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Abstract
Objectives: In this work, the approximate solution of non-linear third order Korteweg-de Vries equation has been studied. Methods: The proposed numerical technique engages finite difference formulation for temporal discretization, whereas, the discretization in space direction is achieved by means of a new cubic B-spline approximation. Findings: In order to corroborate this effort, three test problems have been considered and the computational outcomes are compared with the current methods. It is found that the proposed scheme involves straight forward computations and operates superior to the existing methods. Novelty/Improvements: The proposed numerical scheme is novel for Korteweg-de Vries equation and has never been employed for this purpose before. Keywords: Cubic B-spline Collocation Method, Cubic B-spline Functions, Finite Difference Formulation, Korteweg-de Vries Equation
Highlights
The third order non-linear Korteweg-de Vries (KdV) equation occurs in many physical applications such as non-linear plasma waves which exhibit certain dissipative effects[1], propagation of waves[2] and propagation of bores in shallow water waves[3]
The accuracy and validity of the proposed numerical method is tested by three error norms L∞, L2 and Root Mean Square (RMS), which are calculated as
The approximate results are compared with Multi-Quadratic Radial Basis Functions (MQRBF)[8], Multi-Quadric (MQ) and Inverse Multi-Quadric (IMQ) radial basis functions method[10], Multi-Quadric Quasi-Interpolation (MQQI) approach[11] and integrated multi-quadric quasi-interpolation (IMQQI) method[12]
Summary
The third order non-linear Korteweg-de Vries (KdV) equation occurs in many physical applications such as non-linear plasma waves which exhibit certain dissipative effects[1], propagation of waves[2] and propagation of bores in shallow water waves[3]. Kutluay et al.[4] employed integral methods with heat balance to study the small time solutions to KdV equation. Dehghan and Shokri[8] proposed a numerical method based on multi-quadratic radial basis functions for solving KdV equation. Dag and Dereli[9] explored the numerical solution of KdV equation by means of radial basis functions. A mesh free method based on radial basis functions was presented by Khattak and Tirmizi[10] for approximate solution of KdV equation. Xiao et al.[11] investigated the numerical solution to KdV equation using multi-quadric quasi-interpolation operator. Sarboland and Aminataei[12] proposed a numerical scheme based on integrated radial basis functions and multi-quadric quasi-interpolation operator for solving of KdV equation. Rashid et al.[13] solved Hirota-Satsuma coupled KdV equation by Fourier Pseudo-spectral method
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