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Error Estimates of High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations

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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. [21] for solving time fractional partial differential equation. We prove that this method has the convergence order O(τ3−α) for all α ∈ (0, 1) when the first and second derivatives of the solution are vanish at t = 0, where τ is the time step size and α 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. [21]. We show that this new method also has the convergence order O(τ3−α) for all α ∈ (0, 1). The proofs of the error estimates are based on the energy method developed recently by Lv and Xu [26]. We also consider the space discretization by using the finite element method. Error estimates with convergence order O(τ3−α + h2) 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.

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Authors and Affiliations

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

    Zhiqiang Li

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, P. R. China

    Zhiqiang Li

  3. Department of Mathematics, University of Chester, Parkgate Road, Chester, CH1 4BJ, UK

    Yubin Yan

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  1. Zhiqiang Li
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  2. Yubin Yan
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Correspondence to Zhiqiang Li.

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Li, Z., Yan, Y. Error Estimates of High-Order Numerical Methods for Solving Time Fractional Partial Differential Equations. FCAA 21, 746–774 (2018). https://doi.org/10.1515/fca-2018-0039

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