https://doi.org/10.1140/epjp/s13360-025-06159-6
Regular Article
Explicit wave solutions profile of (3+1)-dimensional Bateman–Burgers equation via bilinear neural network method
1
Department of Mathematics, Shanghai University, No. 99 Shangda Road, 200444, Shanghai, People’s Republic of China
2
Newtouch Center for Mathematics of Shanghai University, 200444, Shanghai, People’s Republic of China
3
Department of Mathematics and Statistics, University of Lahore, Sargodha Campus, Pakistan
Received:
23
January
2025
Accepted:
19
February
2025
Published online:
15
March
2025
This article explores the (3+1)-dimensional Bateman–Burgers equation using the bilinear neural network technique. Single and double layers of neural networks are built to construct different bilinear neural network models such as “4-3-1” and “4-2-2-1” by using specific activation functions. The interaction solution and periodic type-l solutions were extracted for this equation. The (3+1)-dimensional Bateman–Burgers equation has many applications in traffic flow, fluid mechanics, gas dynamics, and nonlinear acoustics. For enhancing the graphical representation of the dynamic behavior and the physical attributes of particular solutions, the computing tool Mathematica 13.1 was used for generating 3D visualizations, 2D graphical representations, and density mappings. The methodology used in this article improves the study of nonlinear partial differential equations that arise in different complex phenomena. In the end, we believe that these solutions will play a role in the understanding of some high-order equation nonlinear phenomena.
Copyright comment Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
