Extended Mittag-Leffler Function Research Articles

Extension of Pochhammer symbol, generalized hypergeometric function and τ-Gauss hypergeometric function

Abstract We introduce new extension of the extended Pochhammer symbol and gamma function by using the extended Mittag-Leffler function. We also present extension of the generalized hypergeometric function as well as some of their special cases by using this extended Pochhammer symbol. Further, we define the extension of the τ-Gauss hypergeometric function. Integral and derivative formulas involving the Mellin transform and fractional calculus techniques associated with this extended τ-Gauss hypergeometric function are also given. Also, new extended τ-Gauss hypergeometric function also provides a few more interesting and well-known results. This enriches the theory of special functions. The obtained results are believed to be newly presented.

Fuzzy differential subordination and superordination results for the Mittag-Leffler type Pascal distribution

In this paper, we derive several fuzzy differential subordination and fuzzy differential superordination results for analytic functions $ \mathcal{M}_{\xi, \beta}^{s, \gamma} $, which involve the extended Mittag-Leffler function and the Pascal distribution series. We also investigate and introduce a class $ \mathcal{MB}_{\xi, \beta}^{F, s, \gamma}(\rho) $ of analytic and univalent functions in the open unit disc $ \mathcal{D} $ by employing the newly defined operator $ \mathcal{M}_{\xi, \beta}^{s, \gamma} $. We determine a specific relationship of inclusion for this class. Further, we establish prerequisites for a function role in serving as both the fuzzy dominant and fuzzy subordinant of the fuzzy differential subordination and superordination, respectively. Some novel results that are sandwich-type can be found here.

Certain Quantum Operator Related to Generalized Mittag–Leffler Function

In this paper, we present a novel class of analytic functions in the form h(z)=zp+∑k=p+1∞akzk in the unit disk. These functions establish a connection between the extended Mittag–Leffler function and the quantum operator presented in this paper, which is denoted by ℵq,pn(L,a,b) and is also an extension of the Raina function that combines with the Jackson derivative. Through the application of differential subordination methods, essential properties like bounds of coefficients and the Fekete–Szegő problem for this class are derived. Additionally, some results of special cases to this study that were previously studied were also highlighted.

Generalized extended Mittag-Leffler function and its properties pertaining to integral transforms and fractional calculus

Generalized extended Mittag-Leffler function and its properties pertaining to integral transforms and fractional calculus

Fractionalized mathematical models for drug diffusion

In this study, an effort is made to develop mathematical models that may be used to explain the distribution of drug administration in the human body after oral and intravenous administration of the drug. The diffusion process was utilized to create three models, applying Fick’s principle and the law of mass action. The Sumudu transform algorithm analyzes the rate of change of concentration in various compartments, such as blood and tissue medium. The general solution of drug concentration is demonstrated in the form of extended Mittag-Leffler function. The amount of drug that is contained in each compartment has been determined via the use of numerical parameters. The effect of the fractional parameter on the drug concentration is shown in graphical form. Using MAPLE software, graphs are created to highlight the change in drug concentration over time. The fractional model gives important and relevant inferences to infer new information about the medical field.

Pathway fractional integral formula involving an extended Mittag–Leffler function

Abstract This work discovers the Laplace transform using a generalized pathway fractional integral formula involving an extended Mittag-Leffler function in the kernel for various parameters. Our findings are fairly broad in scope. Some well-known and novel results can also be obtained here.

Some new generalizations for exponentially (s, m)-preinvex functions considering generalized fractional integral operators

The generalized fractional integral has been one of the most useful operators for modelling non-local behaviors by fractional differential equations. It is considered, for several integral inequalities by introducing the concept of exponentially (s, m)-preinvexity. These variants derived via an extended Mittag-Leffler function based on boundedness, continuity and Hermite-Hadamard type inequalities. The consequences associated with fractional integral operators are more general and also present the results for convexity theory. Moreover, we point out that the variants are useful in solving the problems of science, engineering and technology where the Mittag-Leffler function occurs naturally.

On a Unified Mittag-Leffler Function and Associated Fractional Integral Operator

The aim of this paper is to unify the extended Mittag-Leffler function and generalized Q function and define a unified Mittag-Leffler function. Both the extended Mittag-Leffler function and generalized Q function can be obtained from the unified Mittag-Leffler function. The Laplace, Euler beta, and Whittaker transformations are applied for this function, and generalized formulas are obtained. These formulas reproduce integral transformations of various deduced Mittag-Leffler functions and Q function. Also, the convergence of this unified Mittag-Leffler function is proved, and an associated fractional integral operator is constructed.

Laplace Operator with Caputo-Type Marichev–Saigo–Maeda Fractional Differential Operator of Extended Mittag-Leffler Function

In this paper, the Laplace operator is used with Caputo-Type Marichev–Saigo–Maeda (MSM) fractional differentiation of the extended Mittag-Leffler function in terms of the Laplace function. Further in this paper, some corollaries and consequences are shown which are the special cases of our main findings. We apply the Laplace operator on the right-sided MSM fractional differential operator and on the left-sided MSM fractional differential operator. We also apply the Laplace operator on the right-sided MSM fractional differential operator with the Mittag-Leffler function and the left-sided MSM fractional differential operator with the Mittag-Leffler function.

Inequalities for a Unified Integral Operator for Strongly α , m -Convex Function and Related Results in Fractional Calculus

In this paper, we study integral inequalities which will provide refinements of bounds of unified integral operators established for convex and α , m -convex functions. A new definition of function, namely, strongly α , m -convex function is applied in different forms and an extended Mittag-Leffler function is utilized to get the required results. Moreover, the obtained results in special cases give refinements of fractional integral inequalities published in this decade.

Beta Operator with Caputo Marichev-Saigo-Maeda Fractional Differential Operator of Extended Mittag-Leffler Function

In this paper, a beta operator is used with Caputo Marichev-Saigo-Maeda (MSM) fractional differentiation of extended Mittag-Leffler function in terms of beta function. Further in this paper, some corollaries and consequences are shown that are the special cases of our main findings. We apply the beta operator on the right-sided MSM fractional differential operator and on the left-sided MSM fractional differential operator. We also apply beta operator on the right-sided MSM fractional differential operator with Mittag-Leffler function and the left-sided MSM fractional differential operator with Mittag-Leffler function.

A basic study of a fractional integral operator with extended Mittag-Leffler kernel

<abstract><p>In this present paper, the basic properties of an extended Mittag-Leffler function are studied. We present some fractional integral and differential formulas of an extended Mittag-Leffler function. In addition, we introduce a new extension of Prabhakar type fractional integrals with an extended Mittag-Leffler function in the kernel. Also, we present certain basic properties of the generalized Prabhakar type fractional integrals.</p></abstract>

Further generalizations of Hadamard and Fej\xe9r\u2013Hadamard fractional inequalities and error estimates

The aim of this paper is to generalize the fractional Hadamard and Fejér–Hadamard inequalities. By using a generalized fractional integral operator containing extended Mittag-Leffler function via monotone function, for convex functions we generalize well known fractional Hadamard and Fejér–Hadamard inequalities. Also we study the error bounds of these generalized Hadamard and Fejér–Hadamard inequalities. We also obtain some published results from presented inequalities.

Analysis of fractional blood alcohol model with composite fractional derivative

Abstract The present article deals with certain new and interesting features of fractional blood alcohol model associated with powerful Hilfer fractional operator. The solution of the model depends on three parameters such as (i) the initial concentration of alcohol in stomach after ingestion (ii) the rate of alcohol absorption into the blood stream (ii) the rate at which the alcohol is metabolized by the liver. By employing Sumudu transform algorithm the analytic results of the concentration of alcohol in stomach and the concentration of alcohol in the blood are analyzed. The general solution of concentration of alcohol in stomach and the concentration of alcohol in the blood are demonstrated in the form of extended Mittag-Leffler function. The effect of fractional parameter on concentration of alcohol in stomach and the concentration of alcohol in the blood are shown in graphical form. The comparative study for both the concentrations shows the new features of composite fractional derivative in the discussed model. The discussed fractional blood alcohol model yields important and useful results to interpolate new information in the direction of medical environment.

Generalized Minkowski-type Fractional Inequalities Involving Extended Mittag-leffler Function

In this paper the reverse fractional Minkowski integral inequality using extended Mittag-Leffler function with the corresponding fractional integral operator is proved, as well as several related Minkowskitype inequalities.

Generalized fractional integral inequalities for exponentially (s,m)-convex functions

In this paper we have derived the fractional integral inequalities by defining exponentially (s,m)-convex functions. These inequalities provide upper bounds, boundedness, continuity, and Hadamard type inequality for fractional integrals containing an extended Mittag-Leffler function. The results about fractional integral operators for s-convex, m-convex, (s,m)-convex, exponentially convex, exponentially s-convex, and convex functions are direct consequences of presented results.

Some new fractional integral inequalities for exponentially m-convex functions via extended generalized Mittag-Leffler function

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 certain special cases for our obtained results.

Extended Mittag-Leffler function and associated fractional calculus operators

Abstract Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

Fractional Calculus of Extended Mittag-Leffler Function and Its Applications to Statistical Distribution

Several fractional calculus operators have been introduced and investigated. In this sequence, we aim to establish the Marichev-Saigo-Maeda (MSM) fractional calculus operators and Caputo-type MSM fractional differential operators of extended Mittag-Leffler function (EMLF). We also investigate the statistical distribution associated with the EMLF. Finally, we derive some of the particular cases of the main results.

Fractional integral operator associated with extended Mittag-Leffler function

Fractional integral operator associated with extended Mittag-Leffler function