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The Unified Theory of Shifted Convolution Quadrature for Fractional Calculus

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Abstract

This work devotes to developing a systematic and convenient approach based on the celebrated convolution quadrature theory to design and analyze difference formulas for fractional calculus at an arbitrary shifted point \(x_{n-\theta }\). The developed theory, called shifted convolution quadrature (SCQ), covers most difference formulas from the aspects of characterizing the formation of related generating functions which are convergent with integer orders. For stability reasons, the theoretical determination of shifted parameter \(\theta \) is provided to fill the gap in which the choice of \(\theta \) depends heavily on experiments particularly for non-integer order derivatives. Further, to discuss the effects of \(\theta \) on A(\(\delta \))-stability, stability regions for several generalized popular formulas within SCQ are examined which are crucial to developing robust numerical schemes. Some numerical tests are also considered to demonstrate the necessity of introducing \(\theta \) for theoretical and practical purposes.

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Acknowledgements

The work of the first author was supported in part by the NSFC Grant 12061053, the NSF of Inner Mongolia 2020MS01003. The work of the third author was supported in part by the NSFC Grant 12161063, the NSF of Inner Mongolia 2021MS01018. The work of the fourth author was supported in part by Grants NSFC 11871092 and NSAF U1930402.

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Yang Liu, Baoli Yin & Hong Li

  2. Beijing Computational Science Research Center, Beijing, 100193, China

    Zhimin Zhang

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

    Zhimin Zhang

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  1. Yang Liu
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  2. Baoli Yin
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  3. Hong Li
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  4. Zhimin Zhang
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Correspondence to Yang Liu.

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Liu, Y., Yin, B., Li, H. et al. The Unified Theory of Shifted Convolution Quadrature for Fractional Calculus. J Sci Comput 89, 18 (2021). https://doi.org/10.1007/s10915-021-01630-9

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  • DOI: https://doi.org/10.1007/s10915-021-01630-9

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