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WSGD-OSC Scheme for Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation

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Abstract

In this paper, a new numerical approximation is discussed for the two-dimensional distributed-order time fractional reaction–diffusion equation. Combining with the idea of weighted and shifted Grünwald difference (WSGD) approximation (Tian et al. in Math Comput 84:1703–1727, 2015; Wang and Vong in J Comput Phys 277:1–15, 2014) in time, we establish orthogonal spline collocation (OSC) method in space. A detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order \(\mathscr {O}(\tau ^2+\Delta \alpha ^2+h^{r+1})\), where \(\tau , \Delta \alpha , h\) and r are, respectively the time step size, step size in distributed-order variable, space step size, and polynomial degree of space. Interestingly, we prove that the proposed WSGD-OSC scheme converges with the second-order in time, where OSC schemes proposed previously (Fairweather et al. in J Sci Comput 65:1217–1239, 2015; Yang et al. in J Comput Phys 256:824–837, 2014) can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Some numerical results are also given to confirm our theoretical prediction.

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Acknowledgements

We are grateful to the two anonymous referees for useful comments and suggestions. We also wish to thank Prof. Graeme Fairweather for many useful discussions and for the guidance over the past years.

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Authors and Affiliations

  1. School of Science, Hunan University of Technology, Zhuzhou, 412007, China

    Xuehua Yang & Haixiang Zhang

  2. National Key Laboratory of Science and Technology on Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Xuehua Yang

  3. Department of Mathematics, Hunan Normal University, Changsha, 410081, China

    Da Xu

Authors
  1. Xuehua Yang
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  2. Haixiang Zhang
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  3. Da Xu
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Corresponding authors

Correspondence to Haixiang Zhang or Da Xu.

Additional information

The work is supported by the National Natural Science Foundation of China Grant 11701168, 11601144, 11626096, 11671131; the Scientific Research Fund of Hunan Provincial Education Department Grant YB2016B033, 16K026; the Science Challenge Project Grant TZ2016002.

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Yang, X., Zhang, H. & Xu, D. WSGD-OSC Scheme for Two-Dimensional Distributed Order Fractional Reaction–Diffusion Equation. J Sci Comput 76, 1502–1520 (2018). https://doi.org/10.1007/s10915-018-0672-3

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  • DOI: https://doi.org/10.1007/s10915-018-0672-3

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