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An Improved Algorithm Based on Finite Difference Schemes for Fractional Boundary Value Problems with Nonsmooth Solution

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Abstract

In this paper, an efficient algorithm is presented by the extrapolation technique to improve the accuracy of finite difference schemes for solving the fractional boundary value problems with nonsmooth solution. Two popular finite difference schemes, the weighted shifted Grünwald difference (WSGD) scheme and the fractional centered difference (FCD) scheme, are revisited and stability of the schemes is shown in maximum norm. Based on the analysis of leading singularity of exact solution for the underlying problem, it is demonstrated that, with the use of the proposed algorithm, the improved WSGD and FCD schemes can achieve higher accuracy than the original ones for nonsmooth solution. To further improve the accuracy for solving problems with small fractional order, an extended algorithm dealing with two-term singularities correction is also developed. Several numerical examples are given to validate our theoretical prediction. It is shown that both accuracy and convergence rate of numerical solutions can be significantly improved by using the proposed algorithms.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions that helped us improve our paper. The authors also would like to thank Dr. Sheng Chen for helpful discussion during the Zhaopeng Hao visiting in Purdue University and thank Prof. Zhi-Zhong Sun for proofreading the manuscript.

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Zhaopeng Hao & Wanrong Cao

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  1. Zhaopeng Hao
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  2. Wanrong Cao
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Correspondence to Wanrong Cao.

Additional information

The research is supported by National Natural Science Foundation of China (No. 11671083). Zhaopeng Hao was also partially supported by the National University Student Innovation Program (No. 1410286047) and China Scholarship Council (No. 201506090065).

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Hao, Z., Cao, W. An Improved Algorithm Based on Finite Difference Schemes for Fractional Boundary Value Problems with Nonsmooth Solution. J Sci Comput 73, 395–415 (2017). https://doi.org/10.1007/s10915-017-0417-8

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  • DOI: https://doi.org/10.1007/s10915-017-0417-8

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