@article{Razzaq2026, 
author = {Waseem Razzaq and Asim Zafar and Ahmed Al Nuaim and Abdullah Al Nuaim},
title = {A WAS neural network framework for computing and analyzing solutions of a generalized    (  3  +  1  )-dimensional nonlinear Wave equation: Stability analysis},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {4},
pages = {10694-10715},
keywords = {the (3 + 1)-dimensional wave equation, WAS neural network method, analytical method, neural network, soliton solutions, stability analysis},
url = {https://www.sciopen.com/article/10.3934/math.2026440},
doi = {10.3934/math.2026440},
abstract = {This paper discusses the generalized non-linear    (  3  +  1  )-dimensional wave equation by modeling and analyzing the dynamics of multi-dimensional nonlinear waves with the WAS-neural network technique. The suggested framework accurately models various wave forms such as bright, singular, and bright-dark solitons. Insofar as we know such neural network based solutions of this model are not reported before. To ensure the reliability and proficiency of the WAS neural network technique. The gain solutions are stable or not by executing the stability analysis on them. The graphical visualization in three-dimensional surface and two-dimensional plots are used. The findings validate that the WAS neural network technique is an efficient and strong alternative to classical techniques of higher-dimensional nonlinear wave equations, and has applications in fluid mechanics and engineering systems that have to deal with gas liquid interactions.}
}