The paper presents a mathematical model of radon accumulation in a chamber, which takes into account the hereditary properties of the environment in which radon migrates, and also uses a nonlinear function that is responsible for the mechanisms of radon entering the chamber. The simulation of accumulation is performed in comparison with real data. It is shown that the model presented in this work gives a better agreement between the model and real curves of radon accumulation and can be used for a more accurate description of the processes occurring in the chamber.

The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a nonlocal implicit finite difference scheme (IFDS) are proved. For IFDS, it is shown that the scheme converges with the order corresponding to the estimate for approximating the Gerasimov-Caputo fractional operator. The IFDS scheme is solved by the modified Newton’s method (MNM), for which it is shown that the method is locally stable and converges with the first order of accuracy. In the case of the fractional Riccati equation, approximation, stability, and convergence theorems are proved for a nonlocal explicit finite difference scheme (EFDS). It is shown that EFDS conditionally converges with the first order of accuracy. On specific test examples, the computational accuracy of numerical methods was estimated according to Runge’s rule and compared with the exact solution. It is shown that the order of computational accuracy of numerical methods tends to the theoretical order of accuracy with increasing nodes of the computational grid.

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In this study, the model Riccati equation with variable coefficients as functions, as well as a derivative of a fractional variable order (VO) of the Gerasimov-Caputo type, is used to approximate the data for some physical processes with saturation. In particular, the proposed model is applied to the description of solar activity (SA), namely the number of sunspots observed over the past 25 years. It is also used to describe data from Johns Hopkins University on coronavirus infection COVID-19, in particular data on the Russian Federation and the Republic of Uzbekistan. Finally, it is used to study issues related to seismic activity, in particular, the description of data on the volumetric activity of Radon (RVA). The Riccati equation used in the mathematical model was numerically solved by constructing an implicit finite difference scheme (IFDS) and its implementation by the modified Newton method (MNM). The calculated curves obtained in the study are compared with known experimental data. It is shown that if the model parameters are chosen appropriately, the model curves will give results that correlate well with real experimental data. Moreover, with other parameters of the model, it is possible to make some prediction about the possible course of the considered processes.