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A new high-resolution two-level implicit method based on non-polynomial spline in tension approximations for time-dependent quasi-linear biharmonic equations with engineering applications

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Abstract

In this present work, a new two-level implicit non-polynomial spline in tension method is proposed for the numerical solution of the 1D unsteady quasi-linear biharmonic problem. Using the continuity of the first-order derivative of the spline in tension function, a fourth-order accurate implicit finite-difference method is developed in this manuscript. By considering the linear model biharmonic problem, the given implicit spline method is unconditionally stable. Since the proposed method is based on half-step grid points, so it can be directly applied to 1D singular biharmonic problems without alteration in the scheme. Finally, the numerical experiments of the various biharmonic equation such as the generalized Kuramoto–Sivashinsky equation, extended Fisher–Kolmogorov equation, and 1D linear singular biharmonic problems are carried out to show the efficacy, accuracy, and reliability of the method. From the computational experiments, improved numerical results obtained as compared to the results obtained in earlier research work.

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Acknowledgements

This research work is supported by CSIR-SRF, Grant No: 09/045(1161)/2012-EMR-I. The authors thank the reviewers for their valuable suggestions, which substantially improved the standard of the paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi, 110021, India

    R. K. Mohanty

  2. Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, New Delhi, Delhi, 110007, India

    Sachin Sharma

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  1. R. K. Mohanty
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  2. Sachin Sharma
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Correspondence to R. K. Mohanty.

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Mohanty, R.K., Sharma, S. A new high-resolution two-level implicit method based on non-polynomial spline in tension approximations for time-dependent quasi-linear biharmonic equations with engineering applications. Engineering with Computers 37, 2073–2087 (2021). https://doi.org/10.1007/s00366-019-00928-5

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