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Unconditionally convergent and superconvergent finite element method for nonlinear time-fractional parabolic equations with distributed delay

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Abstract

In this paper, we propose an efficient Newton linearized numerical method for the nonlinear time-fractional parabolic equations with distributed delay based on the Galerkin finite element method in space and the nonuniform L1 scheme in time. The term of distributed delay is approximated by using the compound trapezoidal formula. For the constructed numerical scheme, we mainly focus on the unconditional convergence and superconvergence without any time–space ratio restrictions, the key of which is the use of fractional discrete Grönwall inequality and time–space error splitting technique. Numerical tests for several biological models, including the fractional single-species population model with distributed delay, the fractional diffusive Nicholson’s blowflies equation with distributed delay, and the fractional diffusive Mackey-Glass equation with distributed delay, are conducted to confirm the theoretical results. Finally, combined with the nonunifom Alikhanov scheme in time and the FEM in space, we extend a higher-order Newton linearized numerical scheme for the nonlinear time-fractional parabolic equations with distributed delay and give some numerical tests for some biological models.

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Funding

The work is supported by the China Postdoctoral Science Foundation (2023T160589), National Natural Science Foundation of China (Nos. 11801527, 11971416), Natural Science Foundation of Henan Province (222300420256), Training Plan of Young Backbone Teachers in Colleges of Henan Province (No. 2020GGJS230), Henan University Science and Technology Innovation Talent support program (19HASTIT025).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Shanshan Peng, Meng Li & Fangfang Cao

  2. School of Science, Xuchang University, Xuchang, 461000, China

    Shanshan Peng, Yanmin Zhao & Fangfang Cao

  3. Henan Joint International Research Laboratory of High Performance Computation for Complex Systems, Xuchang, 461000, China

    Yanmin Zhao

  4. Queensland University of Technology, Brisbane, QLD 4001, Australia

    Fawang Liu

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  1. Shanshan Peng
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  2. Meng Li
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  3. Yanmin Zhao
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  4. Fawang Liu
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  5. Fangfang Cao
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Shanshan Peng, Meng Li, Yanmin Zhao, Fawang Liu, and Fangfang Cao. The first draft of the manuscript was written by Shanshan Peng, Meng Li, and Yanmin Zhao, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Meng Li or Yanmin Zhao.

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Peng, S., Li, M., Zhao, Y. et al. Unconditionally convergent and superconvergent finite element method for nonlinear time-fractional parabolic equations with distributed delay. Numer Algor 95, 1643–1714 (2024). https://doi.org/10.1007/s11075-023-01624-8

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