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A fourth-order compact difference method for the nonlinear time-fractional fourth-order reaction–diffusion equation

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Abstract

In this paper, a high-order compact scheme is proposed for solving two-dimensional nonlinear time-fractional fourth-order reaction-diffusion equations. The fractional derivative is the Caputo fractional derivative. A scheme with the second-order accuracy is applied to deal with the time derivative, while the spatial derivatives are discretized by the fourth-order compact numerical differential formulas. The unique solvability of the numerical method is proved in detail. Then by using the energy method, it is proved that the proposed algorithm is convergent with order \(O({\tau ^2} + h_1^4 + h_2^4)\), where \(\tau\) is the temporal step size and \(h_1\), \(h_2\) are the spatial step sizes. Finally, some numerical examples are given to verify the theoretical analysis and efficiency of the developed scheme.

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Acknowledgements

The authors thank the reviewers for their useful comments and suggestions that improved the paper.

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Authors and Affiliations

  1. School of Mathematics, Shahid Rajaee Teacher Training University, Lavizan, 16788, Tehran, Iran

    Majid Haghi

  2. Department of Applied Mathematics, Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran

    Mohammad Ilati

  3. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran

    Mehdi Dehghan

Authors
  1. Majid Haghi
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  2. Mohammad Ilati
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  3. Mehdi Dehghan
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Correspondence to Mohammad Ilati.

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Haghi, M., Ilati, M. & Dehghan, M. A fourth-order compact difference method for the nonlinear time-fractional fourth-order reaction–diffusion equation. Engineering with Computers 39, 1329–1340 (2023). https://doi.org/10.1007/s00366-021-01524-2

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  • DOI: https://doi.org/10.1007/s00366-021-01524-2

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