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A Rational Approximation Scheme for Computing Mittag-Leffler Function with Discrete Elliptic Operator as Input

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Abstract

In this work, we propose a new scheme based on numerical quadrature to calculate the two-parameter Mittag-Leffler function with discrete elliptic operator \(-{\mathcal {L}}_h\) as input. Except pure mathematical interest from approximation theory, our consideration also arises from solving sub-diffusion equations numerically with time-independent diffusion coefficient. We obtain the scheme by applying Gauss-Legendre quadrature rule for the integral representation of the Mittag-Leffler function. Rigorous error analysis is carried out which shows that the scheme converges exponentially with the increase of quadrature nodes. The computational cost of the algorithm is solving K sparse linear systems with K the number of quadrature nodes. It is worth to point out that the scheme is completely parallel which can save much time if the dimension of \({\mathcal {L}}_h\) is very large. Some numerical tests are provided to verify the efficiency and robustness of our scheme.

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Authors and Affiliations

  1. Shenzhen JL Computational Science and Applied Research Institute, Shenzhen, 518027, People’s Republic of China

    Beiping Duan

  2. Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China

    Beiping Duan & Zhimin Zhang

  3. Department of Mathematics, Wayne State University, Detroit, 48202, MI, USA

    Zhimin Zhang

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  1. Beiping Duan
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  2. Zhimin Zhang
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Correspondence to Beiping Duan.

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The work is supported in part by Grants NSFC 11871092 and NSAF U1930402, and the first author is supported by project founded by China Postdoctoral Science Foundation(2020M682895).

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Duan, B., Zhang, Z. A Rational Approximation Scheme for Computing Mittag-Leffler Function with Discrete Elliptic Operator as Input. J Sci Comput 87, 75 (2021). https://doi.org/10.1007/s10915-021-01495-y

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  • DOI: https://doi.org/10.1007/s10915-021-01495-y

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