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

Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, Via per Arnesano, 73100 Lecce, Italy
Mathematics 2018, 6(1), 7; https://doi.org/10.3390/math6010007
Received: 14 December 2017 / Revised: 29 December 2017 / Accepted: 1 January 2018 / Published: 9 January 2018
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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. View Full-Text
Keywords: fractional differential equations; multiterm differential equations; Mittag–Leffler function; matrix function; fractional calculus fractional differential equations; multiterm differential equations; Mittag–Leffler function; matrix function; fractional calculus
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Popolizio, M. Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions. Mathematics 2018, 6, 7.

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