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Numerov type variable mesh approximations for 1D unsteady quasi-linear biharmonic problem: application to Kuramoto-Sivashinsky equation

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Abstract

We present two new two-level compact implicit variable mesh numerical methods of order two in time and two in space, and of order two in time and three in space for the solution of 1D unsteady quasi-linear biharmonic problem subject to suitable initial and boundary conditions. The simplicity of the proposed methods lies in their three-point discretization without requiring any fictitious points for incorporating the boundary conditions. The derived methods are shown to be unconditionally stable for a model linear problem for uniform mesh. We also discuss how our formulation is able to handle linear singular problem and ensure that the developed numerical methods retain their orders and accuracy everywhere in the solution region. The proposed difference methods successfully works for the highly nonlinear Kuramoto-Sivashinsky equation. Many physical problems are solved to demonstrate the accuracy and efficiency of the proposed methods. The numerical results reveal that the obtained solutions not only approximate the exact solutions very well but are also much better than those available in earlier research studies.

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Authors and Affiliations

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

    R. K. Mohanty

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

    Deepti Kaur

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  1. R. K. Mohanty
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  2. Deepti Kaur
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Correspondence to R. K. Mohanty.

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Mohanty, R.K., Kaur, D. Numerov type variable mesh approximations for 1D unsteady quasi-linear biharmonic problem: application to Kuramoto-Sivashinsky equation. Numer Algor 74, 427–459 (2017). https://doi.org/10.1007/s11075-016-0154-3

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