IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v6y2018i1p7-d126109.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/6/1/7/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/6/1/7/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Arenas, Abraham J. & González-Parra, Gilberto & Chen-Charpentier, Benito M., 2016. "Construction of nonstandard finite difference schemes for the SI and SIR epidemic models of fractional order," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 48-63.
    2. Das, Saptarshi & Pan, Indranil & Das, Shantanu, 2016. "Effect of random parameter switching on commensurate fractional order chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 157-173.
    3. Dmytro Sytnyk & Barbara Wohlmuth, 2023. "Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type," Mathematics, MDPI, vol. 11(10), pages 1-35, May.
    4. Wang, Yuan-Ming & Xie, Bo, 2023. "A fourth-order fractional Adams-type implicit–explicit method for nonlinear fractional ordinary differential equations with weakly singular solutions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 21-48.
    5. Rainey Lyons & Aghalaya S. Vatsala & Ross A. Chiquet, 2017. "Picard’s Iterative Method for Caputo Fractional Differential Equations with Numerical Results," Mathematics, MDPI, vol. 5(4), pages 1-9, November.
    6. Yang, Changqing, 2023. "Improved spectral deferred correction methods for fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    7. Jannelli, Alessandra, 2024. "A finite difference method on quasi-uniform grids for the fractional boundary-layer Blasius flow," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 215(C), pages 382-398.
    8. Hamdan, Nur ’Izzati & Kilicman, Adem, 2018. "A fractional order SIR epidemic model for dengue transmission," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 55-62.
    9. Čermák, Jan & Nechvátal, Luděk, 2019. "Stability and chaos in the fractional Chen system," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 24-33.
    10. Bonab, Zahra Farzaneh & Javidi, Mohammad, 2020. "Higher order methods for fractional differential equation based on fractional backward differentiation formula of order three," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 71-89.
    11. Boyadjiev, Lyubomir & Luchko, Yuri, 2017. "Mellin integral transform approach to analyze the multidimensional diffusion-wave equations," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 127-134.
    12. Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
    13. Atangana, Abdon & Gómez-Aguilar, J.F., 2017. "Hyperchaotic behaviour obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 285-294.
    14. Kai Diethelm & Roberto Garrappa & Martin Stynes, 2020. "Good (and Not So Good) Practices in Computational Methods for Fractional Calculus," Mathematics, MDPI, vol. 8(3), pages 1-21, March.
    15. Sowa, Marcin, 2018. "Application of SubIval in solving initial value problems with fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 319(C), pages 86-103.
    16. Cardone, Angelamaria & Conte, Dajana, 2020. "Stability analysis of spline collocation methods for fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 501-514.
    17. Moustafa, Mahmoud & Mohd, Mohd Hafiz & Ismail, Ahmad Izani & Abdullah, Farah Aini, 2018. "Dynamical analysis of a fractional-order Rosenzweig–MacArthur model incorporating a prey refuge," Chaos, Solitons & Fractals, Elsevier, vol. 109(C), pages 1-13.
    18. Elsayed I. Mahmoud & Temirkhan S. Aleroev, 2022. "Boundary Value Problem of Space-Time Fractional Advection Diffusion Equation," Mathematics, MDPI, vol. 10(17), pages 1-12, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:6:y:2018:i:1:p:7-:d:126109. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.