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Time-fractional nonlinear evolution of dynamic wave propagation using the Burgers’ equation

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Abstract

Fractional derivatives are crucial in diverse contexts, offering a means to extend classical derivatives to noninteger orders. This expansion of calculus enables a more detailed understanding of complex behaviors in scientific, engineering, and mathematical disciplines. In this study, we use theoretical and numerical analyses to thoroughly examine the time-fractional Burgers’ equation. Our main emphasis is on deriving time-decay estimates for solutions within a bounded domain. To determine optimal time-decay rates, we introduce a linear compact difference scheme by integrating an \(L_1\) discretization formula for the Caputo derivative and compact difference operators with spatial derivatives. We provide a detailed analysis encompassing existence and uniqueness, and an error estimate of solutions under the \(\Vert \cdot \Vert _{\infty }\) norm for the proposed scheme. Comprehensive numerical experiments highlight the efficiency and robustness of our approach, ensuring reliability for long-time simulations. Furthermore, numerical results are scrutinized to pinpoint the optimal time-decay rate. This work explores the asymptotic behavior of solutions to the time-fractional Burgers’ equation and the development of linear high-order accuracy difference methods for nonlinear time-fractional equations.

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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This research was supported by Chiang Mai University, Thailand.

Funding

This research was supported by Chiang Mai University and Fundamental Fund 2024, Chiang Mai University.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Chiang Mai, 50200, Thailand

    Sivaporn Phumichot, Kanyuta Poochinapan & Ben Wongsaijai

  2. Advanced Research Center for Computational Simulation, Chiang Mai University, Chiang Mai, 50200, Thailand

    Kanyuta Poochinapan & Ben Wongsaijai

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  1. Sivaporn Phumichot
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  2. Kanyuta Poochinapan
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  3. Ben Wongsaijai
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Contributions

Sivaporn Phumichot: Conceptualization, Formal analysis, Investigation, Methodology, Software, Validation, Writing-original draft, and Writing-review and editing; Kanyuta Poochinapan: Formal analysis, Funding acquisition, Investigation, Methodology, Supervision, Validation, Visualization, and Writing-review and editing, Ben Wongsaijai: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Validation, Visualization, Writing-original draft, and Writing-review and editing.

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Correspondence to Ben Wongsaijai.

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Phumichot, S., Poochinapan, K. & Wongsaijai, B. Time-fractional nonlinear evolution of dynamic wave propagation using the Burgers’ equation. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02100-9

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  • DOI: https://doi.org/10.1007/s12190-024-02100-9

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