A Novel Neural Network-Based Symbolic Approach for Shallow-Water Waves with Surface Tension

González-Gaxiola O., Hart-Simmons M., Ahmed H. M., Biswas A.

Fluids, vol.11, no.4, 2026 (ESCI, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 11 Issue: 4
  • Publication Date: 2026
  • Doi Number: 10.3390/fluids11040100
  • Journal Name: Fluids
  • Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus, INSPEC, Directory of Open Access Journals
  • Keywords: Boussinesq equation, Kudryashov R-function, neural networks, symbolic computation
  • Karadeniz Technical University Affiliated: No

Abstract

This paper examines the sixth-order generalized Boussinesq equation, which describes the dynamics of shallow-water waves, including the effects of surface tension. The study introduces Kudryashov’s R-function neural network approach for the first time, aiming to provide exact solutions to the nonlinear differential equation that represents the mathematical model of the sixth-order generalized Boussinesq equation. This technique incorporates the solutions of a nonlinear differential equation into neural networks, using them as an activation function within the hidden layer. In line with previous research on this topic, two categories of solutions are derived: solitary wave and shock wave solutions. Additionally, this paper includes 3D and 2D graphs to visually represent the solutions obtained.