A Computational Approach for Generalized Equal Width Wave Equation Using Cubic Trigonometric B-Spline Differential Quadrature Method

Arshed Ali; Aiman Hayat; Imtiaz Ahmad; Hadia Atta

Authors

Arshed Ali Department of Mathematics, Islamia College Peshawar
Aiman Hayat Department of Mathematics, Islamia College Peshawar, Pakistan
Imtiaz Ahmad Institute of Informatics and Computing in Energy (IICE), Universiti Tenaga Nasional (UNITEN), Kajang, Selangor, Malaysia
Hadia Atta Department of Mathematics, Islamia College Peshawar, Pakistan

Keywords:

Generalized Equal Width Wave Equation, Cubic Trigonometric B Splines, Differential quadrature method, Solitary Wave Interaction, Nonlinear Shallow Water Waves

Abstract

In this paper, a computational method of solving the generalized equal width (GEW) wave equation is proposed, which describes high-order nonlinear shallow water waves. The proposed approach uses cubic trigonometric B-spline (CTBS) basis functions for spatial discretization, together with the classical fourth-order Runge-Kutta scheme for time integration, and the differential quadrature method (DQM) for spatial approximation. This approach leads to the derivation of a set of ordinary differential equations (ODEs) for the CTBS-DQM system, followed by efficient solution of the system with the ODE solver. Numerical experiments are performed for the propagation of a single solitary wave and interaction between two positive solitary waves with different nonlinearity orders (p = 2, 3, 4) and amplitudes, respectively. Error norms ( and ) and conservation invariants ( ) are used to evaluate the accuracy and reliability of the proposed scheme. The present method achieves exceptionally small errors, with and for at , and demonstrates a convergence rate of approximately . A comparative analysis of the present CTBS-DQM with existing cubic B-spline Galerkin and quintic B-spline DQM methods shows that the present approach gives superior results in terms of accuracy with much lesser computational cost, achieving a 97.6% reduction in CPU time (from 297.13 seconds to 7.23 seconds) while maintaining similar or better accuracy levels. The efficiency of the proposed approach is further validated through convergence analysis, and by comparing CPU time, thereby establishing the proposed method as a useful tool for the simulation of the high-order nonlinear wave phenomena.

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