Error estimates of high-order numerical methods for solving time fractional partial differential equations
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Luliang University; Shanghai University; University of ChesterPublication Date
2018-07-12
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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. Error estimates of high-order numerical methods for solving time fractional partial differential equations, Fractional Calculus and Applied Analysis, 3(2018) pp. 746-774Publisher
De GruyterAdditional Links
https://www.degruyter.com/view/j/fca.2018.21.issue-3/fca-2018-0039/fca-2018-0039.xmlType
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enSeries/Report no.
finite difference methodEISSN
1314-2224ae974a485f413a2113503eed53cd6c53
10.1515/fca-2018-0039
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