Mittag-Leffler Research Articles

Mittag–Leffler Fractional Stochastic Integrals and Processes with Applications

We study Mittag–Leffler (ML) fractional integrals involved in the solution processes of a system of coupled fractional stochastic differential equations. We introduce the ML fractional stochastic process as a ML fractional stochastic integral with respect to a standard Brownian motion. We provide some representation formulas of solution processes in terms of Mittag–Leffler fractional integrals and processes. Computable expressions of the mean functions and of the covariances of such processes are specifically given. The application in neuronal modeling is provided, and all involved functions and processes are specifically determined. Numerical evaluations are carried out and some results are shown and discussed.

Computing the Mittag-Leffler function of a matrix argument

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions).

Some Approximate Division and Semigroup Identities for the Mittag-Leffler Function

It is known that the Mittag-Leffler (ML) function, Eα(z), a non-local extension of the Euler exponential function ez, does not enjoy the semigroup property while ez does. The purpose of this note is to show that Eα(z) does, however, for real t, s with s small enough, enjoy the approximate semigroup property, Eα(t+s)≈Eα(t)(1+Eα,α(t)/Eα(t))s. This follows from an approximation of limh→0+Eα(±(u+h)α)/Eα(±uα) , which also yields related expressions for α→1−, and is obtained from a recently proposed universal difference quotient representation for fractional derivatives. Graphical demonstrations are presented to show that the approximations are 'reasonably accurate' for 0h≤0.1, with virtually no distinction from identity for 0h≤0.01.

Mittag-Leffler function based security control for fractional-order complex network system subject to deception attacks via Observer-based AETS and its applications

The goal of this paper is to investigate the security control for uncertain fractional-order delayed complex network systems under deception attacks using the Mittag-Leffler function and observer-based adaptive event-triggered scheme (AETS) with the fractional commensurate order in q ∈ (0, 1). The adaptive event-triggering scheme is used during the data transmission process from the sensors to the observer, where the triggering threshold can be dynamically modified to reduce resource waste. We make a novel model for the estimation error system that takes into account both the effects of the adaptive event-triggered scheme and the effects of deception attacks. A sufficient condition is obtained to guarantee the stochastic mean-square stability of the augmented error system using the Mittag-Leffler (M-L) functions and the Lyapunov functional method and by using the singular value decomposition (SVD) and linear matrix inequality (LMI) techniques, the co-design problem of desired observer and controller gains is found, and it is shown that the solution ensures the stability of a closed-loop uncertain fractional-order complex networked system. At the end of this study, two numerical examples and diesel engine system model are given to show that the above findings are correct.

On Atangana–Baleanu fractional granular calculus and its applications to fuzzy economic models in market equilibrium

Based on relative-distance-measure fuzzy interval arithmetic (RDM-FIA), this paper presents several new concepts for fuzzy functions in fractional granular calculus with Mittag-Leffler (ML) kernels, such as Atangana–Baleanu (AB) fractional granular integrals, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. Then, we present formulas for the fuzzy granular Laplace transform on Caputo fractional granular derivatives, AB fractional granular derivatives, and AB Caputo fractional granular derivatives. In addition to the relationship between AB fractional granular derivatives and AB Caputo fractional granular derivatives established using the granular Laplace transform, new fractional Newton–Leibniz-type formulas for both derivatives and integrals are proved. As applications, we obtain the fundamental solutions for the fuzzy economic models in the framework of Caputo fractional granular derivatives and AB Caputo fractional granular derivatives by utilizing the fuzzy granular Laplace transform. Several graphical explanations are given to show the dynamic behavior of the fundamental solutions. Moreover, we illustrate the advantages of using RDM-FIA to study the mathematical models of uncertain dynamical systems. The results of this study are illustrated using several numerical examples.

Sliding mode control for the stabilization of fractional heat equations subject to boundary uncertainty

By adopting the sliding mode control (SMC) and the generalized Lyapunov method, the boundary feedback stabilization issue is studied for the fractional diffusion system subject to boundary control matched disturbance. The classical sliding surface and the fractional integral sliding function are constructed and sliding mode controllers are designed respectively to realize the Mittag-Leffler (M-L) stabilization of the considered system. The controller based on the newly-introduced fractional integral sliding function not only helps to relax the constraints on the disturbance but also realizes the same stabilization effect as that of the classical one. The well-posedness result of the solution is also obtained for discontinuous fractional heat equations. Besides, a numerical experiment validates the theoretical outcomes.

Unraveling forward and backward source problems for a nonlocal integrodifferential equation: A journey through operational calculus for Dzherbashian‐Nersesian operator

AbstractThis article primarily aims at introducing a novel operational calculus of Mikusiński's type for the Dzherbashian‐Nersesian operator. Using this calculus, we are able to derive exact solutions for the forward and backward source problems of a differential equation that features Dzherbashian‐Nersesian operator in time and intertwined with nonlocal boundary conditions. The initial condition is expressed in terms of Riemann‐Liouville integral. Solutions are presented using Mittag‐Leffler (ML) type functions. The outcomes related to the existence and uniqueness subject to certain conditions of regularity on the input data are developed.

On the analysis of fractional calculus operators with bivariate Mittag Leffler function in the kernel

Bivariate Mittag-Leffler (ML) functions are a substantial generalization of the univariate ML functions, which are widely recognized for their significance in fractional calculus. In the present paper, our initial focus is to investigate the fractional calculus properties of the integral and derivative operators with kernels including the Bivariate ML functions. Further, certain fractional Cauchy-type problems including these operators are considered. Also the numerical approximations of the Caputo type derivative operator are investigated. The theoretical results are justified by applications on examples. Furthermore, the theory of applying the same operators with respect to arbitrary monotonic functions is analyzed in this research.

Controllability of stochastic fractional systems involving state-dependent delay and impulsive effects

In this paper, the controllability concept of a nonlinear fractional stochastic system involving state-dependent delay and impulsive effects is addressed by employing Caputo derivatives and Mittag-Leffler (ML) functions. Based on stochastic analysis theory, novel sufficient conditions are derived for the considered nonlinear system by utilizing Krasnoselkii’s fixed point theorem. Correspondingly, the applicability of the derived theoretical results is indicated by an example.

CALCULUS OPERATORS AND SPECIAL FUNCTIONS ASSOCIATED WITH KOHLRAUSCH–WILLIAMS–WATTS AND MITTAG-LEFFLER FUNCTIONS

In this paper, many important formulas of the subtrigonometric, subhyperbolic, pretrigonometric, prehyperbolic, supertrigonometric, and superhyperbolic functions sin Wiman class are developed for the first time. The subsine, subcosine, subhyperbolic sine, and subhyperbolic cosine associated with Kohlrausch–Williams–Watts (KWW) function and their scaling-law ODEs are proposed. The supersine, supercosine, superhyperbolic sine, and superhyperbolic cosine functions associated with Mittag-Leffler (ML) function and their fractional ODEs are obtained. The conjectures for the supercosine functions containing ML function are presented in detail.

Generalized fractional negative binomial process

In this paper, we introduce a generalized fractional negative binomial process (GFNBP) by time changing the fractional Poisson process with an independent Mittag-Leffler (ML) Lévy subordinator. We study its distributional properties and its connection to PDEs. We examine the long-range dependence (LRD) property of the GFNBP and show that it is not infinitely divisible. The space-fractional and the non-homogeneous variants of the GFNBP are explored. Finally, simulated sample paths for the ML Lévy subordinator and the GFNBP are also presented.

Resonant behaviors of two coupled fluctuating-frequency oscillators with tempered Mittag-Leffler memory kernel

The dynamics of coupled oscillators and their collective behaviors are of great significance in various fields. This paper explores the resonant behaviors in the system composed of two coupled fluctuating-frequency oscillators with tempered Mittag-Leffler (M-L) memory kernel. The study begins by demonstrating that the two particles synchronize in the long-time limit, indicating that the mean-field behavior aligns with that of each individual particle. Accordingly, we derive an exact analytical expression for the steady-state output amplitude gain (OAG) of the mean field using the stochastic average method. Furthermore, the research investigates the diverse phenomena of generalized stochastic resonance (GSR) and examines how various system parameters influence GSR. To validate the theoretical analysis, numerical simulations are conducted to support and verify our findings.

Event-triggered boundary consensus control for multi-agent systems of fractional reaction–diffusion PDEs

The distributed consensus is considered for multi-agent systems (MASs), which characterized by fractional reaction–diffusion partial differential equations (RDPDEs) in this paper. Based on Lyapunov technique and linear matrix inequalities (LMIs) theory, the consensus can be realized via two novel event-triggered boundary control schemes. Firstly, a novel convergence principle subject to finite time is presented for the continuously differentiable function. Secondly, the cooling fin on surface of high-speed aerospace vehicle is remodeled by fractional RDPDEs system, and the well-posedness of presented system is discussed applying the monotone iterative approach. Thirdly, according to the presented static event-triggered boundary control strategy, the consensus criterion in finite time is addressed in the form of LMIs, in addition, the settling time is calculated accurately. Applying the dynamic event-triggered control protocol, the Mittag-Leffler (M-L) consensus condition is achieved. Moreover, the Zeno behaviors are ruled out for proposed event-triggered mechanisms. Finally, the high-speed aerospace vehicle model is presented to verify the effectiveness of the control performance.

On some relations in the class of multi-index Mittag-Leffler functions

In this paper, some properties related to the -parametric Mittag-Leffler (M-L) functions are investigated. More precisely, three families of such functions with different kinds of indices are examined. Appropriate estimates and asymptotic formulae are obtained and which are further used in proving the convergence of series of functions of these families in the whole complex plane, as well as in its compact subsets. Using these auxiliary results, different relations referring to the discussed families are obtained; generalizing the ones already known for particular cases of the parameters. Those are considered different representations of the -parametric M-L functions pertaining to the families discussed, and with which various equalities are established relating different sums that involve matrix coefficients. Some interesting properties are reached for particular choices of the matrices. Finally, bounds of the -parametric M-L function are given under a perturbed matrix argument.

A Hörmander–Fock space

ABSTRACT In a recent paper, we used a basic decomposition property of polyanalytic functions of order 2 in one complex variable to characterize solutions of the classical ∂ ¯ -problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander–Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by η = ( η n ) n ≥ 0 leading to a special entire function E ( z ) that is used to express the kernel function of the Hörmander–Fock space. We present also an example of a special function belonging to the class Mittag-Leffler (ML) introduced recently by Alpay et al. and apply a Bochner–Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes.

Boundary Mittag–Leffler stabilization of a class of time fractional‐order nonlinear reaction–diffusion systems

AbstractBoundary control design of a class of time fractional‐order nonlinear reaction–diffusion systems (FNRDSs) is considered in three cases: domain‐averaged measurement, collocated boundary point measurement, and anti‐collocated boundary point measurement. For domain‐averaged measurement and collocated boundary point measurement, boundary controllers are designed directly based on the measurements, respectively. For the anti‐collocated boundary point measurement, to overcome the difficulty that the measurement cannot be used for the controller design directly, an observer is constructed firstly, and then, an observer‐based boundary controller is designed. Sufficient conditions for Mittag–Leffler (M‐L) stability of the closed‐loop system are all provided in terms of linear matrix inequalities (LMIs). Numerical simulation results are provided to illustrate the feasibility and effectiveness of the proposed methods.

Relatively exact controllability of fractional stochastic delay system driven by Lévy noise

In this article, we consider the relatively exact controllability of fractional stochastic delay system (FSDS) driven by Lévy noise. First, we derive the solution of linear FSDS via delayed matrix functions of Mittag–Leffler (M‐L). Subsequently, by virtue of the controllability Grammian matrix, we explore the relatively exact controllability of linear FSDS. In addition, with the aid of Jensen's inequality, Hölder's inequality, and Itô's isometry, the existence and uniqueness of the considered nonlinear FSDS are investigated by employing Banach contraction principle. Thereafter, the relatively exact controllability of nonlinear FSDS is discussed. Finally, the theoretical results are supported through examples.

Mittag-Leffler stability of fractional-order quaternion-valued memristive neural networks with generalized piecewise constant argument

This paper studies the global Mittag-Leffler (M-L) stability problem for fractional-order quaternion-valued memristive neural networks (FQVMNNs) with generalized piecewise constant argument (GPCA). First, a novel lemma is established, which is used to investigate the dynamic behaviors of quaternion-valued memristive neural networks (QVMNNs). Second, by using the theories of differential inclusion, set-valued mapping, and Banach fixed point, several sufficient criteria are derived to ensure the existence and uniqueness (EU) of the solution and equilibrium point for the associated systems. Then, by constructing Lyapunov functions and employing some inequality techniques, a set of criteria are proposed to ensure the global M-L stability of the considered systems. The obtained results in this paper not only extends previous works, but also provides new algebraic criteria with a larger feasible range. Finally, two numerical examples are introduced to illustrate the effectiveness of the obtained results.

Two discrete Mittag-Leffler extensions of the Cayley-exponential function

<abstract><p>Nabla discrete fractional Mittag-Leffler (ML) functions are the key of discrete fractional calculus within nabla analysis since they extend nabla discrete exponential functions. In this article, we define two new nabla discrete ML functions depending on the Cayley-exponential function on time scales. While, the nabla discrete ML function $ E_{\overline{\gamma}} (\lambda, t) $ converges for $ |\lambda| < 1 $, both of the defined discrete functions converge for more relaxed $ \lambda $. The nabla discrete Laplace transforms of the newly defined functions are calculated and confirmed as well. Some illustrative graphs for the two extensions are provided.</p></abstract>

Controllability of Impulsive Neutral Fractional Stochastic Systems

The study of dynamic systems appears in various aspects of dynamical structures such as decomposition, decoupling, observability, and controllability. In the present research, we study the controllability of fractional stochastic systems (FSF) and examine the Poisson jumps in finite dimensional space where the fractional impulsive neutral stochastic system is controllable. Sufficient conditions are demonstrated with the aid of fixed point theory. The Mittag-Leffler (ML) matrix function defines the controllability of the Grammian matrix (GM). The relation to symmetry is clear since the controllability Grammian is a hermitian matrix (since the integrand in its definition is hermitian) and this is the complex version of a symmetric matrix. In fact, such a Grammian becomes a symmetric matrix in the specific scenario where the controllability Grammian is a real matrix. Some examples are provided to demonstrate the feasibility of the present theory.