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Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial

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  • Roberto Garrappa

    (Dipartimento di Matematica, Università Degli Studi di Bari, Via E. Orabona 4, 70126 Bari, Italy)

Abstract

Solving differential equations of fractional (i.e., non-integer) order in an accurate, reliable and efficient way is much more difficult than in the standard integer-order case; moreover, the majority of the computational tools do not provide built-in functions for this kind of problem. In this paper, we review two of the most effective families of numerical methods for fractional-order problems, and we discuss some of the major computational issues such as the efficient treatment of the persistent memory term and the solution of the nonlinear systems involved in implicit methods. We present therefore a set of MATLAB routines specifically devised for solving three families of fractional-order problems: fractional differential equations (FDEs) (also for the non-scalar case), multi-order systems (MOSs) of FDEs and multi-term FDEs (also for the non-scalar case); some examples are provided to illustrate the use of the routines.

Suggested Citation

  • Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:2:p:16-:d:128173
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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.
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    Cited by:

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    2. Özköse, Fatma & Yavuz, Mehmet & Şenel, M. Tamer & Habbireeh, Rafla, 2022. "Fractional order modelling of omicron SARS-CoV-2 variant containing heart attack effect using real data from the United Kingdom," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    3. Cao, Dewei & Chen, Hu, 2023. "Error analysis of a finite difference method for the distributed order sub-diffusion equation using discrete comparison principle," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 109-117.
    4. Kai Diethelm, 2022. "A New Diffusive Representation for Fractional Derivatives, Part II: Convergence Analysis of the Numerical Scheme," Mathematics, MDPI, vol. 10(8), pages 1-12, April.
    5. J. Alberto Conejero & Jonathan Franceschi & Enric Picó-Marco, 2022. "Fractional vs. Ordinary Control Systems: What Does the Fractional Derivative Provide?," Mathematics, MDPI, vol. 10(15), pages 1-18, August.
    6. Fiaz, Muhammad & Aqeel, Muhammad & Marwan, Muhammad & Sabir, Muhammad, 2022. "Integer and fractional order analysis of a 3D system and generalization of synchronization for a class of chaotic systems," Chaos, Solitons & Fractals, Elsevier, vol. 155(C).
    7. 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.
    8. Faïçal Ndaïrou & Delfim F. M. Torres, 2023. "Pontryagin Maximum Principle for Incommensurate Fractional-Orders Optimal Control Problems," Mathematics, MDPI, vol. 11(19), pages 1-12, October.
    9. Roman Ivanovich Parovik, 2022. "Studies of the Fractional Selkov Dynamical System for Describing the Self-Oscillatory Regime of Microseisms," Mathematics, MDPI, vol. 10(22), pages 1-14, November.
    10. Vsevolod Bohaienko & Fasma Diele & Carmela Marangi & Cristiano Tamborrino & Sebastian Aleksandrowicz & Edyta Woźniak, 2023. "A Novel Fractional-Order RothC Model," Mathematics, MDPI, vol. 11(7), pages 1-16, March.
    11. Daniele Mortari & Roberto Garrappa & Luigi Nicolò, 2023. "Theory of Functional Connections Extended to Fractional Operators," Mathematics, MDPI, vol. 11(7), pages 1-18, April.
    12. El-Mesady, A. & Elsonbaty, Amr & Adel, Waleed, 2022. "On nonlinear dynamics of a fractional order monkeypox virus model," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    13. Li, Hang & Shen, Yongjun & Han, Yanjun & Dong, Jinlu & Li, Jian, 2023. "Determining Lyapunov exponents of fractional-order systems: A general method based on memory principle," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    14. Hansin Bilgili & Chwen Sheu, 2022. "A Bibliometric Review of the Mathematics Journal," Mathematics, MDPI, vol. 10(15), pages 1-17, July.

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