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Convergence Rates of a Finite Difference Method for the Fractional Subdiffusion Equations

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Differential Equations, Mathematical Modeling and Computational Algorithms (DEMMCA 2021)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 423))

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Abstract

We consider the convergence of an effective numerical method of the subdiffusion equation with the Caputo fractional derivative in time. We investigate an implicit difference scheme and an explicit difference scheme by using the projection method in space and a finite difference method which was proposed by Ashyralyev in time. Combining the method of functional analysis and the technique of numerical analysis, we utilize the idea of layering in temporal direction to obtain that the local truncation error is \(O(n^{-\alpha })\). Then we prove that the implicit and explicit numerical methods converge at a rate of \(O(\tau ^\alpha )\) in time. Finally, a numerical experiment is given to confirm the \(\alpha \)-th order accuracy.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11571300 and 11871064) and the Russian Science Foundation (Grant No. N20-11-20085, 23-21-00005).

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Authors and Affiliations

  1. School of Mathematical Sciences, Yangzhou University, Jiangsu, 225002, China

    Li Liu, Zhenbin Fan & Gang Li

  2. Scientific Research Computer Center, Lomonosov Moscow State University, Leninskie Gory, Moscow, 119899, Russia

    Sergey Piskarev

  3. Mari State University, Lenin Square 1, Yoshkar-Ola City, 424000, Russia

    Sergey Piskarev

Authors
  1. Li Liu
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  4. Sergey Piskarev
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Correspondence to Sergey Piskarev .

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Editors and Affiliations

  1. Chair of Appl. Math. Comp. Model., Belgorod State National Research Univers, Belgorod, Russia

    Vladimir Vasilyev

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Liu, L., Fan, Z., Li, G., Piskarev, S. (2023). Convergence Rates of a Finite Difference Method for the Fractional Subdiffusion Equations. In: Vasilyev, V. (eds) Differential Equations, Mathematical Modeling and Computational Algorithms. DEMMCA 2021. Springer Proceedings in Mathematics & Statistics, vol 423. Springer, Cham. https://doi.org/10.1007/978-3-031-28505-9_7

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