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Exponentially Convergent Method for the Approximation of a Differential Equation with Fractional Derivative and Unbounded Operator Coefficient in a Banach Space

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Ukrainian Mathematical Journal Aims and scope

We propose and analyze an exponentially convergent numerical method for solving differential equations with right-hand fractional Riemann–Liouville derivative and unbounded operator coefficient in Banach spaces. We use a representation of the solution in the form of the Danford–Cauchy integral on a hyperbola that covers the spectrum of the operator coefficient with subsequent application of the exponentially convergent quadrature. To this end, we choose the parameters of the hyperbola in order to guarantee the possibility of analytic extension of the integrand in a strip containing the real axis and then apply the Sinc-quadrature. We prove the exponential accuracy of the method and present a numerical example that confirms the obtained a priori estimate.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    V. B. Vasylyk & V. L. Makarov

  2. University Duale Hochschule Gera-Eisenach, Gera-Eisenach, Germany

    I. P. Gavrilyuk

Authors
  1. V. B. Vasylyk
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  2. I. P. Gavrilyuk
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  3. V. L. Makarov
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Correspondence to V. B. Vasylyk.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 2, pp. 151–163, February, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i2.6984.

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Vasylyk, V.B., Gavrilyuk, I.P. & Makarov, V.L. Exponentially Convergent Method for the Approximation of a Differential Equation with Fractional Derivative and Unbounded Operator Coefficient in a Banach Space. Ukr Math J 74, 171–185 (2022). https://doi.org/10.1007/s11253-022-02056-8

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  • DOI: https://doi.org/10.1007/s11253-022-02056-8

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