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

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. Here we study a Petrov–Galerkin method for the model problem, in which element shape functions are quadratic and weight functions are linear B-splines. We 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 = tⁿ. Furthermore, a powerful Fourier analysis has been applied to show that the proposed scheme is unconditionally stable. Finally, propagation of solitary waves and evolution of solitons are analyzed to demonstrate the efficiency and applicability of the proposed scheme. The three invariants (I₁, I₂ and I₃) 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.

Keywords:

• GEW equation
• Petrov–Galerkin
• B-splines
• solitary waves
• soliton

AMS classifications:

• 65N30
• 65D07
• 74S05
• 74J35
• 76B25

Disclosure statement

No potential conflict of interest was reported by the authors.