Two-parameter Mittag-Leffler Function Research Articles

Rational Approximations for the Oscillatory Two-Parameter Mittag–Leffler Function

The two-parameter Mittag–Leffler function Eα,β is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When α>1, the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for α∈(0,1) often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for Eα,β(−t), t≥0, with α∈(1,2). This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants.

INITIAL-BOUNDARY VALUE PROBLEMS TO THE TIME-NONLOCAL DIFFUSION EQUATION

This article investigates a fractional diffusion equation involving Caputo fractional derivative and Riemann-Liouville fractional integral. The equation is supplemented by initial and boundary conditions in the domain defined by the interval by space 0<x<1 and interval by time 0<t<T. The fractional operators are defined rigorously, utilizing the Caputo fractional derivative of order β and the Riemann-Liouville fractional integral of order α, where 0<α<β≤1. The main results include the presentation of well-known properties associated with fractional operators and the establishment of the unique solution to the given problem. The key findings are summarized through a theorem that provides the explicit form of the solution. The solution is expressed as a series involving the two-parameter Mittag-Leffler function and orthonormal eigenfunctions of the Sturm-Liouville operator. The uniqueness of the solution is proven, ensuring that the problem has a single, well-defined solution under specific conditions on the initial function. Furthermore, the article introduces and proves estimates related to the Mittag-Leffler function, providing bounds crucial for the convergence analysis. The convergence of the series is investigated, and conditions for the solution to belong to a specific function space are established. The uniqueness of the solution is demonstrated, emphasizing its singularity within the given problem. Finally, the continuity of the solution in the specified domain is confirmed through the uniform convergence of the series.

Uniqueness and existence for a fractional differential equation with functional boundary condition

Using Banach’s contractive principle and Leray–Schauder’s fixed point theorem, we study the uniqueness and existence for a nonlinear fractional differential equation with functional boundary condition based on the two-parameter Mittag-Leffler function and an implicit integral equation. Several examples are also presented to demonstrate applications of our key theorems. The methods used can also deal with other types of differential equations with various initial or boundary conditions.

A Numerical Scheme for Time-Space Fractional diffusion Models

Time-space fractional models are proved to be highly efficient in characterizing anomalous diffusion in many intricate systems. This study introduces a numerical scheme employing a matrix transform technique to discretize the fractional spectral Laplacian and subsequently applying generalized exponential time differencing to the semi-discrete system. The resulting second-order scheme is implemented efficiently using rational approximations for the two-parameter Mittag-Leffler function and the partial fraction decomposition of these approximations. The advantage of this approach is its applicability to different types of homogeneous boundary conditions including Robin boundary conditions. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed numerical scheme.

Fractional partial differential variational inequality

In this present paper, we introduce and study a dynamical systems involving fractional derivative operator and nonlocal condition, which is constituted of a fractional evolution equation and a time-dependent variational inequality, and is named as fractional partial differential variational inequality (FPDVI, for short). By employing the estimates involving the one-and two-parameter Mittag-Leffler functions, fixed-point theory for set-value mappings, and non-compactness measure theory, we develop a general framework to establish the existence of smooth solutions to (FPDVI).

Thermomechanical Behavior of Functionally Graded Nanoscale Beams Under Fractional Heat Transfer Model with a Two-Parameter Mittag-Leffler Function

Thermomechanical Behavior of Functionally Graded Nanoscale Beams Under Fractional Heat Transfer Model with a Two-Parameter Mittag-Leffler Function

Distributional Representation of a Special Fox–Wright Function with an Application

A review of the literature demonstrates that the Fox–Wright function is not only a mathematical puzzle, but its role is naturally to represent basic physical phenomena. Motivated by this fact, we studied a new representation of this function in terms of complex delta functions. This representation was useful to compute its Laplace transform with respect to the third parameter γ for which it also generalizes the one and two-parameter Mittag-Leffler functions. New identities involving the Fox–Wright function were discussed and used to simplify the results. Different fractional transforms were evaluated and the solution of a fractional kinetic equation was obtained by using its new representation. Several new properties of this function were discussed as a distribution.

Basis Functions for a Transient Analysis of Linear Commensurate Fractional-Order Systems

In this paper, the possibilities of expressing the natural response of a linear commensurate fractional-order system (FOS) as a linear combination of basis functions are analyzed. For all possible types of sα-domain poles, the corresponding basis functions are found, the kernel of which is the two-parameter Mittag–Leffler function Eα,β, β = α. It is pointed out that there are mutually unambiguous correspondences between the basis functions of FOS and the known basis functions of the integer-order system (IOS) for α = 1. This correspondence can be used to algorithmically find analytical formulas for the impulse responses of FOS when the formulas for the characteristics of IOS are known. It is shown that all basis functions of FOS can be generated with Podlubny‘s function of type εk (t, c; α, α), where c and k are the corresponding pole and its multiplicity, respectively.

Fractional Differential Equations and Matrix Bicomplex Two-parameter Mittag-Leffler Functions

Fractional Differential Equations and Matrix Bicomplex Two-parameter Mittag-Leffler Functions

Study of analytic function related to the Le Roy-type Mittag-Leffler function

UDC 517.547 We study some geometric properties (such as univalence, starlikeness, convexity, and close-to-convexity) of Le Roy-type Mittag-Leffler function. In order to achieve our goal, we use new two-sided inequalities for the digamma function. Some examples are also provided to illustrate the obtained results. Interesting consequences are deduced to show that these results improve several results available in the literature for the two-parameter Mittag-Leffler function.

An Analysis of the Fractional-Order Option Pricing Problem for Two Assets by the Generalized Laplace Variational Iteration Approach

An option is the right to buy or sell a good at a predetermined price in the future. For customers or financial companies, knowing an option’s pricing is crucial. It is well recognized that the Black–Scholes model is an effective tool for estimating the cost of an option. The Black–Scholes equation has an explicit analytical solution known as the Black–Scholes formula. In some cases, such as the fractional-order Black–Scholes equation, there is no closed form expression for the modified Black–Scholes equation. This article shows how to find the approximate analytic solutions for the two-dimensional fractional-order Black–Scholes equation based on the generalized Riemann–Liouville fractional derivative. The generalized Laplace variational iteration method, which incorporates the generalized Laplace transform with the variational iteration method, is the methodology used to discover the approximate analytic solutions to such an equation. The expression of the two-parameter Mittag–Leffler function represents the problem’s approximate analytical solution. Numerical investigations demonstrate that the proposed scheme is accurate and extremely effective for the two-dimensional fractional-order Black–Scholes Equation in the perspective of the generalized Riemann–Liouville fractional derivative. This guarantees that the generalized Laplace variational iteration method is one of the effective approaches for discovering approximate analytic solutions to fractional-order differential equations.

Impulse response of commensurate fractional-order systems: multiple complex poles

The impulse response of a fractional-order system with the transfer function s^{delta }/{[(s^{alpha }-a)^2+b^2]^n}, where n in mathbb {N}, a in {mathbb {R}}, b in {mathbb {R}}^+, alpha in {mathbb {R}}^+, delta in {mathbb {R}}, is derived via real and imaginary parts of two-parameter Mittag-Leffler functions and their derivatives. With the aid of a robust algorithm for evaluating these derivatives, the analytic formulas can be used for an effective transient analysis of fractional-order systems with multiple complex poles. By some numerical experiments it is shown that this approach works well also when the popular SPICE-family simulating programs fail to converge to a correct solution.

Computation of the inverse Mittag–Leffler function and its application to modeling ultraslow dynamics

The inverse Mittag–Leffler function has been used to model the logarithmic growth of the mean squared displacement in anomalous diffusion, the restricted mobility of membrane proteins, and the slow viscoelastic creep observed in glasses. These applications are hindered because the inverse Mittag–Leffler function has no explicit form and cannot be approximated by existing methods in the domain $$x \in (0,+\infty )$$ . This study proposes a conversion method to compute the inverse Mittag–Leffler function in terms of the Mittag–Leffler function. The new method uses the one- and two-parameter Mittag–Leffler function to compute the inverse Mittag–Leffler function in the target domain. We apply this method to fit data collected in studies of: (i) the ultraslow mobility of beta-barrel proteins in bacterial membranes, (ii) the ultraslow creep observed in high strength self-compacting concrete, and (iii) the ultraslow relaxation seen in various glasses. The results show that the inverse Mittag–Leffler function can capture ultraslow dynamics in all three cases. This method may also be extended to other generalized logarithmic laws.

Numerical Approaches of the Generalized Time-Fractional Burgers’ Equation with Time-Variable Coefficients

The generalized time-fractional, one-dimensional, nonlinear Burgers equation with time-variable coefficients is numerically investigated. The classical Burgers equation is generalized by considering the generalized Atangana-Baleanu time-fractional derivative. The studied model contains as particular cases the Burgers equation with Atangana-Baleanu, Caputo-Fabrizio, and Caputo time-fractional derivatives. A numerical scheme, based on the finite-difference approximations and some integral representations of the two-parameter Mittag-Leffler functions, has been developed. Numerical solutions of a particular problem with initial and boundary values are determined by employing the proposed method. The numerical results are plotted to compare solutions corresponding to the problems with time-fractional derivatives with different kernels.

An Extension of Caputo Fractional Derivative Operator by Use of Wiman’s Function

The main aim of this work is to study an extension of the Caputo fractional derivative operator by use of the two-parameter Mittag–Leffler function given by Wiman. We have studied some generating relations, Mellin transforms and other relationships with extended hypergeometric functions in order to derive this extended operator. Due to symmetry in the family of special functions, it is easy to study their various properties with the extended fractional derivative operators.

A rheological analog for Brownian motion with hydrodynamic memory

When the density of the fluid surrounding suspended Brownian particles is appreciable, in addition to the forces appearing in the traditional Ornstein and Uhlenbeck theory of Brownian motion, additional forces emerge as the displaced fluid in the vicinity of the randomly moving Brownian particle acts back on the particle giving rise to long-range force correlations which manifest as a “long-time tail” in the decay of the velocity autocorrelation function known as hydrodynamic memory. In this paper, after recognizing that for Brownian particles immersed in a Newtonian, viscous fluid, the hydrodynamic memory term in the generalized Langevin equation is essentially the 1/2 fractional derivative of the velocity of the Brownian particle, we present a rheological analog for Brownian motion with hydrodynamic memory which consists of a linear dashpot of a fractional Scott Blair element and an inerter. The synthesis of the proposed mechanical network that is suggested from the structure of the generalized Langevin equation simplifies appreciably the calculations of the mean square displacement and its time-derivatives which can also be expressed in terms of the two-parameter Mittag–Leffler function.

Fuzzy Differential Subordinations Based upon the Mittag-Leffler Type Borel Distribution

In this paper, we investigate several fuzzy differential subordinations that are connected with the Borel distribution series Bλ,α,β(z) of the Mittag-Leffler type, which involves the two-parameter Mittag-Leffler function Eα,β(z). Using the above-mentioned operator Bλ,α,β, we also introduce and study a class Mλ,α,βFη of holomorphic and univalent functions in the open unit disk Δ. The Mittag-Leffler-type functions, which we have used in the present investigation, belong to the significantly wider family of the Fox-Wright function pΨq(z), whose p numerator parameters and q denominator parameters possess a kind of symmetry behavior in the sense that it remains invariant (or unchanged) when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. Here, in this article, we have used such special functions in our study of a general Borel-type probability distribution, which may be symmetric or asymmetric. As symmetry is generally present in most works involving fuzzy sets and fuzzy systems, our usages here of fuzzy subordinations and fuzzy membership functions potentially possess local or non-local symmetry features.

A Rational Approximation Scheme for Computing Mittag-Leffler Function with Discrete Elliptic Operator as Input

In this work, we propose a new scheme based on numerical quadrature to calculate the two-parameter Mittag-Leffler function with discrete elliptic operator $$-{\mathcal {L}}_h$$ as input. Except pure mathematical interest from approximation theory, our consideration also arises from solving sub-diffusion equations numerically with time-independent diffusion coefficient. We obtain the scheme by applying Gauss-Legendre quadrature rule for the integral representation of the Mittag-Leffler function. Rigorous error analysis is carried out which shows that the scheme converges exponentially with the increase of quadrature nodes. The computational cost of the algorithm is solving K sparse linear systems with K the number of quadrature nodes. It is worth to point out that the scheme is completely parallel which can save much time if the dimension of $${\mathcal {L}}_h$$ is very large. Some numerical tests are provided to verify the efficiency and robustness of our scheme.

The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α<q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.

Singular points of the integral representation of the Mittag-Leffler function

The paper presents an integral representation of the two-parameter Mittag-Leffler function $E_{\rho,\mu}(z)$ and singular points of this representation have been studied. It has been found that there are two singular points for this integral representation: $\zeta=1$ and $\zeta=0$. The point $\zeta=1$ is a pole of the first order and the point $\zeta=0$, depending on the values of parameters $\rho,\mu$ is either a pole or a branch point, or a regular point. The subsequent study showed that at some values of parameters $\rho,\mu$ with the help of the residue theory one can calculate the integral included in the studied integral representation and express the function $E_{\rho,\mu}(z)$ through elementary functions.