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A fast second-order predictor-corrector method for a nonlinear time-fractional Benjamin-Bona-Mahony-Burgers equation

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Abstract

A second-order predictor-corrector method of Nguyen and Jang (Fract. Calc. Appl. Anal. 20(2), 447–476, 2017) is generalised to graded meshes to solve nonlinear fractional initial-value problems whose typical solutions have a weak singularity at the initial time. In comparison with existing predictor-corrector methods in the literature, this new method significantly improves the numerical accuracy while reducing the computational cost. Moreover, to increase its computational efficiency still further, a corresponding fast algorithm based on the sum-of-exponentials approximation to the kernel of the scheme is described. An error analysis is given for problems whose right-hand sides satisfy a Lipschitz condition. The method (and its fast variant) is then extended to solve the nonlinear time-fractional Benjamin-Bona-Mahony-Burgers (BBMB) initial-boundary value problem, combined with a standard discretisation of the spatial derivatives on a uniform mesh. Estimates are derived for the discrete \(H^1\)-norm errors in the computed solution for the BBMB problem; to enable this analysis, a new Gronwall inequality is proved. Finally, several numerical experiments show the sharpness of our theoretical error bounds for both problems.

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Acknowledgements

We thank Professor Fanhai Zeng for pointing out an error in the original version of this paper and providing some references that helped us correct this error.

Funding

(The work of Yongtao Zhou is supported in part by the China Postdoctoral Science Foundation under grant 2021M690322 and the National Natural Science Foundation of China for Young Scientists under grant 12101037. The work of Cui Li is supported in part by the National Natural Science Foundation of China for Young Scientists under grant 12001163, the Key Scientific Research Projects for Colleges and Universities of Henan under grant 20A110012 and the Science and Technology Program of Henan under grant 212102310552. The work of Martin Stynes is supported in part by the National Natural Science Foundation of China under grants 12171025 and NSAF-U2230402.)

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Authors and Affiliations

  1. School of Science, Qingdao University of Technology, 266520, Qingdao, China

    Yongtao Zhou

  2. School of Mathematics and Information Science, Henan University of Economics and Law, 450046, Zhengzhou, China

    Cui Li

  3. Applied and Computational Mathematics Division, Beijing Computational Science Research Center, 100193, Beijing, China

    Martin Stynes

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  1. Yongtao Zhou
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  2. Cui Li
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  3. Martin Stynes
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Contributions

Yongtao Zhou and Martin Stynes wrote the main manuscript text. Yongtao Zhou and Cui Li prepared the numerical results. All authors reviewed the manuscript.

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Correspondence to Martin Stynes.

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Zhou, Y., Li, C. & Stynes, M. A fast second-order predictor-corrector method for a nonlinear time-fractional Benjamin-Bona-Mahony-Burgers equation. Numer Algor 95, 693–720 (2024). https://doi.org/10.1007/s11075-023-01586-x

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  • DOI: https://doi.org/10.1007/s11075-023-01586-x

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