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A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions

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Abstract

In this paper, a compact finite difference scheme with global convergence order \(O\big (\tau ^{2-\alpha }+h^4\big )\) is derived for fractional sub-diffusion equations with the spatially variable coefficient subject to Neumann boundary conditions. The difficulty caused by the variable coefficient and the Neumann boundary conditions is overcome by subtle decomposition of the coefficient matrices. The stability and convergence of the proposed scheme are studied using its matrix form by the energy method. The theoretical results are supported by numerical experiments.

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Acknowledgments

The authors would like to thank the editor and the referees for their helpful comments. In particular, we would like to thank the referees for pointing out that the scheme in [30] may still work even though the equations have variable coefficients.

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Authors and Affiliations

  1. Department of Mathematics, University of Macau, Avenida da Universidade, Taipa, Macau, China

    Seakweng Vong, Pin Lyu & Zhibo Wang

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, Sichuan, China

    Zhibo Wang

Authors
  1. Seakweng Vong
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  2. Pin Lyu
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  3. Zhibo Wang
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Corresponding author

Correspondence to Seakweng Vong.

Additional information

S. Vong: This research is supported by the Macao Science and Technology Development Fund FDCT/001/2013/A and the Grant MYRG2015-00064-FST from University of Macau.

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Vong, S., Lyu, P. & Wang, Z. A Compact Difference Scheme for Fractional Sub-diffusion Equations with the Spatially Variable Coefficient Under Neumann Boundary Conditions. J Sci Comput 66, 725–739 (2016). https://doi.org/10.1007/s10915-015-0040-5

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  • DOI: https://doi.org/10.1007/s10915-015-0040-5

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