Abstract
In the field of mathematical modeling and numerical analysis, singular perturbation problems occupy an important place due to their wide range of applications in various scientific and engineering disciplines. These problems arise in systems with multiscale dynamics, where the complex interaction of fast and slow processes significantly affects the behavior of the solution. Sobolev-type equations are widely used in mathematical physics and fluid mechanics to describe phenomena of various nature, including the propagation of plasma and ion-acoustic waves, filtration processes in porous media, compaction of clay materials, second-order fluid models, thermodynamic processes, and the dynamics of long waves of small amplitude. This work focuses on the study of nonlinear singularly perturbed Sobolev initial-boundary value problems and the development of new difference schemes for their numerical solution. The paper explores the mathematical aspects of the singularly perturbed problem.