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Two High-Order Time Discretization Schemes for Subdiffusion Problems with Nonsmooth Data

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Abstract

Two new high-order time discretization schemes for solving subdiffusion problems with nonsmooth data are developed based on the corrections of the existing time discretization schemes in literature. Without the corrections, the schemes have only a first order of accuracy for both smooth and nonsmooth data. After correcting some starting steps and some weights of the schemes, the optimal convergence orders O(k3-α) and O(k4-α) with 0 > α > 1 can be restored for any fixed time t for both smooth and nonsmooth data, respectively. The error estimates for these two new high-order schemes are proved by using Laplace transform method for both homogeneous and inhomogeneous problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

MSC 2010: Primary 65M06; Secondary 65M12, 65M15, 26A33, 35R11

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Authors and Affiliations

  1. Department of Mathematics, Luliang University, Lishi, 033000, P.R. China

    Yanyong Wang

  2. Department of Mathematical and Physical Sciences Fac. of Science and Engineering, University of Chester Thornton Science Park, Pool Lane, Ince, CH2 4NU, UK

    Yubin Yan

  3. Department of Mathematics, Luliang University, Lishi, 033000, P.R. China

    Yan Yang

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  1. Yanyong Wang
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  2. Yubin Yan
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  3. Yan Yang
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Correspondence to Yanyong Wang.

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Wang, Y., Yan, Y. & Yang, Y. Two High-Order Time Discretization Schemes for Subdiffusion Problems with Nonsmooth Data. Fract Calc Appl Anal 23, 1349–1380 (2020). https://doi.org/10.1515/fca-2020-0067

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  • DOI: https://doi.org/10.1515/fca-2020-0067

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