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Optimal error analysis of the Alikhanov formula for a time-fractional Schrödinger equation

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Abstract

In this paper, we have developed a fully discrete Alikhanov finite element method to solve the time-fractional Schrödinger equation with non-smooth solution. The proposed scheme uses the Alikhanov formula on graded meshes to approximate the Caputo fractional derivative in temporal direction and the standard finite element method in spatial direction. Furthermore, the \(L^2(\Omega )\)-norm stability and the optimal convergent result for the computed solution are derived. Finally, a numerical example is presented to verify the accuracy of the proposed scheme.

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Acknowledgements

The research of Chaobao Huang is supported in part by the National Natural Science Foundation of China under Grant 12101360 and the Natural Science Foundation of Shandong Province under Grant ZR2020QA031. The research of Na An is supported by the National Natural Science Foundation of China under Grant 11801332.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Guoye Zhao & Na An

  2. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, 250014, China

    Chaobao Huang

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  1. Guoye Zhao
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  2. Na An
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  3. Chaobao Huang
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Correspondence to Na An.

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Zhao, G., An, N. & Huang, C. Optimal error analysis of the Alikhanov formula for a time-fractional Schrödinger equation. J. Appl. Math. Comput. 69, 159–170 (2023). https://doi.org/10.1007/s12190-022-01733-y

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  • DOI: https://doi.org/10.1007/s12190-022-01733-y

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