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An ADI Crank–Nicolson Orthogonal Spline Collocation Method for the Two-Dimensional Fractional Diffusion-Wave Equation

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Abstract

A new method is formulated and analyzed for the approximate solution of a two-dimensional time-fractional diffusion-wave equation. In this method, orthogonal spline collocation is used for the spatial discretization and, for the time-stepping, a novel alternating direction implicit method based on the Crank–Nicolson method combined with the \(L1\)-approximation of the time Caputo derivative of order \(\alpha \in (1,2)\). It is proved that this scheme is stable, and of optimal accuracy in various norms. Numerical experiments demonstrate the predicted global convergence rates and also superconvergence.

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Acknowledgments

The authors wish to thank the referees for their constructive criticisms and helpful suggestions. This project was funded in part by the National Nature Science Foundation of China (contract Grant 11271123) and the Research and Innovation Project for College Graduates of Hunan Province (contract Grant CX2012B196).

Conflict of interest

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. Mathematical Reviews, American Mathematical Society, 416 Fourth Street, Ann Arbor, MI, 48103, USA

    Graeme Fairweather

  2. School of Science, Hunan University of Technology, Zhuzhou, 412008, Hunan, People’s Republic of China

    Xuehua Yang & Haixiang Zhang

  3. College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Da Xu

Authors
  1. Graeme Fairweather
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  2. Xuehua Yang
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  3. Da Xu
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  4. Haixiang Zhang
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Correspondence to Graeme Fairweather.

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Fairweather, G., Yang, X., Xu, D. et al. An ADI Crank–Nicolson Orthogonal Spline Collocation Method for the Two-Dimensional Fractional Diffusion-Wave Equation. J Sci Comput 65, 1217–1239 (2015). https://doi.org/10.1007/s10915-015-0003-x

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  • DOI: https://doi.org/10.1007/s10915-015-0003-x

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