This paper presents a probabilistic–statistical approach to the analysis of diffusion processes in randomly nonhomogeneous multilayered bodies under conditions of incomplete experimental information on the boundary. The boundary condition is reconstructed from experimental data using linear regression, while the solution of the corresponding contact initial-boundary value problem is obtained in the form of a Neumann series and averaged over an ensemble of phase configurations. A system of statistical estimates for the solution is developed, including confidence intervals and two-sided critical regions, which provide complementary characteristics of uncertainty. Numerical experiments are performed for six representative samples differing in sample size, variance, and observation interval. It is shown that, despite significant differences in the statistical properties of the input data, the averaged concentration field preserves a qualitatively stable spatio-temporal structure. The results of the article address gaps in existing research by applying a probabilistic-statistical approach that consistently integrates two key elements for the analysis of diffusion processes in multilayer media. The first of these is the reconstruction of boundary conditions using linear regression to recover the conditions at the body boundary based on incomplete experimental data. The second key point is the analysis of uncertainty propagation by combining the regression model with a probabilistic analysis of the corresponding contact initial-boundary value problem, which allows us to quantitatively assess how the errors in the experimental data affect the final solution. From the point of view of mathematical modeling methods, the novelty of the approach lies in the creation of a structural-hierarchical scheme that synthesizes the approaches of mathematical statistics and the theory of random fields. The developed method is a theoretical and computational innovative basis for the analysis of specific physical and technological processes.
Chernukha et al. (Thu,) studied this question.