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Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations

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Abstract

Two alternating direction implicit difference schemes are derived for two-dimensional distributed-order fractional diffusion equations. It is proved that the schemes are unconditionally stable and convergent in a discrete \(L^1(L^\infty )\) norm with the convergence orders \(O(\tau ^2|\ln \tau |+h_1^2+h_2^2+\Delta \alpha ^2)\) and \(O(\tau ^2|\ln \tau |+h_1^4+h_2^4+\Delta \alpha ^4),\) respectively, where \( \tau , h_i \;(i=1,2)\) and \(\Delta \alpha \) are the step sizes in time, space and distributed order. Several numerical examples are given to confirm the theoretical results.

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Acknowledgments

The authors thank the anonymous referees for their valuable comments and suggestions to improve this work.

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Authors and Affiliations

  1. College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, People’s Republic of China

    Guang-hua Gao

  2. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Zhi-zhong Sun

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  1. Guang-hua Gao
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  2. Zhi-zhong Sun
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Correspondence to Zhi-zhong Sun.

Additional information

The research is supported by the research Grants 11271068, 11401319, 11426137, 61304169 from National Natural Science Foundation of China, BK20130860 from Natural Science Youth Foundation of Jiangsu Province.

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Gao, Gh., Sun, Zz. Two Alternating Direction Implicit Difference Schemes for Two-Dimensional Distributed-Order Fractional Diffusion Equations. J Sci Comput 66, 1281–1312 (2016). https://doi.org/10.1007/s10915-015-0064-x

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  • DOI: https://doi.org/10.1007/s10915-015-0064-x

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