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An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space

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Abstract

This paper addresses the numerical solution of the three-dimensional nonlocal evolution equation with a weakly singular kernel. The first order fractional convolution quadrature scheme and backward Euler (BE) alternating direction implicit (ADI) method, are proposed to approximate and discretize the Riemann-Liouville (R-L) fractional integral term and temporal derivative, respectively. In order to obtain a fully discrete method, the standard central finite difference approximation is used to discretize the second-order spatial derivative. By using ADI scheme for the three-dimensional problem, the overall computational cost is reduced significantly. Two new approaches are adopted for theoretical stability analysis. The convergence behaviour of the proposed method is provided and the error bounds are proved. In addition, two test problems illustrate the validity and effectiveness of the methods. The CPU time of our scheme is extremely little.

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Authors and Affiliations

  1. School of Science, Hunan University of Technology, Zhuzhou, 412007, China

    Haixiang Zhang, Yuan Liu & Xuehua Yang

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  1. Haixiang Zhang
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  2. Yuan Liu
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  3. Xuehua Yang
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Correspondence to Xuehua Yang.

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The work was supported by National Natural Science Foundation of China (12126321), Scientific Research Fund of Hunan Provincial Education Department (21B0550), Hunan Provincial Natural Science Foundation of China (2022JJ50083, 2021JJ30209)

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Zhang, H., Liu, Y. & Yang, X. An efficient ADI difference scheme for the nonlocal evolution problem in three-dimensional space. J. Appl. Math. Comput. 69, 651–674 (2023). https://doi.org/10.1007/s12190-022-01760-9

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  • DOI: https://doi.org/10.1007/s12190-022-01760-9

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