Mittag-Leffler Function Research Articles

Controllability of Impulsive Damped Fractional Order Systems Involving State Dependent Delay

In this article, the concept of controllability on fractional order impulsive systems involving state dependent delay and damping behavior is analysed by utilizing Caputo fractional derivative. The main motivation is to derive the sufficient conditions for the controllability of the considered systems. Based on the Laplace transform and inverse Laplace transform, the solution of fractional-order dynamical systems are obtained. The results are established by utilizing basic ideas of fractional calculus, Mittag-Leffler function and Banach fixed point theorem. Finally, an application is provided to illustrate the derived result.

Exponential synchronization of fractional-order T–S fuzzy complex multi-links networks with intermittent dynamic event-triggered control

In this paper, the exponential synchronization problem for fractional-order T–S fuzzy complex multi-links networks under an intermittent dynamic event-triggered control (IDE-TC) strategy is discussed. To this end, a new triggering rule with a dynamical variable is first introduced, including some available static triggering rules as its particular form. Then, it is proved that the applied dynamical variable is positive by using some properties of the Mittag-Leffler function. Hence, the inter-event time between any two successive triggering moments can be enlarged than static event-triggered results. Additionally, there is a guaranteed positive minimum inter-event time for each sample path solution of systems. Based on the Lyapunov method and the graph theory, exponential synchronization criteria of the proposed networks under the IDE-TC strategy is deduced. Finally, the validity and feasibility of the derived theoretical results are investigated by an application with simulations.

Efficient second-order accurate exponential time differencing for time-fractional advection-diffusion-reaction equations with variable coefficients

Time-fractional advection-diffusion-reaction type equations are useful for characterizing anomalous transport processes. In this paper, linearly implicit as well as explicit generalized exponential time differencing (GETD) schemes are proposed for solving a class of such equations having time-space dependent coefficients. The implicit scheme, being unconditionally stable, is robust in handling the numerical instabilities in problems where the advection term is dominant. Regarding the error analysis, uniformly optimal second-order convergence rates are derived using time-graded meshes to counter the effect of the inherent singularity of the continuous solution. Implementation of generalized exponential integrators requires computing the action of Mittag-Leffler function of matrices on a vector, or on a matrix in the case of the implicit scheme. For cost-effective implementation, using global Padé approximants these computation tasks get reduced to solving linear systems. A new approach based on Sylvester equation formulation of the resulting linear systems is developed in this paper. This technique leads to significantly faster algorithms for implementing the GETD schemes. Numerical experiments are provided to illustrate the theoretical findings and to assert the efficiency of the Sylvester equation based approach. Application of this approach to an existing GETD scheme for solving a nonlinear subdiffusion problem is also discussed.

A new self-organization of complex networks structure generalized by a new class of fractional differential equations generated by 3D-gamma function

In this note, we suggest a generalization of gamma function to be 3D-gamma function. As a consequence, some special functions are generalized including the Mittag-Leffler function. Moreover, we utilize the generalized Mittag-Leffler to extend the AB-fractional calculus. Examples are introduced to cover our theory. The solvability of abstract Riccati equation is considered, and Hyers-Ulam stability is discovered in the sequel. Based on the new study, we design new stable of Self-organization in complex networks (SOCNs).

New Integral Transforms of the Extended k- Generalized Mittag-Leffler Function with Graphical Representations

Different integral transforms have extensive applications in the various areas of science and engineering. Some of the new integral transforms of extended k-generalized Mittag-Leffler function are discussed in this paper. We examine integral transforms such as the Euler-Beta, Laplace, Mohand, Aboodh, SEE, and Sadik transform. Moreover, we also tried to establish the graphical representations of these transforms.

Analytical Analysis for Space Fractional Helmholtz Equations by Using The Hybrid Efficient Approach

Abstract The Helmholtz equation is an important differential equation. It has a wide range of uses in physics, including acoustics, electro-statics, optics, and quantum mechanics. In this article, a hybrid approach called the Shehu transform decomposition method (STDM) is implemented to solve space-fractional-order Helmholtz equations with initial boundary conditions. The fractional-order derivative is regarded in the Caputo sense. The solutions are provided as series, and then we use the Mittag-Leffler function to identify the exact solutions to the Helmholtz equations. The accuracy of the considered problem is examined graphically and numerically by the absolute, relative, and recurrence errors of the three problems. For different values of fractional-order derivatives, graphs are also developed. The results show that our approach can be a suitable alternative to the approximate methods that exist in the literature to solve fractional differential equations.

Bounds on Initial Coefficients for Bi-Univalent Functions Linked to q-Analog of Le Roy-Type Mittag-Leffler Function

This study introduces a new class of bi-univalent functions by incorporating the q-analog of Le Roy-type Mittag-Leffler functions alongside q-Ultraspherical polynomials. We formulate and solve the Fekete-Szegö functional problems for this newly defined class of functions, providing estimates for the coefficients |α2| and |α3| in their Taylor-Maclaurin series. Additionally, our investigation produces novel results by adapting the parameters in our initial discoveries.

On the computation of the Mittag-Leffler function of fractional powers of accretive operators

On the computation of the Mittag-Leffler function of fractional powers of accretive operators

On the Fekete-Szegö Inequalities of the Generalized Mittag-Leffler Function Associated with a Lambert Series

This study aims to investigate the Fekete-Szegö problem for the linear operator generated by the convolution (the Hadamard product) involving one of the generalized forms of the Mittag-Leffler function and the well-known Lambert series. The findings will mainly apply on some subclasses of starlike and convex functions.

Analysis of the multi-term fractional Bateman equations in radioactive decay by means of Mikusiński algebraic calculus

Fractional Bateman equations model the concentrations of nuclear species through a radioactive decay chain, by considering non-Markovian behavior. Cruz-López and Espinosa-Paredes (Nuclear Engineering and Technology, 2022) solved the mathematical problem with a single memory coefficient, based on the Laplace-transform technique. Cruz-López, Espinosa-Paredes and François (Computer Physics Communications, 2022) extended the analysis to several fractional orders, with Laplace transforms and series expansions checked by induction. In this paper, we solve such a case of distinct memory effects, via the more general method of Mikusiński operational calculus. We do not distinguish between equal and different decay rates. Multivariate Mittag-Leffler functions appear in the closed-form expression.

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.

Hamilton energy, competitive modes and ultimate bound estimation of a new 3D chaotic system, and its application in chaos synchronization

Abstract This paper discusses the dynamical behavior of a new 3D chaotic system of integer and fractional order. To get a comprehensive knowledge of the dynamics of the proposed system, we have studied competitive modes and Hamilton energy for different parameter values. In order to get the ultimate bound set for the proposed system, we employed the Lagrange coefficient approach to solve the optimization problem. We have also explored the use of the bound set in synchronization. Furthermore, we have examined the Hamilton energy, time series, bifurcation diagrams, and Lyapunov exponents for the fractional version of the proposed chaotic system. Finally, we looked at the Mittage-Leffler positive invariant sets and global attractive sets by merging the Lyapunov function approach with the Mittage-Leffler function. Numerical simulations have shown the obtained bound sets and other analytical outcomes.

Generalized Mittag-Leffler-confluent hypergeometric functions in fractional calculus integral operator with numerical solutions

The Mittag-Leffler and confluent hypergeometric functions were originally developed to extend the exponential function and its area of applications. This study aims to examine some operators involving generalized Mittag-Leffler-type functions in the kernels, employing the generalized Fox-Wright function in specific circumstances. Furthermore, we investigate some of the commonly utilized generalized fractional integral operators in fractional calculus. Moreover, a numerical technique is developed to solve fractional differential equations of both kinds, linear and nonlinear. The graphic results of the examples show how effective this method is at solving fractional differential equations. Lastly, various effects and implications of these results are thoroughly examined.

The boundary value problem for an ordinary linear half-order differential equation

This study is devoted to the study of the solution of a boundary value problem for an ordinary linear differential equation of half order with constant coefficients. Using of the fundamental solution of the main part of the considered equation, we obtained the principal relations, from which we obtain the necessary conditions for the Fredholm property of the original problem. Further, using the Mittag-Leffler function, a general solution of the homogeneous equation is obtained. Finally, the problem under consideration is reduced to an integral Fredholm equation of the second kind with a non-singular kernel, i.e., the Fredholm property of the stated problem is proved.

Impact response of fractionally damped rectangular plates made of viscoelastic composite materials

The paper aims to find dynamic response of fractionally damped rectangular viscoelastic plates under low-velocity impact of a mass. The general fractional (Rubbery, Transition and Glassy) RTG model for the plate material was considered. First, the governing equation of the plate is derived and solved. Next, using the modified Hertz contact law, the governing integral equation of the problem in terms of indentation is derived and solved. Finally, relation between indentation and impact force is discretized and using the Mittag-Leffler function, history of impact force is obtained. The work is validated by comparison of the results with some particular cases and available experimental data.

On a new extension of extended (p,q) Beta function

The main objective of this paper is to introduce a further extension of extended (p, q)-Beta function by considering product of two Mittag-Leffler functions in the kernel. We investigate various properties of this newly defined Beta function such as integral representations, summation formulas and Mellin transform. Further, some known and new relations for various forms of extended Beta functions are obtained as special cases of the main results.

Special functions with general kernel: Properties and applications to fractional partial differential equations

Abstract In this paper, we reconstruct the gamma and beta functions using a general kernel function in their integral representations. We also reconstruct the Gauss and confluent hypergeometric functions using the beta function with general kernel in their series representations. The general kernel function we use here can be chosen as any special function such as exponential function, Mittag-Leffler function, Wright function, Fox-Wright function, Kummer function or M-series. Using different choices of this general kernel function, various of the generalized gamma, beta, Gauss hypergeometric and confluent hypergeometric functions in the literature can be obtained. In this paper, we first obtain the integral representations, functional relations, summation, derivative and transformation formulas and double Laplace transforms of the special functions we construct. Furthermore, we compute the solutions of some fractional partial differential equations involving special functions with general kernel via the double Laplace transform and graph some of the solutions for specific values. Finally, we obtain the incomplete beta function with general kernel by defining the beta distribution with general kernel.

A Note on Some Novel Laplace and Stieltjes Transforms Associated with the Relaxation Modulus of the Andrade Model

In the framework of linear viscoelasticity, the authors have previously calculated a novel inverse Laplace transform involving the Mittag–Leffler function in order to calculate the relaxation modulus in the Andrade model. Here, we generalize this result, calculating the inverse Laplace transform of a given function Fα,βs by using two different approaches: the Bromwich integral and the decomposition of Fα,βs in simple fractions. From both calculations, we obtain a set of novel Laplace and Stieltjes transforms.

Oscillatory integrals for Mittag-Leffler functions with two variables

Oscillatory integrals for Mittag-Leffler functions with two variables

Optimization of parameters for image denoising algorithm pertaining to generalized Caputo-Fabrizio fractional operator

The aim of the present paper is to optimize the values of different parameters related to the image denoising algorithm involving Caputo Fabrizio fractional integral operator of non-singular type with the Mittag-Leffler function in generalized form. The algorithm aims to find the coefficients of a kernel to remove the noise from images. The optimization of kernel coefficients are done on the basis of different numerical parameters like Mean Square Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Structure Similarity Index measure (SSIM) and Image Enhancement Factor (IEF). The performance of the proposed algorithm is investigated through above-mentioned numeric parameters and visual perception with the other prevailed algorithms. Experimental results demonstrate that the proposed optimized kernel based on generalized fractional operator performs favorably compared to state of the art methods. The uniqueness of the paper is to highlight the optimized values of performance parameters for different values of fractional order. The novelty of the presented work lies in the development of a kernel utilizing coefficients from a fractional integral operator, specifically involving the Mittag-Leffler function in a more generalized form.