The nonlinear wave dynamics of fractional foam drainage and Boussinesq equations with Atangana’s beta derivative through analytical solutions

The space–time fractional foam drainage and the Boussinesq equations are valuable mathematical models to describe different complex physical phenomena, including the drainage of soap films in bubbles and foams, the behavior of surface water waves in oceans and rivers, the propagation of long waves, etc. In this research, we explore several unrivaled and some existing analytical soliton solutions to the above-stated equations using an effective expansion scheme, known as the G′/(G′+G+A)-expansion scheme within the framework of Atangana’s beta derivative. A fractional composite transformation is used to interpret the considered equations into nonlinear equations of a single independent variable. The wave dynamics of the outcomes are illustrated through three- and two-dimensional graphical representations and contour plots. The solutions include kink, flat kink-shaped, the parabolic-shaped, the bell-shaped, and the compacton solitons. The study reveals that the fractional-order derivative in the projected equations influences the wave behavior, providing deeper insights into wave composition when the fractional derivative approaches larger. The comparison table demonstrates the effectiveness of the method across various fractional orders, with results approaching accuracy as the fractional-order approaches integer values. This study emphasizes the computational robustness and adaptability of the proposed method for investigating various phenomena across a wide range of physical science and engineering disciplines