Some notes on properties of the matrix Mittag-Leffler function

Sadeghi, Amir; Cardoso, João R.

doi:10.1016/j.amc.2018.06.037

Cited by 19 publications

(8 citation statements)

References 8 publications

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“…MLFs of matrix argument arise naturally in solutions of systems of fractional differential equations. The two-parametric MLF of a matrix A ∈ C n×n (see [22]) is given by:…”

Section: Applications With Matrix Argumentsmentioning

confidence: 99%

Highly Accurate Global Padé Approximations of Generalized Mittag–Leffler Function and Its Inverse

Sarumi

Furati

2020

J Sci Comput

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The two-parametric Mittag-Leffler function (MLF), E α,β , is fundamental to the study and simulation of fractional differential and integral equations. However, these functions are computationally expensive and their numerical implementations are challenging. In this paper, we present a unified framework for developing global rationalThis framework is based on the series definition and the asymptotic expansion at infinity. In particular, we develop three types of fourth-order global rational approximations and discuss how they could be used to approximate the inverse function. Unlike existing approximations which are either limited to MLF of one parameter or of low accuracy for the two-parametric MLF, our rational approximants are of fourth order accuracy and have low percentage error globally. For efficient utilization, we study the partial fraction decomposition and use them to approximate the two-parametric MLF with a matrix argument which arise in the solutions of fractional evolution differential and integral equations.

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“…MLFs of matrix argument arise naturally in solutions of systems of fractional differential equations. The two-parametric MLF of a matrix A ∈ C n×n (see [22]) is given by:…”

Section: Applications With Matrix Argumentsmentioning

confidence: 99%

Highly Accurate Global Padé Approximations of Generalized Mittag–Leffler Function and Its Inverse

Sarumi

Furati

2020

J Sci Comput

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“…To solve equation (43), we can use Theorem 5.15 of book [25], p.323. Fractional differential equation (43) can be written in the form…”

Section: Solution Of Equation For X µ (τ )mentioning

confidence: 99%

Dirac particle with memory: Proper time non-locality

Tarasov

2020

Physics Letters A

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A generalization of the standard model of Dirac particle in external electromagnetic field is proposed. In the generalization we take into account interactions of this particle with environment, which is described by the memory function. This function takes into account that the behavior of the particle at proper time can depend not only at the present time, but also on the history of changes on finite time interval. In this case the Dirac particle can be considered an open quantum system with non-Markovian dynamics. The violation of the semigroup property of dynamic maps is a characteristic property of dynamics with memory. We use the Fock-Schwinger proper time method and derivatives of non-integer orders with respect to proper time. The fractional differential equation, which describes the Dirac particle with memory, and the expression of its exact solution are suggested. The asymptotic behavior of the proposed solutions is described. PACS 45.10.Hj Perturbation and fractional calculus methods PACS 03.65.Yz Decoherence; open systems; quantum statistical methods PACS 11.90.+t Other topics in general theory of fields and particles

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“…See more information about Mittag-Leffler functions e.g., in [27]. The Mittag-Leffler function includes the following special cases:…”

Section: Mittag-leffler Functionmentioning

confidence: 99%

General Solutions for Descriptor Systems of Coupled Generalized Sylvester Matrix Fractional Differential Equations via Canonical Forms

Tansri

Chansangiam

2020

Symmetry

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We investigate a descriptor system of coupled generalized Sylvester matrix fractional differential equations in both non-homogeneous and homogeneous cases. All fractional derivatives considered here are taken in Caputo’s sense. We explain a 4-step procedure to solve the descriptor system, consisting of vectorization, a matrix canonical form concerning ranks, and matrix partitioning. The procedure aims to reduce the descriptor system to a descriptor system of fractional differential equations. We also impose a condition on coefficient matrices, related to the symmetry of the solution for descriptor systems. It follows that an explicit form of its general solution is given in terms of matrix power series concerning Mittag–Leffler functions. The main system includes certain systems of coupled matrix/vector differential equations, and single matrix differential equations as special cases. In particular, we obtain an alternative procedure to solve linear continuous-time descriptor systems via a matrix canonical form.

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Some notes on properties of the matrix Mittag-Leffler function

Cited by 19 publications

References 8 publications

Highly Accurate Global Padé Approximations of Generalized Mittag–Leffler Function and Its Inverse

Highly Accurate Global Padé Approximations of Generalized Mittag–Leffler Function and Its Inverse

Dirac particle with memory: Proper time non-locality

General Solutions for Descriptor Systems of Coupled Generalized Sylvester Matrix Fractional Differential Equations via Canonical Forms

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