Mittag-Leffler Function Research Articles

A new generalized beta function associated with statistical distribution and fractional kinetic equation

Several authors have extensively investigated beta function, hypergeometric function and confluent hypergeometric function, their extensions and generalizations due to their several application in many areas of engineering, probability theory and science. The main purpose of this paper is to present a new generalization of extended beta function, hypergeometric function and confluent hypergeometric function with the help of m-parameter Mittag-Leffler function, as well as examine some important properties like integral representations, differential formula and summation formulas. We also examine generalized Caputo fractional derivative operator with the help of m-parameter Mittag-Leffler function with associated properties using the generalized beta function. We define a new beta distribution involving the new generalized beta function. The mean, variance, coefficient of variance, moment generating function and characteristic function and cumulative distribution are derived. Further, we derive the solution of fractional Kinetic equation involving generalized hypergeometric function.

Investigation of MHD fractionalized viscous fluid and thermal memory with slip and Newtonian heating effect: a fractional model based on Mittag-Leffler kernel

This paper introduces an innovative approach for modelling unsteady incompressible natural convection flow over an inclined oscillating plate with an inclined magnetic effect that employs the Atangana-Baleanu time-fractional derivative (having a non-singular and non-local kernel) and the Mittag-Leffler function. The fractional model, which includes Fourier and Fick's equations, investigates memory effects and is solved using the Laplace transform. The Mittag-Leffler function captures power-law relaxation dynamics, which improves our understanding of thermal and fluid behaviour. Graphical examination shows the influence of fractional and physically involved parameters, leading to the conclusion that concentration, temperature, and velocity profiles initially grow and then decrease asymptotically with time. Moreover, the study emphasizes the impact of effective Prandtl and Schmidt numbers on temperature, concentration, and velocity levels in the fluid.

The well-posedness of semilinear fractional dissipative equations on [formula omitted]

In this paper, we study the well-posedness of time-space fractional dissipative equations. Combining the Hörmander-multipliers theory and asymptotic property of the Mittag-Leffler functions, we give a useful method to estimate the solution operator which is independent of the C0-semigroup generated by the fractional Laplace operator (−Δ)β2 and the Mainardi's Wright type function. Furthermore, we discuss the global/local well-posedness in some appropriate time-space Lebesgue space and Besov spaces. We also obtain several results about blow-up alternative and asymptotic behavior. The approaches are based on the Hörmander-multipliers theory, Gagliardo-Nirenberg inequalities, fixed point techniques, Sobolev embedding and some methods in harmonic analysis.

Multivariate Mittag-Leffler Solution for a Forced Fractional-Order Harmonic Oscillator

The harmonic oscillator is a fundamental physical–mathematical system that allows for the description of a variety of models in many fields of physics. Utilizing fractional derivatives instead of traditional derivatives enables the modeling of a more diverse array of behaviors. Furthermore, if the effect of the fractional derivative is applied to each of the terms of the differential equation, this will involve greater complexity in the description of the analytical solutions of the fractional differential equation. In this work, by using the Laplace method, the solutions to the multiple-term forced fractional harmonic oscillator are presented, described through multivariate Mittag-Leffler functions. Additionally, the cases of damped and undamped free fractional harmonic oscillators are addressed. Finally, through simulations, the effect of the fractional non-integer derivative is demonstrated, and the consistency of the result is verified when recovering the integer case.

Dynamics of propagation patterns: An analytical investigation into fractional systems

Recent years have seen a growing interest in fractional differential equations, particularly the fractional Chaffee-Infante ([Formula: see text]) equation, pivotal for understanding dynamics governed by fractional orders in specific physical systems. Exploring solitary wave solutions, this study employs the extended Khater method and truncated Mittag-Leffler function properties to formulate tailored solutions for the ([Formula: see text]) model. Through a traveling wave ansatz, the equation transforms into a nonlinear ordinary differential equation, revealing intricate propagation patterns of solitary waves. Visual representations aid comprehension, while rigorous validation ensures solution precision, ultimately providing a comprehensive understanding of system responses to external stimuli. This study effectively integrates analytical and numerical methodologies to derive precise solitary wave solutions, with significant implications for advancing comprehension of complex phenomena in various disciplines governed by fractional-order dynamics.

On Uniformly Starlike Functions with Respect to Symmetrical Points Involving the Mittag-Leffler Function and the Lambert Series

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions ΤJ(αi). Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions f∈ΤJ(αi). In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass.

Geometric properties of an integral operator associated with Mittag-Leffler functions

The main object of this paper is to introduce a new integral operator associated with Mittag-Leffler function. Further, we obtain some sufficient condition for this integral operator belonging to certain classes of starlike functions.

Some Integral Representations, Transformations, and Differential Relations for Mittag-Leffler Function of Double Variables

The fundamental purpose of this research is to offer a natural progression in the mathematical properties and presentation of specific Mittag-Leffler-type functions with two variables. Integral representations, differential relations, and integral transforms are established. Special cases and the consequences of our main findings are also highlighted.

Solvability and stability of a class of fractional Langevin differential equations with the Mittag–Leffler function

Solvability and stability of a class of fractional Langevin differential equations with the Mittag–Leffler function

A study on the fractional Black–Scholes option pricing model of the financial market via the Yang-Abdel-Aty-Cattani operator

PurposeFinancial mathematics is one of the most rapidly evolving fields in today’s banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF).Design/methodology/approachThe homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity.FindingsIn this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function.Originality/valueThe Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives.

Quasi-Projective Synchronization of Discrete-Time Fractional-Order Complex-Valued BAM Fuzzy Neural Networks via Quantized Control

In this paper, we ponder a kind of discrete-time fractional-order complex-valued fuzzy BAM neural network. Firstly, in order to guarantee the quasi-projective synchronization of the considered networks, an original quantitative control strategy is designed. Next, by virtue of the relevant definitions and properties of the Mittag-Leffler function, we propose a novel discrete-time fractional-order Halanay inequality, which is more efficient for disposing of the discrete-time fractional-order models with time delays. Then, based on the new lemma, fractional-order h-difference theory, and comparison principle, we obtain some easy-to-verify synchronization criteria in terms of algebraic inequalities. Finally, numerical simulations are provided to check the accuracy of the proposed theoretical results.

Representations of Solutions of Time-Fractional Multi-Order Systems of Differential-Operator Equations

This paper is devoted to the general theory of systems of linear time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix-valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single-order systems, and, hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations, the representation formulas are still unknown, even in the case of fractional-order ordinary differential equations. In this paper, we obtain representation formulas for the solutions of arbitrary fractional multi-order systems of differential-operator equations. The existence and uniqueness theorems in appropriate topological vector spaces are also provided. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.

On the fractional q-integral operators involving q-analogue of Mittag-Leffler function

Abstract This paper aims to derive the images of the product of the generalized q-analogue of Mittag-Leffler functions and the basic binomial function, under the fractional q-integral operators of Kober and Weyl type. The derived findings are of a general nature and include, as specific examples, the Weyl q-integral operator, the q-derivative operators, and the Riemann–Liouville q-integral operator.

Polynomial decay of a linear system of PDEs via Caputo fractional‐time derivative

An in‐depth study and analysis of the stability of one‐dimensional Linear System of PDEs via Caputo time fractional derivative (LSCFD) was presented. We proved some stability results for the LSCFD in different Hilbert spaces. Indeed, by using Fourier analysis method and the properties of the Mittag–Leffler Function (MLF), some polynomial stability results for LSCFD have been established. Finally, as an application, we used finite difference methods well suited to integer and fractional order derivatives, and performed some numerical experiments to confirm the theoretical results.

The Solution of Caputo Fractional Partial Differential Equations By Sumudu Transform

In this paper, we obtained an explicit solutions of the fractional diffusion-wave equations involving partial Caputo fractional derivative by using Sumudu transform. The solutions of the fractional diffusion-wave equations obtained in terms of Mittag-Leffler function and generalized Wright function.Some illustrative examples are also given.

Uniqueness and Hyers–Ulam’s stability for a fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition

Abstract Introducing a pair-parameter matrix Mittag–Leffler function, we study the uniqueness and Hyers–Ulam stability to a new fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition using Banach’s contractive principle as well as Babenko’s approach in a Banach space. These investigations have serious applications since uniqueness and stability analysis are essential topics in various research fields. The techniques used also work for different types of differential equations with initial or boundary conditions, as well as integral equations. Moreover, we present a Python code to compute approximate values of our newly established pair-parameter matrix Mittag–Leffler functions, which extend the multivariate Mittag–Leffler function. A few examples are given to show applications of the key results obtained.

Constructive fractional models through Mittag-Leffler functions

Constructive fractional models through Mittag-Leffler functions

Application of Caputo fractional approach to MHD Casson nanofluid with alumina nanoparticles of various shape factors on an inclined quadratic translated plate

The present study aims to examine the effect of convective heat transfer of nanofluid past an inclined plate. This paper scrutinizes heat transfer over an inclined surface of alumina nanofluids under the influence of radiation and magnetic fields with temperature-dependent thermophysical properties. The mathematical findings of velocity and temperature distributions are examined and visualized using non-dimensional flow parameters. The core of the research is to know the impact of using shape effects of alumina nanoparticles with a variety of base fluids like water, ethylene glycol, and engine oil. The outcomes concluded that the friction of the non-Newtonian viscous fluid decreased with increasing the inclination of the surface. The governing equations of the flow have been developed by using a few approximations such as Boussinesq's, and Roseland's. Using the dimensional analysis, the initial governing equations are simplified to dimensionless partial differential equations. A fractional model is developed using the Caputo fractional derivative. The solution of the proposed fractional partial differential equations is obtained in terms of special functions, Wright function, and Mittag-Leffler function by using the Laplace transformation method. Based on the interpretation, it is found that a reduction in velocity is observed with an increase in the angle of inclination i.e., γ = π/3 to γ = π/2. From the analysis, it is found that alumina/engine oil can carry higher temperatures than compared to other nanofluids and thus is a good option for heat transfer devices.

Analyzing Observability of Fractional Dynamical Systems Employing ψ-Caputo Fractional Derivative

In this present article, we inquire into the observability of linear and nonlinear fractional dynamical systems in terms of ψ-Caputo fractional derivative. The observability Grammian matrix, which is positive definite and expressed by the Mittag-Leffler functions, is used to obtain the necessary and sufficient conditions of observability of linear fractional dynamical systems, and Banach’s fixed point theorem is used to get the sufficient conditions for the observability of nonlinear fractional systems. Three numerical examples are given to demonstrate the applicability of theoretical results for linear and nonlinear cases.

Asymptotic stability of nonlinear fractional delay differential equations with α ∈ (1, 2): An application to fractional delay neural networks.

We introduce a theorem on linearized asymptotic stability for nonlinear fractional delay differential equations (FDDEs) with a Caputo order α∈(1,2), which can be directly used for fractional delay neural networks. It relies on three technical tools: a detailed root analysis for the characteristic equation, estimation for the generalized Mittag-Leffler function, and Lyapunov's first method. We propose coefficient-type criteria to ensure the stability of linear FDDEs through a detailed root analysis for the characteristic equation obtained by the Laplace transform. Further, under the criteria, we provide a wise expression of the generalized Mittag-Leffler functions and prove their polynomial long-time decay rates. Utilizing the well-established Lyapunov's first method, we establish that an equilibrium of a nonlinear Caputo FDDE attains asymptotically stability if its linearization system around the equilibrium solution is asymptotically stable. Finally, as a by-product of our results, we explicitly describe the asymptotic properties of fractional delay neural networks. To illustrate the effectiveness of our theoretical results, numerical simulations are also presented.