Generalized Mittag-Leffler Function Research Articles

A study on unification of generalized hypergeometric function and Mittag-Leffler function with certain integral transforms of generalized basic hypergeometric function

This research article explores some new properties of generalized hypergeometric function and its q-analogue. The connections between ${}_{2}{{R}_{1}}^{\upsilon }(\mathfrak{z})$, the Wright function, and generalized Mittag-Leffler functions are explored. The authors introduce the q-analogue of generalized hypergeometric function denoted by ${}_{2}{{R}_{1}}^{\upsilon ,q}(\mathfrak{z})$ and discuss its properties and connections with q-Wright function and q-versions of generalized Mittag-Leffler functions. We get the q-integral transforms such as q-Mellin, q-Euler (beta), q-Laplace, q-sumudu, and q-natural transforms of Wright-type generalized q-hypergeometric function. This article contributes to the understanding of hypergeometric functions in q-calculus.

Analysis of the nonlinear Fitzhugh–Nagumo equation and its derivative based on the Rabotnov fractional exponential function

The Fitzhugh–Nagumo equation is an essential governing equation that explains the process of impulse transmission from the nerves. This work suggests a novel generalized arbitrary-order Fitzhugh–Nagumo equation that is based on the recently developed nonsingular fractional operator. The modified Fitzhugh–Nagumo equation has been generalized using fractional Yang–Abdel–Cattani operator to provide a more accurate representation of the equation with memory effects and non-local behavior. Further, we discussed some interesting and new properties of the proposed fractional operator and the Rabotnov exponential fractional kernel with a modified Sumudu integral transform. In order to solve the proposed nonlinear differential equation, the homotopy perturbation method coupled with the modified Sumudu integral transform technique is implemented to examine the result of a newly proposed Yang–Abdel–Cattani fractional Fitzhugh–Nagumo equation, which converges to the exact solution. The nonlinear terms of this equation are handled in terms of He’s polynomials. The existence and uniqueness of the solution have been demonstrated using the Banach space fixed point theorem. The convergence and error analysis of the suggested technique are also discussed. The findings are expressed in terms of generalized Mittag-Leffler functions. The obtained results are plotted with the help of MATLAB R2016a mathematical software. The results in this work are more accurate, and it is proposed that the new Yang–Abdel–Cattani fractional derivative operator is an effective tool for finding the results of any other nonlinear problems arising in science and engineering. All three problems have been computed for different orders to confirm the significance of the fractionalizing concept in the proposed equation. The increase of the fractional order value ℘ to 1 confirms that our proposed time-fractional Fitzhugh–Nagumo equation in the Yang–Abdel–Cattani sense gets close to the analytic solution.

On Uniformly Starlike Functions with Respect to Symmetrical Points Involving the Mittag-Leffler Function and the Lambert Series

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions ΤJ(αi). Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions f∈ΤJ(αi). In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass.

A study on the fractional Black–Scholes option pricing model of the financial market via the Yang-Abdel-Aty-Cattani operator

PurposeFinancial mathematics is one of the most rapidly evolving fields in today’s banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF).Design/methodology/approachThe homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity.FindingsIn this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function.Originality/valueThe Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives.

Asymptotic stability of nonlinear fractional delay differential equations with α ∈ (1, 2): An application to fractional delay neural networks.

We introduce a theorem on linearized asymptotic stability for nonlinear fractional delay differential equations (FDDEs) with a Caputo order α∈(1,2), which can be directly used for fractional delay neural networks. It relies on three technical tools: a detailed root analysis for the characteristic equation, estimation for the generalized Mittag-Leffler function, and Lyapunov's first method. We propose coefficient-type criteria to ensure the stability of linear FDDEs through a detailed root analysis for the characteristic equation obtained by the Laplace transform. Further, under the criteria, we provide a wise expression of the generalized Mittag-Leffler functions and prove their polynomial long-time decay rates. Utilizing the well-established Lyapunov's first method, we establish that an equilibrium of a nonlinear Caputo FDDE attains asymptotically stability if its linearization system around the equilibrium solution is asymptotically stable. Finally, as a by-product of our results, we explicitly describe the asymptotic properties of fractional delay neural networks. To illustrate the effectiveness of our theoretical results, numerical simulations are also presented.

Fuzzy differential subordination associated with generalized Mittag-Leffler type Poisson distribution

This article introduces a new operator defined by the convolution of the Poisson distribution series and the generalized Mittag-Leffler function. Applying this operator, we study a novel subclass of analytic functions using the technique of fuzzy differential subordination. Also, we prove some significant results related to the fuzzy differential relation for this novel class. These results are applied to particular functions to showcase the significance of our findings.

On Janowski Type Harmonic Functions Associated with the Wright Hypergeometric Functions

In our present study we consider Janowski type harmonic functions class introduced and studied by Dziok, whose members are given by $h(z) = z + \sum_{n=2}^{\infty} h_n z^n$ and $g(z) = \sum_{n=1}^{\infty} g_n z^n$, such that $\mathcal{ST}_{H}(F,G)=\big\{ f = h + \bar{g} \in {H}:\frac{\mathfrak{D}_H f(z)}{f(z)}\prec\frac{1+Fz}{1+G z};\, (-G \leq F < G \leq 1, \text{ with } g_1=0)\big\},$ where $\mathfrak{D}_H f(z) = zh'(z)-\overline{zg'(z)}\,$ and $z\in \mathbb{U}=\{z:z\in \mathbb{C} \text{ and }|z| < 1 \}.$ We investigate an~association between these subclasses of harmonic univalent functions by applying certain convolution operator concerning Wright's generalized hypergeometric functions and several special cases are given as a corollary. Moreover we pointed out certain connections between Janowski-type harmonic functions class involving the generalized Mittag–Leffler functions. Relevant connections of the results presented herewith various well-known results are briefly indicated.

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.

Goufo-Caputo fractional viscoelastic photothermal model of an unbounded semiconductor material with a cylindrical cavity

In order to describe the photothermal dynamic behavior of viscoelastic semiconductor materials, the authors of this work propose a fractional modification of the Kelvin-Voigt type. The generalized Mittag-Leffler function method with two parameters was utilized as the non-local, nonsingular kernel of the Goufo-Caputo fractional operator, which is similar to the AB fractional derivative. Also, a new concept of photovoltaic heat transfer has been developed to describe the process of photothermal heating in semiconductors, in which light energy is converted into thermal energy by absorption. This unique model is grounded in the theoretical framework of the Moore-Gibson-Thompson (MGT) equation, and it offers additional insights into the underlying process of light sensitive heat transfer, as well as the intricate interaction between heating signals, elasticity, and plasma. As a direct application of the proposed model, the photothermal interactions that occur when an infinitely rotating semiconductor material with a cylindrical hole is subjected to a laser thermal flux and immersed in a constant axial magnetic field have been investigated. These findings may have implications for developing optically absorbent nanostructures and their use in photothermal energy generation. For this reason, the comparison of modified photoelasticity models with spherical derivatives and classical models applied to semiconducting materials seems interesting.

A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law

In this paper, new generalized weighted fractional derivatives with respect to another function are derived in the sense of Caputo and Riemann–Liouville involving a new modified version of a generalized Mittag–Leffler function with three parameters, as well as their corresponding fractional integrals. In addition, several new and existing operators of nonsingular kernels are obtained as special cases of our operator. Many important properties related to our new operator are introduced, such as a series version involving Riemann–Liouville fractional integrals, weighted Laplace transforms with respect to another function, etc. Finally, an example is given to illustrate the effectiveness of the new results.

Remarks on a fractional nonlinear partial integro-differential equation via the new generalized multivariate Mittag-Leffler function

Introducing a new generalized multivariate Mittag-Leffler function which is a generalization of the multivariate Mittag-Leffler function, we derive a sufficient condition for the uniqueness of solutions to a brand new boundary value problem of the fractional nonlinear partial integro-differential equation using Banach’s fixed point theorem and Babenko’s technique. This has many potential applications since uniqueness is an important topic in many scientific areas, and the method used clearly opens directions for studying other types of equations and corresponding initial or boundary value problems. In addition, we use Python which is a high-level programming language efficiently dealing with the summation of multi-indices to compute approximate values of the generalized Mittag-Leffler function (it seems impossible to do so by any existing integral representation of the Mittag-Leffler function), and provide an example showing applications of key results derived.

Analytic solutions to the fractional kinetic equation involving the generalized Mittag-Leffler function using the degenerate Laplace type integral approach

Analytic solutions to the fractional kinetic equation involving the generalized Mittag-Leffler function using the degenerate Laplace type integral approach

A New Method for Solving Sequential Fractional Wave Equations

In this article, we focus on two classes of fractional wave equations in the context of the sequential Caputo derivative. For the first class, we derive the closed-form solution in terms of generalized Mittag–Leffler functions. Subsequently, we consider a more general class of nonhomogeneous fractional wave equations. Due to the complexity of finding exact solutions for these problems, we employ a numerical technique based on the operational matrix method to approximate the solution. We provide several theoretical and numerical examples to validate the effectiveness of this numerical approach. The results demonstrate the accuracy and efficiency of the proposed method.

Modified Atangana-Baleanu fractional operators involving generalized Mittag-Leffler function

In this paper, we are going to deal with fractional operators (FOs) with non-singular kernels which is not an easy task because of its restriction at the origin. In this work, we first show the boundedness of the extended form of the modified Atangana-Baleanu (A-B) Caputo fractional derivative operator. The generalized Laplace transform is evaluated for the introduced operator. By using the generalized Laplace transform, we solve some fractional differential equations. The corresponding form of the Atangana-Baleanu Caputo fractional integral operator is also established. This integral operator is proved bounded and obtained its Laplace transform. The existence and Hyers-Ulam stability is explored. In the last results, we studied the relation between our defined operators. The operators in the literature are obtained as special cases for these newly explored FOs.

Existence in the Large for Caputo Fractional Multi-Order Systems with Initial Conditions

One of the key applications of the Caputo fractional derivative is that the fractional order of the derivative can be utilized as a parameter to improve the mathematical model by comparing it to real data. To do so, we must first establish that the solution to the fractional dynamic equations exists and is unique on its interval of existence. The vast majority of existence and uniqueness results available in the literature, including Picard’s method, for ordinary and/or fractional dynamic equations will result in only local existence results. In this work, we generalize Picard’s method to obtain the existence and uniqueness of the solution of the nonlinear multi-order Caputo derivative system with initial conditions, on the interval where the solution is bounded. The challenge presented to establish our main result is in developing a generalized form of the Mittag–Leffler function that will cooperate with all the different fractional derivative orders involved in the multi-order nonlinear Caputo fractional differential system. In our work, we have developed the generalized Mittag–Leffler function that suffices to establish the generalized Picard’s method for the nonlinear multi-order system. As a result, we have obtained the existence and uniqueness of the nonlinear multi-order Caputo derivative system with initial conditions in the large. In short, the solution exists and is unique on the interval where the norm of the solution is bounded. The generalized Picard’s method we have developed is both a theoretical and a computational method of computing the unique solution on the interval of its existence.

Analysis of Cauchy Problems and Diffusion Equations Associated with the Hilfer–Prabhakar Fractional Derivative via Kharrat–Toma Transform

In this paper, the Kharrat–Toma transforms of the Prabhakar integral, a Hilfer–Prabhakar (HP) fractional derivative, and the regularized version of the HP fractional derivative are derived. Moreover, we also compute the solution of some Cauchy problems and diffusion equations modeled with the HP fractional derivative via Kharrat–Toma transform. The solutions of Cauchy problems and the diffusion equations modeled with the HP fractional derivative are computed in the form of the generalized Mittag–Leffler function.

Jackson Differential Operator Associated with Generalized Mittag–Leffler Function

Quantum calculus plays a significant role in many different branches such as quantum physics, hypergeometric series theory, and other physical phenomena. In our paper and using quantitative calculus, we introduce a new family of normalized analytic functions in the open unit disk, which relates to both the generalized Mittag–Leffler function and the Jackson differential operator. By using a differential subordination virtue, we obtain some important properties such as coefficient bounds and the Fekete–Szegő problem. Some results that represent special cases of this family that have been studied before are also highlighted.

The Cauchy Problem for the Delay Differential Equation with Dzhrbashyan – Nersesyan Fractional Derivative

Последние десятилетия количество работ, посвященных исследованию задач для дифференциальных уравнений дробного порядка, заметно растет. Интерес исследователей вызван тем, что количество областей науки, в которых используются уравнения, содержащие дробные производные, варьируется от биологии и медицины до теории управления, инженерии, финансов, а также оптики, физики и так далее. Включение запаздывания в уравнение дробного порядка существенно влияет на ход процесса, описываемого этим уравнением, так как неизвестная функция задается при различных значениях аргумента, что вносит эффект предыстории в уравнение. Поэтому, математические модели, содержащие дробный оператор и запаздывающий аргумент, более точны, чем модели, содержащие производные целого порядка. В данной работе исследуется задача Коши для линейного обыкновенного дифференциального уравнения с запаздывающим аргументом c оператором дробного дифференцирования Джрбашяна – Нерсесяна, обобщающим известные дробные операторы Римана – Лиувилля и Герасимова – Капуто. Результаты работы получены с использованием методов теории целого и дробного исчислений, методов теории дифференциальных уравнений с запаздывающим аргументом, метода специальных функций. В работе доказывается теорема о справедливости аналога формулы Лагранжа. Также доказано, что специальная функция Wγm​​(t), которая, в свою очередь, определяется через обобщенную функцию Миттаг – Леффлера (или функция Прабхакара), удовлетворяет уравнению и условиям, сопряженным исследуемому, и является фундаментальным решением рассматриваемого уравнения. Сформулирована и доказана теорема существования и единственности решения начальной задачи. Решение поставленной задачи выписано в терминах специальной функции Wν​(t). In recent, the number of works devoted to the study of problems for fractional order differential equations is growing noticeably. The interest of researchers is due to the fact that the number of areas of science in which equations containing fractional derivatives are used varies from biology and medicine to control theory, engineering, finance, as well as optics, physics, and so on. The inclusion of delay in the fractional order equation significantly affects the course of the process described by this equation, since the unknown function is given for different values of the argument, which includes a history effect into the equation. Therefore, mathematical models containing a fractional operator and a delay argument are more accurate than models containing integer derivatives. In this paper, we study the Cauchy problem for a linear ordinary delay differential equation with the Dzhrbashyan – Nersesyan fractional differentiation operator, which is generalizing the Riemann – Liouville and Gerasimov – Caputo fractional operators. The results of the work are obtained using the methods of the theory of integer and fractional calculus, methods of the theory of delay differential equations, the method of special functions. In this paper proves a theorem on the validity of an analogue of the Lagrange formula. It is also proved that the special function Wγm​​(t), which is defined in terms of the generalized Mittag-Leffler function (or the Prabhakar function), satisfies the equation and conditions associated with the one under study, and is the fundamental solution of the considered equation. The main result is that the existence and uniqueness theorem to the initial value problem is proved. The solution to the problem is written out in terms of the special function Wν​(t).

Two effective methods for solving nonlinear coupled time-fractional Schrödinger equations

The objective of this work is to implement two efficient techniques, namely, the Laplace Adomian decomposition method (LADM) and the modified generalized Mittag–Leffler function method (MGMLFM) on a system of nonlinear fractional partial differential equations (NFPDEs) to get an analytic-approximate solution. The nonlinear time-fractional Schrödinger equation (TFSE) and coupled fractional order Schrödinger-Korteweg-de Vries (Sch-KdV) equation are found in various areas such as quantum mechanics and physics. These equations describe different types of wave propagation like dust-acoustic waves, Langmuir and electromagnetic waves in plasma physics. Using the proposed methods, a convenient solution is established for the considered nonlinear fractional order models. The obtained analytic-approximate travelling-waves solutions and the effect of the fractional order α on the behaviour of these projected solutions are presented in some figures and tables along with the exact solution. We compare the approximate values with their corresponding values of the known exact solution and compute the absolute error. Consequently, we can deduce that the used methods are very efficient, reliable and simple to construct a series form that rapidly convergent to the exact solution, which indicates the advantages of the methods.

Third-Order Differential Subordination for Meromorphic Functions Associated with Generalized Mittag-Leffler Function

Using the results of third-order differential subordination, we introduce certain families of admissible functions and discuss some applications of third-order differential subordination for meromorphic functions associated with a linear operator containing a generalized Mittag-Leffler function.