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Error estimates of high-order numerical methods for solving time fractional partial differential equations
Li, Zhiqiang Yan, Yubin
Li, Zhiqiang
Yan, Yubin
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2018-07-12
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Abstract
Error estimates of some high-order numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a high-order numerical method proposed recently by Li et al. \cite{liwudin} for solving time fractional partial differential equation. We prove that this method has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$ when the first and second derivatives of the solution are vanish at $t=0$, where $\tau$ is the time step size and $\alpha$ is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. \cite{liwudin}. We show that this new method also has the convergence order $O(\tau^{3- \alpha})$ for all $\alpha \in (0, 1)$. The proofs of the error estimates are based on the energy method developed recently by Lv and Xu \cite{lvxu}. We also consider the space discretization by using the finite element method. Error estimates with convergence order $O(\tau^{3- \alpha} + h^2)$ are proved in the fully discrete case, where $h$ is the space step size. Numerical examples in both one- and two-dimensional cases are given to show that the numerical results are consistent with the theoretical results.
Citation
Li, Z., Yan, Y. (2018). Error estimates of high-order numerical methods for solving time fractional partial differential equations. Fractional Calculus and Applied Analysis, 21(3), 746-774. https://doi.org/10.1515/fca-2018-0039
Publisher
Springer
Journal
Fractional Calculus and Applied Analysis
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DOI
10.1515/fca-2018-0039
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Article
Language
en
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finite difference method
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EISSN
1314-2224