Estimation of the Solution of a Spatial Parabolic Equation Describing Anisotropic Geological Systems

Abstract

Partial differential equations have a wide practical application. For the mathematical description of various physical processes and phenomena, initial-boundary problems are constructed. The construction and analysis of their solutions is an urgent task of modern scientific research. The article considers the engineering production problem of compaction of subsiding soils by deep explosions. The compaction process can be implemented in two ways: by concentrated and elongated explosive charges, which determines the type of function of the gas source resulting from the explosion. An anisotropic geological system is described by a spatial parabolic equation with given initial and boundary conditions. The solution of the initial-boundary problem is the density of the soil as a result of its compaction by deep explosions and is determined at each point of the three-dimensional Euclidean space over time. The conditions for the existence of a solution to the stated initial-boundary problems are the continuity and differentiability of the functions included in the parabolic equation. An estimate for the solution of the initial-boundary problem within the framework of the anisotropic geological system under study is constructed and proved.