Mittag-Leffler function
View article: Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels
Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels Open
In this manuscript we propose the discrete versions for the recently introduced fractional derivatives with nonsingular Mittag-Leffler function. The properties of such fractional differences are studied and the discrete integration by part…
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Analytical Solutions of the Electrical RLC Circuit via Liouville–Caputo Operators with Local and Non-Local Kernels Open
In this work we obtain analytical solutions for the electrical RLC circuit model defined with Liouville–Caputo, Caputo–Fabrizio and the new fractional derivative based in the Mittag-Leffler function. Numerical simulations of alternative mo…
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Some applications of Mittag-Leffler function in the unit disk Open
In this paper we introduce an operator associated with generalized Mittag-Leffler function in the unit disk U = {z : |z| < 1}. By using this operator and the virtue of differential subordination, we obtain interesting results. Some applica…
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Mittag-Leffler stability analysis of fractional discrete-time neural networks via fixed point technique Open
A class of semilinear fractional difference equations is introduced in this paper. The fixed point theorem is adopted to find stability conditions for fractional difference equations. The complete solution space is constructed and the cont…
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Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel Open
In this manuscript, we have proposed a comparison based on newly defined fractional derivative operators which are called as Caputo-Fabrizio (CF) and Atangana-Baleanu (AB). In 2015, Caputo and Fabrizio established a new fractional operator…
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The Role of the Mittag-Leffler Function in Fractional Modeling Open
This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by …
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Mittag-Leffler stability of impulsive differential equations of fractional order Open
In this paper we consider a nonlinear system of impulsive differential equations of fractional order. Applying the definition of Mittag-Leffler stability introduced by Podlubny and his co-authors and the fractional Lyapunov method, we give…
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The Mittag-Leffler function Open
We review the function theoretical properties of the Mittag-Leffler function $E_{a,b}\left( z\right) $ in a self-contained manner, but also add new results; more than half is new!
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Ulam–Hyers–Mittag-Leffler stability for ψ-Hilfer fractional-order delay differential equations Open
In this paper, we present results on the existence, uniqueness, and Ulam–Hyers–Mittag-Leffler stability of solutions to a class of ψ-Hilfer fractional-order delay differential equations. We use the Picard operator method and a generalized …
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Analytical study for time and time-space fractional Burgers’ equation Open
In this paper, the variational iteration method (VIM) is applied to solve the time and space-time fractional Burgers’ equation for various initial conditions. VIM solutions are computed for the fractional Burgers’ equation to show the beha…
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Epidemiological analysis of fractional order COVID-19 model with Mittag-Leffler kernel Open
This paper derived fractional derivatives with Atangana-Baleanu, Atangana-Toufik scheme and fractal fractional Atangana-Baleanu sense for the COVID-19 model. These are advanced techniques that provide effective results to analyze the COVI…
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An Inverse Problem for a Generalized Fractional Derivative with an Application in Reconstruction of Time- and Space-Dependent Sources in Fractional Diffusion and Wave Equations Open
In this article, we consider two inverse problems with a generalized fractional derivative. The first problem, IP1, is to reconstruct the function u based on its value and the value of its fractional derivative in the neighborhood of the f…
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Some New Fractional-Calculus Connections between Mittag–Leffler Functions Open
We consider the well-known Mittag–Leffler functions of one, two and three parameters, and establish some new connections between them using fractional calculus. In particular, we express the three-parameter Mittag–Leffler function as a fra…
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A Comparative Analysis of the Fractional‐Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law Open
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 nonlin…
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Some properties of a linear operator involving generalized Mittag-Leffler function Open
This paper introduces a new class T +1;; ;k() of analytic functions which is dened by means of a linear operator involving generalized Mittag-Leer function H;;k(f). The results investigated in this paper include, some inclusion relation fo…
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An Efficient Analytical Technique, for The Solution of Fractional-Order Telegraph Equations Open
In the present article, fractional-order telegraph equations are solved by using the Laplace-Adomian decomposition method. The Caputo operator is used to define the fractional derivative. Series form solutions are obtained for fractional-o…
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Analysis of the fractional tumour-immune-vitamins model with Mittag–Leffler kernel Open
Recently, Atangana-Baleanu fractional derivative has got much attention of the researchers due to its non-locality and non-singularity. This operator contains an accurate kernel that describes the better dynamics of systems with a memory e…
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On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function Open
The aim of the present paper is to develop a new generalized form of the fractional kinetic equation involving a generalized k-Mittag-Leffler function Ek,ζ,ηγ,ρ(⋅) $E^{\\gamma,\\rho}_{k,\\zeta,\\eta}(\\cdot)$. The solutions of fractional k…
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A note on fractional integral operator associated with multiindex Mittag-Leffler functions Open
Recently Kiryakova and several other ones have investigated so-called multiindex Mittag-Leffler functions associated with fractional calculus. Here, in this paper, we aim at establishing a new fractional integration formula (of pathway typ…
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Fractional generalized Hadamard and Fejér-Hadamard inequalities for <i>m</i>-convex functions Open
The objective of this paper is to present the fractional Hadamard and Fejér-Hadamard inequalities in generalized forms. By employing a generalized fractional integral operator containing extended generalized Mittag-Leffler function involvi…
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A modified exp-function method for fractional partial differential equations Open
This paper proposes a novel exponential rational function method, a modification of the well-known exp-function method, to find exact solutions of the time fractional Cahn-Allen equation and the time fractional Phi-4 equation. The solution…
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An Application of Improved Bernoulli Sub-Equation Function Method to The Nonlinear Time-Fractional Burgers Equation Open
In this work, we study on the improved Bernoulli sub-equation function method. We apply this method&nbsp;to the nonlinear time-fractional Burgers equation. We obtain new analytical solutions to this model for values of&nbsp;n, m an…
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Theory and application for the time fractional Gardner equation with Mittag-Leffler kernel Open
In this work, the time fractional Gardner equation is presented as a new fractional model for Atangana–Baleanu fractional derivative with Mittag-Leffler kernel. The approximate consequences are analysed by applying a recurrent process. The…
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Generalized Mittag-Leffler Input Stability of the Fractional Differential Equations Open
The behavior of the analytical solutions of the fractional differential equation described by the fractional order derivative operators is the main subject in many stability problems. In this paper, we present a new stability notion of the…
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Some New Extensions on Fractional Differential and Integral Properties for Mittag-Leffler Confluent Hypergeometric Function Open
This article uses fractional calculus to create novel links between the well-known Mittag-Leffler functions of one, two, three, and four parameters. Hence, this paper studies several new analytical properties using fractional integration a…
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Mittag-Leffler function for discrete fractional modelling Open
From the difference equations on discrete time scales, this paper numerically investigates one discrete fractional difference equation in the Caputo delta’s sense which has an explicit solution in form of the discrete Mittag-Leffler functi…
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Analytic solution of fractional order Pseudo-Hyperbolic Telegraph equation using modified double Laplace transform method Open
The Pseudo-Hyperbolic Telegraph partial differential equation (PHTPDE) based on the Caputo fractional derivative is investigated in this paper. The modified double Laplace transform method (MDLTM) is constructed for the proposed model. The…
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Some fractional integral formulas for the Mittag-Leffler type function with four parameters Open
In this paper we present some results from the theory of fractional integration operators (of Marichev- Saigo-Maeda type) involving the Mittag-Leffler type function with four parameters ζ , γ, E μ, ν [z] which has been recently introduced …
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Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function Open
In the article, we establish some new general fractional integral inequalities for exponentially m -convex functions involving an extended Mittag-Leffler function, provide several kinds of fractional integral operator inequalities and give…
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An Application of an Operator Associated with Generalized Mittag-Leffler Function Open
The main object of this paper is to give an application of an operator associated with generalized Mittag-Leffler function in the unit disk $% \\mathcal{U}=\\{z\\in \\mathbb{C}:\\left\\vert z\\right\\vert &lt;1\\}$ to the differential …