Third-Order Differential Subordinations for Analytic Functions Associated with Generalized Mittag-Leffler Functions

In this paper, third-order differential subordination results are obtained for analytic functions associated with an operator defined by the normalized form of the generalized Mittag-Leffler functions. Some particular cases involving Mittag-Leffler and hyperbolic functions are also considered.     

On M-Wright transforms and time-fractional diffusion equations

In this paper, we consider the Mittag-Leffler operator as an analytical solution of time-fractional diffusion equation in the Caputo sense. This solution is presented by an integral representation in terms of the M-Wright functions and the exponential operators. Further, we study the Mittag-Leffler operators associated with the Legendre and Bessel diffusion equations. Finally, we extend the obtained integral representation for the time-fractional diffusion equation of distributed order.     

Ultimate boundedness of impulsive fractional differential equations

This article investigates nonlinear impulsive Caputo fractional differential equations. Utilizing Lyapunov functions, Laplace transforms of fractional derivatives and boundedness of Mittag-Leffler functions, several sufficient conditions are derived to ensure the global ultimate boundedness and the exponential stability of the systems. An example is given to explain the obtained results.     

无穷积分余项与其被积函数比值的收敛阶

对一类函数的无穷积分余项与该函数的比值得到当x趋于无穷大时的收敛阶,这类函数是幂函数与指数函数的乘积函数,并将其应用到Mittag-Leffler函数.同时考虑了对应的级数情形.     

Ultimate boundedness of impulsive fractional delay differential equations

This paper concerns with the ultimate boundedness problem for impulsive fractional delay differential equations. Based on the impulsive fractional differential inequality, the boundedness of Mittag-Leffler functions, and the successful construction of suitable Lyapunov functionals, some algebraic criteria are derived for testing the global ultimate boundedness of the equations, and the estimations of the global attractive sets are provided as well. One example is also given to show the effectiveness of the obtained theoretical results.     

Green's function of two-dimensional time-fractional diffusion equation using addition formula of Wright function

In this paper, using the Mellin transform of Wright function we derive an addition formula for the Wright function. In some special cases, addition formulas for the Hermite, Bessel and Mittag-Leffler functions are also given and the Green's function of two-dimensional time-fractional diffusion equation is presented in the whole plane.     

A numerical-analytical solution of multi-term fractional-order differential equations

In this paper, a solution to initial value problems for fractional-order linear commensurate multi-term differential equations with Caputo derivatives is presented. The solution is obtained in the form of a finite sum of the Mittag-Leffler–type functions and the meta-trigonometric cosine function by using a numerical-analytical method. The results of presented numerical experiments show that for high accuracy calculations of these functions, the multi-precision arithmetic must be applied. The approach for solving of the initial value problems for generalized Basset equation, generalized Bagley-Torvik equation, and multi-term fractional equation is demonstrated.     

$(p,q)$-Analogue of Mittag-Leffler Function with $(p,q)$-Laplace Transform

The aim of this paper is to define $(p, q)$-analogue of Mittag-Leffler Function, by using $(p, q)$-Gamma function. Some transformation formulae are also derived by using the $(p, q)$-derivative. The $(p, q)$-analogue for this function provides elegant generalization of $q$-analogue of Mittag-Leffler function in connection with $q$-calculus. Moreover, the $(p, q)$-Laplace Transform of the Mittag-Leffler function has been obtained. Some special cases have also been discussed.     

A linear operator associated with the Mittag-Leffer function and related conformal mappings

In the present paper, we introduce a linear operator associated with the Mittag-Leffler function. Some convolution properties of meromorphic functions involving this operator are given.     

On a Multivariable Class of Mittag-Leffler Type Functions

The present paper studies and investigates a class of Mittag-Leffler type multivariable functions. We derive the necessary convergence conditions and establish several properties associated with this class and those related with the corresponding class of fractional integral operators. New extensions of the introduced definitions and special cases of some of the results are also pointed out.     

On the numerical computation of the Mittag-Leffler function

Recently simple limiting functions establishing upper and lower bounds on the Mittag-Leffler function were found. This paper follows those expressions to design an efficient algorithm for the approximate calculation of expressions usual in fractional-order control systems. The numerical experiments demonstrate the superior efficiency of the proposed method.     

Mittag-Leffler vector random fields with Mittag-Leffler direct and cross covariance functions

In terms of the two-parameter Mittag-Leffler function with specified parameters, this paper introduces the Mittag-Leffler vector random field through its finite-dimensional characteristic functions, which is essentially an elliptically contoured one and reduces to a Gaussian one when the two parameters of the Mittag-Leffler function equal 1. Having second-order moments, a Mittag-Leffler vector random field is characterized by its mean function and its covariance matrix function, just like a Gaussian one. In particular, we construct direct and cross covariances of Mittag-Leffler type for such vector random fields.     

On a sum with a three-parameter Mittag-Leffler function

The bounding of the two-parameter Mittag-Leffler function is discussed. As a particular consequence, we present two proofs for a sum involving a three-parameter Mittag-Leffler function. This sum can be seen as a generalization of a relation involving a product of two exponential functions. Particular cases are recovered.     

On One Class of Operators Associated with Differential Equations of Fractional Order

We carry out spectral analysis of one class of integral operators associated with fractional order differential equations applicable in mechanics. We establish connection between the eigenvalues of these operators and the zeros of Mittag-Leffler type functions. We give sufficient conditions for complete nonselfadjointness.     

分数阶常微分方程的改进精细积分法

基于Mittag-Leffler函数的定义式,构造Mittag-Leffler矩阵函数的精细迭代计算格式.与常规指数函数的迭代格式相比,迭代递推中多了修正项,其表达式与分数阶导数的阶次有关.对于以Caputo分数导数定义的动力学分数阶常微分方程,使用基于Mittag-Leffler函数的精细积分法可计算方程解在各时间段端点对应函数值.算例表明了所提计算方法的有效性,其精度可由所增加修正项的阶次控制.     

Correlation structure of time-changed Pearson diffusions

The stochastic solution to diffusion equations with polynomial coefficients is called a Pearson diffusion. If the time derivative is replaced by a distributed order fractional derivative, the stochastic solution is called a distributed order fractional Pearson diffusion. This paper develops a formula for the covariance function of distributed order fractional Pearson diffusion in the steady state, in terms of generalized Mittag-Leffler functions. The correlation function decays like a power law. That formula shows that distributed order fractional Pearson diffusions exhibits long range dependence.     

Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field

Fractional order quaternion-valued neural networks are a type of fractional order neural networks for which neuron state, synaptic connection strengths, and neuron activation functions are quaternion. This paper is dealing with the Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field. The fractional order quaternion-valued neural networks are separated into four real-valued systems forming an equivalent four real-valued fractional order neural networks, which decreases the computational complexity by avoiding the noncommutativity of quaternion multiplication. Via some fractional inequality techniques and suitable Lyapunov functional, a brand new criterion is proposed first to ensure the Mittag-Leffler stability for the addressed neural networks. Besides, the combination of quaternion-valued adaptive and impulsive control is intended to realize the asymptotically synchronization between two fractional order quaternion-valued neural networks. Ultimately, two numerical simulations are provided to check the accuracy and validity of our obtained theoretical results.     

The Regularity of Stochastic Convolution Driven by Tempered Fractional Brownian Motion and Its Application to Mean-field Stochastic Differential Equations

In this paper, some properties of a stochastic convolution driven by tempered fractional Brownian motion are obtained. Based on this result, we get the existence and uniqueness of stochastic mean-field equation driven by tempered fractional Brownian motion. Furthermore, combining with the Banach fixed point theorem and the properties of Mittag-Leffler functions, we study the existence and uniqueness of mild solution for a kind of time fractional mean-field stochastic differential equation driven by tempered fractional Brownian motion.     

A generalization of the concept of <Emphasis Type="Italic">q</Emphasis>-fractional integrals

In this paper, we consider the fractional q-integral with variable lower limit of integration. We prove the semigroup property of these integrals, and a formula of Leibniz type. Finally, we evaluate fractional q-integrals of some functions. The consideration of q-exponential function in that sense leads to q-analogs of Mittag-Leffier function.     

The Fox-Wright functions and Laguerre multiplier sequences

Linear operators which (1) preserve the reality of zeros of polynomials having only real zeros and (2) map stable polynomials into stable polynomials are investigated using recently established results concerning the zeros of certain Fox-Wright functions and generalized Mittag-Leffler functions. The paper includes several open problems and questions.