Using a finite difference scheme together with neural networks analysis for propagation of the long waves equation

In this paper, we explored the Benjamin-Bona-Mahony-Burger (BBMB) equation in the context of the conformable fractional derivative. The major results are the existence, the uniqueness, and inaddition, the Ulam-Hyers (UH) stability of the solutions that are obtained, which guarantee the mathematical soundness of the proposed model. For the numerical study, the standard finite difference method (SFDM) and the non-standard finite difference method (NSFDM) were developed, and their performance was assessed through a comparison with the exact solution. The conclusions showed that NSFDM is more accurate and stable compared to SFDM. Additionally, a neural network (NN) scheme was used as a further validation tool, which was complemented by regression analysis and error distribution measures. The change of fractional order significantly affects the solution profiles, as shown by two or three-dimensional plots in numerical simulations. The fractional dynamics, therefore, play a crucial role in modifying wave propagation in more dimensions. The unique feature of this research is the joint use of conformable fractional calculus with NSFDM and neural computing for the BBMB equation, providing a new way for the treatment of nonlinear dispersive wave models.