Generalized Mittag-Leffler Function Research Articles

The Laplace transform Induced by the Deformed Exponential Function of Two Variables

Abstract Based on the easy computation of the direct transform and its inversion, the Laplace transform was used as an effective method for solving differential and integral equations. Its various generalizations appeared in order to be used for treating some new problems. They were based on the generalizations and deformations of the kernel function and of the notion of integral. Here, we expose our generalization of the Laplace transform based on the so-called deformed exponential function of two variables. We point out on some of its properties which hold on in the same or similar manner as in the case of the classical Laplace transform. Relations to a generalized Mittag-Leffler function and to a kind of fractional Riemann-Liouville type integral and derivative are exhibited.

Novel analytical solutions of the fractional Drude model

Alternative solutions for the equation of motion of the electrons in a metal in the presence of an external electric field are presented. Novel fractional derivatives of type Atangana–Koca–Caputo with constant and variable-order and fractional conformable derivative in the Liouville–Caputo sense were considered. The variable-order fractional derivative can be set as a smooth function, bounded on (0;1], while, the constant-order fractional derivative can be set as a fractional equation, bounded on (0;1]. We presented the exact solution using the properties of Laplace transform operator with its inversions. Numerical simulations are presented for evaluating the difference between these operators. Based on the generalized Mittag–Leffler function and power-law function with the conformable derivative, new behaviors for the optical properties in metals were obtained.

Fractional driven-damped oscillator and its general closed form exact solution

New questions in fundamental physics and in other fields, which cannot be formulated adequately using traditional integral and differential calculus emerged recently. Fractional calculus was shown to describe phenomena where conventional approaches have been unsatisfactory. The driven damped fractional oscillator entails a rich set of important features, including loss of energy to the environment and resonances. In this paper, this oscillator with Caputo fractional derivatives is solved analytically in closed form. The exact solution is expressed in terms of generalized Mittag-Leffler functions. The standard driven-damped Harmonic Oscillator is recovered as a special case of non-fractional derivatives. In contradistinction to the standard oscillator, the solution of the fractional oscillator is shown to decay algebraically and to possess a finite number of zeros. Several decay patterns are uncovered and are a direct consequence of the asymptotic properties of the generalized Mittag-Leffler functions. Other interesting properties of the fractional oscillator like the momentum–position phase-plane diagrams and the time dependence of the energy terms are discussed as well.

Hadamard and Fejér Hadamard Inequalities for (h - m)- Convex Functions Via Fractional Integral Containing the Generalized Mittag-Leffler Function

Hadamard and Fejér Hadamard Inequalities for (h - m)- Convex Functions Via Fractional Integral Containing the Generalized Mittag-Leffler Function

Generalizations of some fractional integral inequalities for m-convex functions via generalized Mittag-Leffler function

In this paper we are interested to present some general fractional integral inequalities for m-convex functions by involving generalized Mittag-Leffler function. In particular we produce inequalities for several kinds of fractional integrals. Also these inequalities have some connections with known integral inequalities.

A Linearized Stability Theorem for Nonlinear Delay Fractional Differential Equations

In this paper, we prove a theorem of linearized asymptotic stability for fractional differential equations with a time delay. More precisely, using the method of linearization of a nonlinear equation along an orbit (Lyapunov's first method), we show that an equilibrium of a nonlinear Caputo fractional differential equation with a time delay is asymptotically stable if its linearization at the equilibrium is asymptotically stable. Our approach based on a technique which converts the linear part of the equation into a diagonal one. Then using properties of generalized Mittag-Leffler functions, the construction of an associated Lyapunov--Perron operator and the Banach contraction mapping theorem, we obtain the desired result.

Fractional differential equations for the generalized Mittag-Leffler function

In this paper, we establish some (presumably new) differential equation formulas for the extended Mittag-Leffler-type function by using the Saigo-Maeda fractional differential operators involving the Appell function F_{3}(cdot) and results in terms of the Wright generalized hypergeometric-type function {}_{m+1}psi^{({kappa_{l}}_{linmathbb{N}_{0}} )}_{n+1}(z; p) recently established by Agarwal. Some interesting special cases are also pointed out.

Differential subordination associated with generalized Mittag-Leffler function

A new operator \(\mathcal {E}^\gamma _{\alpha ,\beta }f(z)\) associated with the generalized Mittag-Leffler function \(E^\gamma _{\alpha ,\beta }(z)\) is defined in the unit disk. Second-order differential subordination results for operator \(\mathcal {E}^\gamma _{\alpha ,\beta }f(z)\) are obtained by investigating appropriate classes of admissible functions. Inequalities for normalized Mitttag-Leffler function \(\mathbb {E}^\gamma _{\alpha ,\beta }(z)\) and hyperbolic functions are also obtained.

Weighted Hardy-type inequalities involving convex function for fractional calculus operators

The aim of this paper is to establish some new weighted Hardy-type inequalities involving convex and monotone convex functions using Hilfer fractional derivative and fractional integral operator with generalized Mittag-Leffler function in its kernel. We also discuss one dimensional cases of our related results. As a special case of our general results we obtain the results of Iqbal et al. (2017). Moreover, the refinement of Hardy-type inequalities for Hilfer fractional derivative is also included.

Generalized integral inequalities for fractional calculus

In this paper, we present a variety of integral inequalities in Lp and Lp,r spaces for the integral operator involving generalized Mittag-Leffler function in its kernel, Hilfer fractional derivative, generalized Riemann-Liouville and Riemann-Liouville k-fractional integral operators.

A family of meromorphic functions involving generalized Mittag-Leffler function

A family of meromorphic functions involving generalized Mittag-Leffler function

Analytical Solutions of Fractional Walter’s B Fluid with Applications

Fractional Walter’s Liquid Model-B has been used in this work to study the combined analysis of heat and mass transfer together with magnetohydrodynamic (MHD) flow over a vertically oscillating plate embedded in a porous medium. A newly defined approach of Caputo-Fabrizio fractional derivative (CFFD) has been used in the mathematical formulation of the problem. By employing the dimensional analysis, the dimensional governing partial differential equations have been transformed into dimensionless form. The problem is solved analytically and solutions of mass concentration, temperature distribution, and velocity field are obtained in the presence and absence of porous and magnetic field impacts. The general solutions are expressed in the format of generalized Mittag-Leffler function MΩ2,Ω3Ω1χ and Fox-H function Hp,q+11,p satisfying imposed conditions on the problem. These solutions have combined effects of heat and mass transfer; this is due to free convections differences between mass concentration and temperature distribution. Graphical illustration is depicted in order to bring out the effects of various physical parameters on flow. From investigated general solutions, the well-known previously published results in the literature have been recovered. Graphs are plotted and discussed for rheological parameters.

Generalized Fractional Calculus Formulas for a Product of Mittag-Leffler Function and Multivariable Polynomials

The object of this paper is to establish the left and right-sided generalized fractional calculus formulas involving the product of generalized Mittag-Leffler function and general class of multivariable polynomials. The results obtained in this paper are presented in terms of Wright generalized hypergeometric function. Further, we point out also their relevance.

Explicit formulas for the Dunkl dihedral kernel and the (κ,a)-generalized Fourier kernel

In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the (κ,a)-generalized Fourier transform for κ=0. In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag–Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition. New bounds for the kernel of the (κ,a)-generalized Fourier transform are obtained as well.

Analytical and numerical solutions of the telegraph equation using the Atangana–Caputo fractional order derivative

This paper describes the telegraph equation using the Atangana–Caputo’s fractional derivative with two fractional orders and . The new definition is based on the concept of the power law and the generalized Mittag-Leffler function. The first order of the derivative equation was included in the power law function and the second was included in the generalized Mittag-Leffler function. This approach considers media which have two different properties. The fractional spatial derivative equation and the fractional temporal derivative equation were analyzed separately. The generalization of these equations exhibit different cases of anomalous behavior. Numerical solutions using an iterative scheme were obtained.

Fractional Hermite-Hadamard inequalities containing generalized Mittag-Leffler function.

The objective of this paper is to establish some new refinements of fractional Hermite-Hadamard inequalities via a harmonically convex function with a kernel containing the generalized Mittag-Leffler function.

New nonlinear model of population growth.

The model of population growth is revised in this paper. A new model is proposed based on the concept of fractional differentiation that uses the generalized Mittag-Leffler function as kernel of differentiation. The new model includes the choice of sexuality. The existence of unique solution is investigated and numerical solution is provided.

A real distinct poles rational approximation of generalized Mittag-Leffler functions and their inverses: Applications to fractional calculus

The generalized Mittag-Leffler function and its inverse is very useful in solving fractional differential equations and structural derivatives, respectively. However, their computational complexities have made them difficult to deal with numerically. We propose a real distinct pole rational approximation of the generalized Mittag-Leffler function. Under some mild conditions, this approximation is proven and empirically shown to be L-Acceptable. Due to the complete monotonicity property of the Mittag-Leffler function, we derive a rational approximation for the inverse generalized Mittag-Leffler function. These approximations are especially useful in developing efficient and accurate numerical schemes for partial differential equations of fractional order. Several applications are presented such as complementary error function, solution of fractional differential equations, and the ultraslow diffusion model using the structural derivative.

Tur\xe1n type inequalities for generalized Mittag-Leffler function

In this paper, we show several Turán type inequalities for a generalized Mittag-Leffler function with four parameters via the (p,k)-gamma function.

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.