An approximation of one-dimensional nonlinear Kortweg de Vries equation of order nine.

Sidra Saleem,Malik Zawwar Hussain,Imran Aziz,Shou-Fu Tian

doi:10.1371/journal.pone.0262157

Sidra Saleem, Malik Zawwar Hussain + Show 2 more

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https://doi.org/10.1371/journal.pone.0262157

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Abstract

This research presents the approximate solution of nonlinear Korteweg-de Vries equation of order nine by a hybrid staggered one-dimensional Haar wavelet collocation method. In literature, the underlying equation is derived by generalizing the bilinear form of the standard nonlinear KdV equation. The highest order derivative is approximated by Haar series, whereas the lower order derivatives are attained by integration formula introduced by Chen and Hsiao in 1997. The findings are shown in the form of tables and a figure, demonstrating the proposed technique’s convergence, robustness, and ease of application in a small number of collocation points.

Highlights

Many problems that have arisen in different eras of science and engineering have been described using linear and nonlinear phenomena
Numerous semi-analytic techniques, such as Adomian decomposition method (ADM), homotopy analysis method (HAM), and homotopy perturbation method, have been utilized to produce series solutions, convergence of the series has been a challenge in these solutions which was solved by many semi-analytic techniques like Adomian decomposition method (ADM), homotopy analysis method (HAM), homotopy perturbation method (HPM) and modified variational iteration method (MVIM) [2, 3]
We identify Θ = 2J and Γ = 2Θ, where J is defined before

Summary

Introduction

Many problems that have arisen in different eras of science and engineering have been described using linear and nonlinear phenomena. While some of these issues can be solved immediately, a considerable number of them remain at the cutting edge of mathematical modelling. The authors were driven to find analytic, semi analytic, or numerical solutions to these models after studying the nature of their solutions. When finding analytic solutions to these PDEs proved difficult to come by, the authors became interested in semi-analytic and numerical solutions. Numerous semi-analytic techniques, such as Adomian decomposition method (ADM), homotopy analysis method (HAM), and homotopy perturbation method, have been utilized to produce series solutions, convergence of the series has been a challenge in these solutions which was solved by many semi-analytic techniques like Adomian decomposition method (ADM), homotopy analysis method (HAM), homotopy perturbation method (HPM) and modified variational iteration method (MVIM) [2, 3]

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