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On Semi-Analytical Solutions for Linearized Dispersive KdV Equations

Author

Listed:
  • Appanah Rao Appadu

    (Department of Mathematics and Applied Mathematics, Nelson Mandela University, Port Elizabeth 6031, South Africa)

  • Abey Sherif Kelil

    (Department of Mathematics and Applied Mathematics, Nelson Mandela University, Port Elizabeth 6031, South Africa)

Abstract

The most well-known equations both in the theory of nonlinearity and dispersion, KdV equations, have received tremendous attention over the years and have been used as model equations for the advancement of the theory of solitons. In this paper, some semi-analytic methods are applied to solve linearized dispersive KdV equations with homogeneous and inhomogeneous source terms. These methods are the Laplace-Adomian decomposition method (LADM), Homotopy perturbation method (HPM), Bernstein-Laplace-Adomian Method (BALDM), and Reduced Differential Transform Method (RDTM). Three numerical experiments are considered. As the main contribution, we proposed a new scheme, known as BALDM, which involves Bernstein polynomials, Laplace transform and Adomian decomposition method to solve inhomogeneous linearized dispersive KdV equations. Besides, some modifications of HPM are also considered to solve certain inhomogeneous KdV equations by first constructing a newly modified homotopy on the source term and secondly by modifying Laplace’s transform with HPM to build HPTM. Both modifications of HPM numerically confirm the efficiency and validity of the methods for some test problems of dispersive KdV-like equations. We also applied LADM and RDTM to both homogeneous as well as inhomogeneous KdV equations to compare the obtained results and extended to higher dimensions. As a result, RDTM is applied to a 3D-dispersive KdV equation. The proposed iterative schemes determined the approximate solution without any discretization, linearization, or restrictive assumptions. The performance of the four methods is gauged over short and long propagation times and we compute absolute and relative errors at a given time for some spatial nodes.

Suggested Citation

  • Appanah Rao Appadu & Abey Sherif Kelil, 2020. "On Semi-Analytical Solutions for Linearized Dispersive KdV Equations," Mathematics, MDPI, vol. 8(10), pages 1-34, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1769-:d:427519
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    References listed on IDEAS

    as
    1. Ali, Ahmad T. & Khater, Mostafa M.A. & Attia, Raghda A.M. & Abdel-Aty, Abdel-Haleem & Lu, Dianchen, 2020. "Abundant numerical and analytical solutions of the generalized formula of Hirota-Satsuma coupled KdV system," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    2. Ahmed Farooq Qasim & Ekhlass S. AL-Rawi, 2018. "Adomian Decomposition Method with Modified Bernstein Polynomials for Solving Ordinary and Partial Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2018, pages 1-9, October.
    3. Helal, M.A. & Mehanna, M.S., 2007. "A comparative study between two different methods for solving the general Korteweg–de Vries equation (GKdV)," Chaos, Solitons & Fractals, Elsevier, vol. 33(3), pages 725-739.
    4. Ahmed Farooq Qasim & Almutasim Abdulmuhsin Hamed, 2019. "Treating Transcendental Functions in Partial Differential Equations Using the Variational Iteration Method with Bernstein Polynomials," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2019, pages 1-8, March.
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