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Superconvergence and Postprocessing of Collocation Methods for Fractional Differential Equations

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Abstract

This paper aims to propose a complete superconvergence analysis for a postprocessing technique based on collocation methods for fractional differential equations (FDEs). We start with the simple linear FDEs with Caputo derivative of order \(0<\alpha <1\). The problem is reformulated as a weakly singular Volterra integral equation (VIE), and based on the resolvent theory of VIEs, the existence, uniqueness and regularity for the exact solution for the original FDE are obtained. Then the piecewise polynomial collocation method is adopted to solve the reformulated VIE, and based on the regularity of the original FDE, the convergence for the collocation method and the superconvergence for the iterated collocation method are investigated in detail, respectively. Further, based on the obtained collocation solution, the interpolation postprocessing approximation of higher accuracy is constructed on graded mesh, and the superconvergence is obtained. Compared with classical iterated collocation method, the cost on computation of interpolation postprocessing technique is less. Numerical experiments are given to illustrate the theoretical results, and it is also shown that the proposed postprocessing technique can be extended to certain nonlinear and systems of FDEs.

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  1. School of Science, Harbin Institute of Technology, Shenzhen, 518055, China

    Lu Wang & Hui Liang

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  1. Lu Wang
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  2. Hui Liang
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Correspondence to Hui Liang.

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The work was supported by the National Natural Science Foundation of China (No. 12171122), Guangdong Provincial Natural Science Foundation of China (2023A1515010818), Fundamental Research Project of Shenzhen (No. JCYJ20190806143201649) and Shenzhen Science and Technology Program (No. RCJC20210609103755110).

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Wang, L., Liang, H. Superconvergence and Postprocessing of Collocation Methods for Fractional Differential Equations. J Sci Comput 97, 29 (2023). https://doi.org/10.1007/s10915-023-02339-7

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  • DOI: https://doi.org/10.1007/s10915-023-02339-7

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