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Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

Author

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  • Dmytro Sytnyk

    (Department of Mathematics, Technical University of Munich, 85748 Garching, Germany
    Department of Numerical Mathematics, Institute of Mathematics, National Academy of Sciences, 01024 Kyiv, Ukraine
    The authors contributed equally to this work.)

  • Barbara Wohlmuth

    (Department of Mathematics, Technical University of Munich, 85748 Garching, Germany
    The authors contributed equally to this work.)

Abstract

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A , fractional order α and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side f ( t ) . The resulting method possesses exponential convergence for positive sectorial A , any finite t , including t = 0 and the whole range α ∈ ( 0 , 2 ) . It is suitable for a practically important case, when no knowledge of f ( t ) is available outside the considered interval t ∈ [ 0 , T ] . The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:10:p:2312-:d:1147774
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    References listed on IDEAS

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    1. Roberto Garrappa, 2018. "Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial," Mathematics, MDPI, vol. 6(2), pages 1-23, January.
    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.
    3. 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.
    4. Valentin Keyantuo & Carlos Lizama & Mahamadi Warma, 2013. "Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-11, November.
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