# Numerical solutions of the generalized equal width wave equation using Petrov Galerkin method

## TL;DR

This paper develops a Petrov-Galerkin numerical scheme with quadratic shape functions and linear B-spline weights to solve the generalized equal width wave equation, analyzing its stability, error bounds, and effectiveness in modeling soliton propagation.

## Contribution

The paper introduces a novel Petrov-Galerkin method for the GEW equation, providing theoretical error analysis and demonstrating superior performance over existing schemes.

## Key findings

- The scheme is unconditionally stable.
- It accurately models solitary wave propagation.
- It outperforms other numerical methods in efficiency and accuracy.

## Abstract

In this article we consider a generalized equal width wave (GEW) equation which is a significant nonlinear wave equation as it can be used to model many problems occurring in applied sciences. As the analytic solution of the (GEW) equation of this kind can be obtained hardly, developing numerical solutions for this type of equations is of enormous importance and interest. Here we are interested in a Petrov-Galerkin method, in which element shape functions are quadratic and weight functions are linear B-splines. We firstly investigate the existence and uniqueness of solutions of the weak form of the equation. Then we establish the theoretical bound of the error in the semi-discrete spatial scheme as well as of a full discrete scheme at $t=t^{n}$. Furthermore, a powerful Fourier analysis has been applied to show that the proposed scheme is unconditionally stable. Finally, propagation of single and double solitary waves and evolution of solitons are analyzed to demonstrate the efficiency and applicability of the proposed numerical scheme by calculating the error norms (in $L_{2}(\Omega)$ and $L_{\infty}(\Omega)$). The three invariants ($% I_{1},I_{2}$ and $I_{3})$ of motion have been commented to verify the conservation features of the proposed algorithms. Our proposed numerical scheme has been compared with other published schemes and demonstrated to be valid, effective and it outperforms the others.