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On the Numerical Solution of 1D and 2D KdV Equations Using Variational Homotopy Perturbation and Finite Difference Methods

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  • Abey Sherif Kelil

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

  • Appanah Rao Appadu

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

Abstract

The KdV equation has special significance as it describes various physical phenomena. In this paper, we use two methods, namely, a variational homotopy perturbation method and a classical finite-difference method, to solve 1D and 2D KdV equations with homogeneous and non-homogeneous source terms by considering five numerical experiments with initial and boundary conditions. The variational homotopy perturbation method is a semi-analytic technique for handling linear as well as non-linear problems. We derive classical finite difference methods to solve the five numerical experiments. We compare the performance of the two classes of methods for these numerical experiments by computing absolute and relative errors at some spatial nodes for short, medium and long time propagation. The logarithm of maximum error vs. time from the numerical methods is also obtained for the experiments undertaken. The stability and consistency of the finite difference scheme is obtained. To the best of our knowledge, a comparison between the variational homotopy perturbation method and the classical finite difference method to solve these five numerical experiments has not been undertaken before. The ideal extension of this work would be an application of the employed methods for fractional and stochastic KdV type equations and their variants.

Suggested Citation

  • Abey Sherif Kelil & Appanah Rao Appadu, 2022. "On the Numerical Solution of 1D and 2D KdV Equations Using Variational Homotopy Perturbation and Finite Difference Methods," Mathematics, MDPI, vol. 10(23), pages 1-36, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4443-:d:983450
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    1. Mesfin Mekuria Woldaregay & Gemechis File Duressa, 2021. "Uniformly Convergent Hybrid Numerical Method for Singularly Perturbed Delay Convection-Diffusion Problems," International Journal of Differential Equations, Hindawi, vol. 2021, pages 1-20, September.
    2. He, Ji-Huan & Wu, Xu-Hong, 2006. "Construction of solitary solution and compacton-like solution by variational iteration method," Chaos, Solitons & Fractals, Elsevier, vol. 29(1), pages 108-113.
    3. Samuel, F.M. & Motsa, S.S., 2019. "A highly accurate trivariate spectral collocation method of solution for two-dimensional nonlinear initial-boundary value problems," Applied Mathematics and Computation, Elsevier, vol. 360(C), pages 221-235.
    4. Fasika Wondimu Gelu & Gemechis File Duressa, 2021. "A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem," Abstract and Applied Analysis, Hindawi, vol. 2021, pages 1-11, March.
    5. Syed Tauseef Mohyud-Din & Ahmet Yildirim & Sefa Anıl Sezer & Muhammad Usman, 2010. "Modified Variational Iteration Method for Free-Convective Boundary-Layer Equation Using Padé Approximation," Mathematical Problems in Engineering, Hindawi, vol. 2010, pages 1-11, April.
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