Mathematical study of BLUES function method for KdV Burgers’ and BBM-Burgers’ equations

Abstract

The Korteweg–De Vries (KdV) Burgers’ and Benjamin–Bona–Mohoney (BBM) Burgers’ equations are crucial in understanding wave dynamics, heat transfer, and plasma waves. It is essential to solve these models over a long time domain to study how energy will transmit and dissipate, or whether waves will remain coherent or disperse due to dissipation effects. Researchers study various semi-analytical and numerical methods to solve these models. However, numerical methods come with the drawback of discretizing the domain, which leads to some errors in the solutions. In a recent paper (Berx and Indekeu, 2021), the authors introduced a new semi-analytical technique, namely the beyond linear use of the superposition (BLUES) function method for partial differential equations, and showed that the proposed method provides better accuracy compared to existing methods. Therefore, the purpose of this article is to describe the BLUES function method for the KdV and BBM Burgers’ equations. The absence of assumptions, convergence control parameters, linearization, and discretization demonstrates the method’s superiority over conventional numerical and semi-analytical techniques. The article mainly focuses on the stability and convergence analysis of the method. Additionally, the numerical validation of the results includes two instances of KdV-Burgers equations and two instances of BBM-Burgers equations. The efficacy and precision of the suggested methodology are illustrated through the utilization of graphical representations and tabular data.