Matrix 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.

Uniqueness and Hyers–Ulam’s stability for a fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition

Abstract Introducing a pair-parameter matrix Mittag–Leffler function, we study the uniqueness and Hyers–Ulam stability to a new fractional nonlinear partial integro-differential equation with variable coefficients and a mixed boundary condition using Banach’s contractive principle as well as Babenko’s approach in a Banach space. These investigations have serious applications since uniqueness and stability analysis are essential topics in various research fields. The techniques used also work for different types of differential equations with initial or boundary conditions, as well as integral equations. Moreover, we present a Python code to compute approximate values of our newly established pair-parameter matrix Mittag–Leffler functions, which extend the multivariate Mittag–Leffler function. A few examples are given to show applications of the key results obtained.

Application of a matrix Mittag-Leffler function to the fractional partial integro-differential equation in R n

Application of a matrix Mittag-Leffler function to the fractional partial integro-differential equation in R n

A Matrix Mittag–Leffler Function and the Fractional Nonlinear Partial Integro-Differential Equation in ℝn

In this paper, we introduce the matrix Mittag–Leffler function, which is a generalization of the multivariate Mittag–Leffler function, in order to investigate the uniqueness of the solutions to a fractional nonlinear partial integro-differential equation in Rn with a boundary condition based on Banach’s contractive principle and Babenko’s approach. In addition, we present an example demonstrating applications of the key results derived using a Python code that computes the approximate value of our matrix Mittag–Leffler function.

On controllability and observability of a class of fractional-order switched systems with impulse

This paper is devoted to investigating two structural properties for a class of fractional-order switched and impulsive systems. The structural properties, i.e., observability and controllability, are explored mainly based on an algebraic approach. More specifically, firstly, according to the Laplace transform and mathematical induction, the general solution of such hybrid fractional-order systems is obtained over every impulsive interval. Next, applying the solution derived and relevant matrix theory, several necessary and sufficient controllability and observability criteria that take the form of a row of Gramian matrices are analytically established in terms of a deterministic impulse-switching time sequence. Resorting to the property of matrix Mittag-Leffler function, the developed controllability and observability Gramian criteria are further converted to some easy-test Kalman-type rank conditions. Finally, a numerical example illustrating the theoretical controllability and observability conditions is given.

Finite-Interval Stability Analysis of Impulsive Fractional-Delay Dynamical System

Stability analysis over a finite time interval is a well-formulated technique to study the dynamical behaviour of a system. This article provides a novel analysis on the finite-time stability of a fractional-order system using the approach of the delayed-type matrix Mittag-Leffler function. At first, we discuss the solution’s existence and uniqueness for our considered fractional model. Then standard form of integral inequality of Gronwall’s type is used along with the application of the delayed Mittag-Leffler argument to derive the sufficient bounds for the stability of the dynamical system. The analysis of the system is extended and studied with impulsive perturbations. Further, we illustrate the numerical simulations of our analytical study using relevant examples.

Formula omitted]-asymptotically periodic fractional functional differential equations with off-diagonal matrix Mittag-Leffler function kernels

By employing off-diagonal matrix Mittag-Leffler functions and stability theory for line fractional functional differential equations, a new technique is proposed to investigate the existence, uniqueness and global asymptotical stability of S-asymptotically periodic solution for a class of semilinear Caputo fractional functional differential equations. Some better results are derived, which improve and extend the existing research findings in recent years. As an application of the general theory, some decision theorems are established for the asymptotically dynamical behaviors for fractional four-neuron BAM neural networks. The methods used in this paper could be applied to the study of other fractional differential systems in the areas of science and engineering.

Computing Enclosures for the Matrix Mittag–Leffler Function

We propose two algorithms for numerically calculating interval matrices including two-parameter matrix Mittag–Leffler (ML) functions. We first present an algorithm for computing enclosures for scalar ML functions. Then, the two proposed algorithms are developed by exploiting the scalar algorithm and verified block diagonalization. The first algorithm relies on a numerical spectral decomposition. The cost of this algorithm is only cubic plus that of the scalar algorithm if the second parameter is not too small. The second algorithm is based on a numerical Jordan decomposition, and can also be applied to defective matrices. The cost of this algorithm is quartic plus that of the scalar algorithm. A numerical experiment illustrates an application to a fractional differential equation.

Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations

Linear systems of fractional differential equations have been studied from various points of view: applications to electric circuit theory, approximate solutions by numerical methods, and recently exact solutions by analytical methods. We discover here that, to obtain a fully closed-form solution in all cases, it is necessary to introduce a new type of Mittag-Leffler function involving triple series, and also to construct the associated fractional calculus operators, which we introduce and study in this paper. We then complete the rigorous analytical solutions for the aforesaid systems of fractional differential equations. As a consequence, comparing the solutions found here with the vector-matrix solutions known in the literature, we obtain explicit formulae for the elements of the 2×2 matrix Mittag-Leffler function.

Matrix Mittag-Leffler function and solution of multi-term fractional differential equations

In this paper, we first derive the solution of fractional differential equation system expressed in matrix Mittag-Leffler function by using the Adomian decomposition method. Then we consider the initial value problem (IVP) for multi-term fractional differential equation. By introducing new unknown functions, we rewrite the IVP for multi-term fractional differential equation into the IVP for a fractional differential equation system. Thus the solution can be given in terms of matrix Mittag-Leffler functions. We demonstrate the method using four numerical examples and the results are simulated using MATHEMATICA 8.

Matrix Mittag-Leffler function and solution of multi-term fractional differential equations

Matrix Mittag-Leffler function and solution of multi-term fractional differential equations

On the Matrix Mittag–Leffler Function: Theoretical Properties and Numerical Computation

Many situations, as for example within the context of Fractional Calculus theory, require computing the Mittag–Leffler (ML) function with matrix arguments. In this paper, we collect theoretical properties of the matrix ML function. Moreover, we describe the available numerical methods aimed at this purpose by stressing advantages and weaknesses.

Solution of Fractional Differential Equation Systems and Computation of Matrix Mittag–Leffler Functions

In this paper, solutions for systems of linear fractional differential equations are considered. For the commensurate order case, solutions in terms of matrix Mittag–Leffler functions were derived by the Picard iterative process. For the incommensurate order case, the system was converted to a commensurate order case by newly introducing unknown functions. Computation of matrix Mittag–Leffler functions was considered using the methods of the Jordan canonical matrix and minimal polynomial or eigenpolynomial, respectively. Finally, numerical examples were solved using the proposed methods.

Some notes on properties of the matrix Mittag-Leffler function

We have come across with publications where some properties of the matrix exponential were incorrectly extended to the matrix Mittag-Leffler function, and then used as a key tool to solve certain linear matrix fractional differential equations. The main purpose of these notes is to give some clarifications on properties of the matrix Mittag-Leffler function, by explaining in detail why some identities do not hold and by providing a list of (valid) properties of this function. A sufficient condition to identify very special cases of pairs of matrices satisfying the semigroup property is given as well as examples illustrating the results.

Analytic Solution of a System of Linear Distributed Order Differential Equations in the Reimann-Liouville Sense

In this paper, solution of a system of linear differential equations of distributed order in the Riemann-Liouville sense is analytically obtained. The distributed order relaxation equation is a special case of the system investigated in this paper. The solution of the mentioned system is introduced on the basis of a function which can be considered as the distributed order generalization of the matrix Mittag-Leffler functions. It is shown that this generalized function in two special cases of the weight function can be expressed in terms of the multivariate Mittag-Leffler functions and the Wright functions.

Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus

The computation of the Mittag-Leffler (ML) function with matrix arguments, and some applications in fractional calculus, are discussed. In general the evaluation of a scalar function in matrix arguments may require the computation of derivatives of possible high order depending on the matrix spectrum. Regarding the ML function, the numerical computation of its derivatives of arbitrary order is a completely unexplored topic; in this paper we address this issue and three different methods are tailored and investigated. The methods are combined together with an original derivatives balancing technique in order to devise an algorithm capable of providing high accuracy. The conditioning of the evaluation of matrix ML functions is also studied. The numerical experiments presented in the paper show that the proposed algorithm provides high accuracy, very often close to the machine precision.

Optimal control of linear systems with fractional derivatives

Abstract Problem of time-optimal control of linear systems with fractional Caputo derivatives is examined using technique of attainability sets and their support functions. A method to construct a control function that brings trajectory of the system to a strictly convex terminal set in the shortest time is elaborated. The proposed method uses technique of set-valued maps and represents a fractional version of Pontryagin’s maximum principle. A special emphasis is placed upon the problem of computing of the matrix Mittag-Leffler function, which plays a key role in the proposed methods. A technique for computing matrix Mittag-Leffler function using Jordan canonical form is discussed, which is implemented in the form of a MATLAB routine. Theoretical results are supported by examples, in which the optimal control functions, in particular of the “bang-bang” type, are obtained.

Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

Multiterm fractional differential equations (MTFDEs) nowadays represent a widely used tool to model many important processes, particularly for multirate systems. Their numerical solution is then a compelling subject that deserves great attention, not least because of the difficulties to apply general purpose methods for fractional differential equations (FDEs) to this case. In this paper, we first transform the MTFDEs into equivalent systems of FDEs, as done by Diethelm and Ford; in this way, the solution can be expressed in terms of Mittag–Leffler (ML) functions evaluated at matrix arguments. We then propose to compute it by resorting to the matrix approach proposed by Garrappa and Popolizio. Several numerical tests are presented that clearly show that this matrix approach is very accurate and fast, also in comparison with other numerical methods.

On the time-fractional Schrödinger equation: Theoretical analysis and numerical solution by matrix Mittag-Leffler functions

This paper presents a deep analysis of a time-dependent Schrödinger equation with fractional time derivative. After the discretization of the spatial operator the equation is reformulated in terms of a special system of fractional differential equations; it occurs that the eigenvalues of the coefficient matrix lay on the boundary of the stability region of fractional differential equations. The main difficulties in solving this system are hence related to the simultaneous presence of persisting oscillations (possibly with high frequency as it is typical with Schrödinger equations) and a persisting memory (as a consequence of the fractional order); moreover, an accurate spatial discretization gives rise to systems of large to very large size, involving a noteworthy computational complexity. By means of a theoretical analysis the exact solution is split into two or three terms (depending on the order of the fractional derivative), thus to face the numerical computation by different and suitably selected methods: direct evaluation of matrix functions for the terms characterized by smooth behaviour but with persistent memory and a step-by-step strategy, in conjunction with matrix function, for the oscillating term. In both cases, Krylov subspace methods are employed for the computation of matrix functions and convergence results are presented.

On the Convergence of Krylov Subspace Methods for Matrix Mittag–Leffler Functions

In this paper we analyze the convergence of some commonly used Krylov subspace methods for computing the action of matrix Mittag-Leffler functions. As is well known, such functions find application in the solution of fractional differential equations. We illustrate the theoretical results by some numerical experiments.