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

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Abstract

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.

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  1. www.mathworks.com/matlabcentral/fileexchange/66272-mittag-leffler-function-with-matrix-arguments.

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Authors and Affiliations

  1. Università degli Studi di Bari, Dipartimento di Matematica, Via E. Orabona n.4, 70125, Bari, Italy

    Roberto Garrappa

  2. Università del Salento, Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Via per Arnesano, 73100, Lecce, Italy

    Marina Popolizio

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  1. Roberto Garrappa
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  2. Marina Popolizio
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Correspondence to Marina Popolizio.

Additional information

This work is supported under the GNCS-INdAM 2017 Project “Analisi numerica per modelli descritti da operatori frazionari”.

R. Garrappa \(\cdot \) M. Popolizio: Member of the INdAM Research group GNCS.

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Garrappa, R., Popolizio, M. Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus. J Sci Comput 77, 129–153 (2018). https://doi.org/10.1007/s10915-018-0699-5

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  • DOI: https://doi.org/10.1007/s10915-018-0699-5

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