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

Moret, Igor; Novati, Paolo

doi:10.1137/080738374

Cited by 50 publications

(54 citation statements)

References 27 publications

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“…However, for a long time, this has been considered only as a theoretical tool because of the lack of effective methods to numerically approximate this function. Only recently have many advances been made for the numerical evaluation of the scalar ML function [26][27][28][29]; the case of matrix arguments has since been analyzed [30,31], and finally a numerical algorithm has been accomplished, which reaches very high accuracies [32]. In this paper, we show the effectiveness of the matrix approach when solving MTFDEs, both in terms of execution time and in terms of accuracy, and also in comparison with some well-established numerical methods.…”

Section: Introductionmentioning

confidence: 99%

“…The numerical computation of matrix functions is an extensively studied topic that has deserved great attention during the last decades (we refer to Higham [37] for a complete treatise and a full list of references). Only recent studies have considered matrix arguments for the ML function (see, e.g., [30][31][32]38,39]). To be precise, even the numerical scalar case has received poor attention, and only recently has Garrappa [29] developed a powerful Matlab routine (ml.m, available on Matlab website) that gives very accurate results for arguments all over the complex plane.…”

Section: Matrix Approach For the Solution Of Linear Mtfdesmentioning

confidence: 99%

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

Popolizio

2018

Mathematics

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Abstract: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.

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Section: Introductionmentioning

confidence: 99%

Section: Matrix Approach For the Solution Of Linear Mtfdesmentioning

confidence: 99%

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

Popolizio

2018

Mathematics

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“…Other approaches based on mixed techniques (Taylor series, asymptotic series, and integral representations) were discussed in [18,32]. The only existing Matlab code [30] (which implements some of the ideas introduced in [16]) shows a great variability in the amount of computation required to achieve a prescribed accuracy and in some regions of the complex plane turns out to be poorly accurate.The recent introduction [11,12,26] of new methods for fractional differential equations involving a large number of evaluations of the ML function, also with matrix arguments [27], motivates the investigation of different techniques to perform the computation, over the whole complex plane, in an accurate and fast way.In this paper we consider an approach based on the inversion of the Laplace transform (LT) in which a quadrature rule is applied on a suitable complex contour, namely a parabola. Methods of this kind have been successfully applied [13,40] to the ML function restricted to some very special cases (0 < α < 1, β = 1 or real z).…”

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confidence: 99%

Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions

Garrappa¹

2015

SIAM J. Numer. Anal.

242

172

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Abstract. The Mittag-Leffler (ML) function plays a fundamental role in fractional calculus but very few methods are available for its numerical evaluation. In this work we present a method for the efficient computation of the ML function based on the numerical inversion of its Laplace transform (LT): an optimal parabolic contour is selected on the basis of the distance and the strength of the singularities of the LT, with the aim of minimizing the computational effort and reduce the propagation of errors. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. The application to the three parameter ML (also known as Prabhakar) function is also presented.Key words. Mittag-Leffler function, Laplace transform, trapezoidal rule, fractional calculus, Prabhakar function, special function.AMS subject classifications. 33E12, 44A10, 65D30, 33F05, 26A33, 1. Introduction. The Mittag-Leffler (ML) function was introduced, at the beginning of the twentieth century, by the Swedish mathematician Magnus Gustaf Mittag-Leffler [24,25] while studying summation of divergent series. Extensions to two [41] and three [31] parameters of the original single parameter function were successively considered; all these functions can be regarded as special instances of the generalized hypergeometric function investigated by Fox [9] and Wright [43].Until the 1960s, few authors (e.g., [19]) recognized the importance of the ML function in fractional calculus and, in particular, in describing anomalous processes with hereditary effects [1,4,5,8]. For an historical outline and a review of the main properties of the ML function we refer to [17,23] and to the recent monograph [15].For any argument z ∈ C, the ML function with two parameters α, β ∈ C, with ℜ(α) > 0, is defined by means of the series expansion where Γ(z) = ∞ 0 t z−1 e −t dt is the Euler's gamma function; since the integral representation of Γ(z) holds only for ℜ(z) > 0, the extension to the half-plane ℜ(z) ≤ 0, with z ∈ {0, −1, −2, . . .}, is accomplished by means of the relationship Γ(z + n) = z(z + 1) · · · (z + n − 1)Γ(z), n ∈ N, [20,22].As a special case, the ML function with one parameter is obtained for β = 1, i.e. E α (z) = E α,1 (z), whilst the generalization to a third parameter γ E γ α,β (z) = 1 Γ(γ)is recently receiving an increasing attention due to the applications in modeling polarization processes in anomalous or inhomogeneous materials [3].In this work we restrict our attention to real parameters α, β and γ, with α > 0 and γ > 0, since they are of more practical interest. * This work has been supported under the GNCS -INdAM Project 2014.† Università degli Studi di Bari "Aldo Moro", Dipartimento di Matematica Via E. Orabona n.4, 70125 Bari, Italy (roberto.garrappa@uniba.it) 1 2 R. GARRAPPAWith the exception of a few special cases in which the ML function can be represented in terms of other elementary and special functions, as for instance E 1,1 (z) = e z , E 2,1 (z 2 ) = cosh(z), E 2,1 (−z 2 ) = cos(z) and E 1 2 ,1 (±z 1/2 ) = e z ...

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“…Future work will extend the approaches here to larger scales by exploiting the parallelism that the Cauchy integral offers-the shifted linear systems (z k I−A)u k = v of (3) may be solved independently and via iterative methods such as Krylov methods [28]. Conclusion Cauchy integrals are effective in computationally solving bimolecular master equations.…”

Section: Discussionmentioning

confidence: 99%