Mathematical Modeling of Variations Radon Volumetric Activity Considering Heredity

The issue of seismicity in the Kamchatka region underscores the importance of fundamental research that has practical applications and contributes to a deeper understanding of the processes occurring in the Earth's crust during the preparation of a future earthquake focus. One such relevant direction is the use of experimental data from subsurface gas monitoring in comparison with seismicity information, combined with the results of studies on subsurface gas migration processes using mathematical modeling methods. The object of the study is the volumetric activity of radon (RVA) in an accumulation chamber. The aim of the study is to develop mathematical models of RVA considering heredity and methods for identifying certain parameters of the geological environment based on experimental data, utilizing methods and efficient algorithms for solving direct and inverse problems. The scientific novelty consists of the following: 1) mathematical hereditary models RVA are proposed based on an equation with variable coefficients and a Gerasimov-Caputo type fractional derivative of both constant and variable order; 2) efficient parallel hybrid algorithms have been developed that implement non-local explicit and implicit finite-difference schemes for solving direct problems for the proposed models; 3) for the developed parallel algorithms, estimates of the efficiency and complexity of the algorithms are given; 4) a new numerical algorithm based on the Levenberg-Marquardt iterative method has been developed to solve inverse problems of identifying the order of the fractional derivative and the coefficient of air exchange included in the equation of the hereditary models of the dynamics of RVA; 5) The FEVO v1.0 software package has been developed in C for GNU/Linux operating systems, which allows mathematical modeling of variations in the dynamics of the RVA, solving corresponding direct and inverse problems based on experimental data, and processing and visualizing data. The scientific significance of the research results lies in the development of new hereditary models of the dynamics of RVA in the accumulation chamber, which take into account the temporal non-locality of the radon transport process using fractional calculus. Inverse problems based on these models contribute to the further advancement of mathematical modeling in geophysics, particularly in solving problems related to the identification of certain parameters of the geological medium through which radon gas enters the accumulation chamber. The practical significance lies in the fact that the developed algorithms and software tools for their implementation enable solving direct and inverse problems of mathematical modeling of RVA dynamics in the accumulation chamber in order to determine the type of RVA regime and identify anomaly signs in experimental data, which may be short-term precursors of earthquakes, as well as for estimating radon flux density, which can assist in more accurately selecting locations for radon monitoring stations.