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This analysis finds soliton-like solutions and singular solutions in Sobolev-type equations, indicating practical implications for fluid dynamics applications.
The current study deals with Sobolev‐type equations, which are utilized in numerous disciplines, including thermodynamics, fluid dynamics, ecology, and soil mechanics. In the present manuscript, we investigate Sobolev‐type models by means of the modified auxiliary equation (MAE) and the new modified extended direct algebraic (NMEDA) approaches. The results obtained in this report include families of analytical solutions, such as singular solutions, mixed singular solutions, trigonometric solutions, mixed dark‐bright solutions, shock solutions, and mixed periodic solutions. The stability of the model has been investigated herein. Additionally, a variety of physical problems were selected to provide a range of solutions. Graphs in 2D, 3D, and line formats corresponding to various parameter values are plotted.
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Shahzad et al. (2025) studied this question.
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