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Good (and Not So Good) Practices in Computational Methods for Fractional Calculus

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  • Kai Diethelm

    (Fakultät Angewandte Natur- und Geisteswissenschaften, University of Applied Sciences Würzburg-Schweinfurt, Ignaz-Schön-Str. 11, 97421 Schweinfurt, Germany
    GNS mbH Gesellschaft für Numerische Simulation mbH, Am Gaußberg 2, 38114 Braunschweig, Germany)

  • Roberto Garrappa

    (Department of Mathematics, University of Bari, Via E. Orabona 4, 70126 Bari, Italy
    INdAM Research Group GNCS, Piazzale Aldo Moro 5, 00185 Rome, Italy)

  • Martin Stynes

    (Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, China)

Abstract

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:324-:d:327120
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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:

    1. Zhang, Yufeng & Li, Jing & Zhu, Shaotao & Ma, Zerui, 2024. "Harmonic resonance and bifurcation of fractional Rayleigh oscillator with distributed time delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 281-297.
    2. 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.

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