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

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  • Marina Popolizio

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

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.

Suggested Citation

  • Marina Popolizio, 2018. "Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions," Mathematics, MDPI, vol. 6(1), pages 1-13, January.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:1:p:7-:d:126109
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    References listed on IDEAS

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    1. Garrappa, Roberto, 2015. "Trapezoidal methods for fractional differential equations: Theoretical and computational aspects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 110(C), pages 96-112.
    2. Al-Refai, Mohammed & Luchko, Yuri, 2015. "Maximum principle for the multi-term time-fractional diffusion equations with the Riemann–Liouville fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 40-51.
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