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Numerical simulations of Rosenau–Burgers equations via Crank–Nicolson spectral Pell matrix algorithm

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Abstract

The current work addresses the use of the numerical Crank–Nicolson technique along with spectral collocation to seek the approximate solution of a class of high-order nonlinear partial differential equations known as the Rosenau–Burgers equation. First, the Crank–Nicolson approach is applied to reduce the Rosenau–Burgers equation to a set of linear equations. The solvability and uniqueness of the semi-discretized form is established. Then, the spectral based collocation method by using the Pell functions is employed to approximate the derived system of linearized equations in terms of unknown coefficients. The theoretical part and error analysis are comprehensively studied in the weighted \(L^2\)-norm. Finally, four illustrative test cases are conducted to showcase the efficiency of the suggested combined technique. The results are well compared with existing results and exact solutions in the literature to demonstrate the advantages of the applied novel approach.

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Acknowledgements

The author would like to express his thank to the anonymous referees whose constructive comments improved the quality of this paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

    Mohammad Izadi

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 3R4, Canada

    Hari Mohan Srivastava

  3. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, 40402, Taiwan

    Hari Mohan Srivastava

  4. Center for Converging Humanities, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, 02447, Republic of Korea

    Hari Mohan Srivastava

  5. Department of Applied Mathematics, Chung Yuan Christian University, Chung-Li, Taoyuan City, 320314, Taiwan

    Hari Mohan Srivastava

  6. Department of Mathematics and Informatics, Azerbaijan University, 71 Jeyhun Hajibeyli Street, AZ1007, Baku, Azerbaijan

    Hari Mohan Srivastava

  7. Section of Mathematics, International Telematic University Uninettuno, Corso Vittorio Emanuele II, 39, 00186, Roma, Italy

    Hari Mohan Srivastava

  8. Mathematics Unit, School of Science and Engineering, University of Kurdistan-Hewlêr, Erbil, Kurdistan Region, Iraq

    Kamal Mamehrashi

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  1. Mohammad Izadi
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Correspondence to Mohammad Izadi.

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Izadi, M., Srivastava, H.M. & Mamehrashi, K. Numerical simulations of Rosenau–Burgers equations via Crank–Nicolson spectral Pell matrix algorithm. J. Appl. Math. Comput. 71, 1009–1033 (2025). https://doi.org/10.1007/s12190-024-02273-3

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