution using an iterative scheme via Laplace transformation. Uniqueness and existence of a solution using the Banach fixed point theorem are studied. A detailed analysis of the stability of the special solution is presented. Finally, our generalized model in the derivative sense is solved numerically by the Adams-Bashforth-Moulton method.

Analysis of MHD Couette flow by fractal-fractional differential operators

Abstract In this paper, an analysis is carried out to study the MHD Couette flow (flow between two parallel plates such that the upper plate is moving with constant velocity while the lower plate is at rest) for an incompressible viscous fluid under isothermal conditions. The governing equations are developed from the problem, formulated with the recently presented fractal-fractional operators in Riemann–Liouville sense with power law, exponential decay and the Mittag–Leffler law kernels. For each operator, we present a comprehensive analysis including, the numerical solutions, stability analysis and error analysis. We apply very accurate method to get the desired results. We demonstrate the numerical simulations to prove the efficiency of the proposed method.

An analytic study on the approximate solution of a nonlinear time‐fractional Cauchy reaction–diffusion equation with the Mittag–Leffler law

The main aim of the current article is considering a nonlinear time‐fractional Cauchy reaction–diffusion equation with the Mittag–Leffler law and deriving its approximate analytical solution in a systematic way. More precisely, after reformulating the nonlinear time‐fractional Cauchy reaction–diffusion equation with the Mittag–Leffler law, its approximate analytical solution is derived formally through the use of the homotopy analysis transform method (HATM) which is based on the homotopy method and the Laplace transform. The existence and uniqueness of the solution of the nonlinear time‐fractional Cauchy reaction–diffusion equation with the Mittag–Leffler law are also studied by adopting the fixed‐point theorem. In the end, by considering some two‐ and three‐dimensional graphs, the influence of the order of time‐fractional operator on the displacement is examined in detail.

An efficient approach for fractional nonlinear chaotic model with Mittag-Leffler law

In this work, we exemplify the behaviour of the nonlinear model of arbitrary order differential equations by adopting q-homotopy analysis transform method (q-HATM). In the present study, the illustrated scheme is a graceful amalgamation of Laplace transform with q-homotopy analysis algorithm and we considered arbitrary order derivative using Atangana-Baleanu (AB) operator. The suggested nonlinear system exhibits chaotic behaviour in nature with respect to considered initial conditions. Fixed point hypothesis heard present the existence and uniqueness for the attained solution. We exemplified suggested arbitrary order system with to illustrate and confirm the efficiency of the projected solution procedure. Further, the numerical simulation is illustrated and also the chaotic behaviour of the obtained result captured with respect to arbitrary order in terms of plots. The obtained results confirm the projected scheme is highly methodical, easy to implement and very powerful to exemplify the nature of the dynamical system of arbitrary order.