STATIONARY DIFFUSION MODEL IN HETEROGENEOUS MEDIA WITH SMALL PARAMETERS - Akhmetova, О.; Issayev, S.A. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2025)
This study presents a theoretical and applied investigation of a stationary diffusion problem in heterogeneous media incorporating small parameters and modified boundary conditions. The relevance of this work stems from the necessity to accurately represent boundary layer effects in complex physical systems with variable internal structures, such as multilayered materials, biological tissues, and natural geophysical environments. Traditional models often oversimplify such effects, resulting in distortions in the solution near domain boundaries. The primary objective of the research is to construct a rigorous mathematical framework that accounts for both the presence of small parameters and altered boundary conditions. The analysis includes deriving a priori stability estimates and constructing asymptotic expansions for the solution. The methodology relies on functional formulation in Sobolev spaces, the Galerkin method, asymptotic techniques, and weak convergence theory. Special attention is given to the existence and uniqueness of both strong and generalized solutions, as well as to their convergence properties as the small parameters tend to zero. The results confirm the solvability of the problem, provide robust energy estimates, and establish solution stability under various types of boundary modifications. It is demonstrated that changes in boundary conditions primarily affect solution behavior near the domain edges, while preserving the global structure of the field. The model is applied in various fields, including climate modeling, thermal diffusion, and hydrophysical systems. This work lays a foundation for the development of adaptive numerical schemes and for extending the analysis to non-stationary problems.