Radiative transfer (RT) in spectral lines in plasmas and gases under complete redistribution of the photon frequency in the emission-absorption act is known as a superdiffusion transport characterized by the irreducibility of the integral (in the space coordinates) equation for the atomic excitation density to a diffusion-type differential equation. The dominant role of distant rare flights (Lévy flights, introduced by Mandelbrot for trajectories generated by the Lévy stable distribution) is well known and is used to construct approximate analytic solutions in the theory of stationary RT (the escape probability method is the best example). In the theory of nonstationary RT, progress based on similar principles has been made recently. This includes approximate self-similar solutions for the Green’s function (i) at an infinite velocity of carriers (no retardation effects) to cover the Biberman–Holstein equation for various spectral line shapes; (ii) for a finite fixed velocity of carriers to cover a wide class of superdiffusion equations dominated by Lévy walks with rests; (iii) verification of the accuracy of above solutions by comparison with direct numerical solutions obtained using distributed computing. The article provides an overview of the above results with an emphasis on the role of distant rare flights in the discovery of nonstationary self-similar solutions.

A mathematical model is a reflection of knowledge on the real object studied. The paper shows how the accumulation of data (statistical data and knowledge) about the COVID-19 pandemic lead to gradual refinement of mathematical models, to the expansion of the scope of their use. The resulting model satisfactorily describes the dynamics of COVID-19 in Moscow from 19.03.2020 to 01.09.2021 and can be used for forecasting with a horizon of several months. The dynamics of the model is mainly determined by herd immunity. Monitoring the situation in Moscow has not yet (as of 01.09.2021) revealed noticeable seasonality of the disease nor an increase in infectivity (due to the Delta strain). The results of using balanced identification technology to monitor the COVID-19 pandemic are: models corresponding to the data available at different points in time (from March 2020 to August 2021); new knowledge (dependencies) acquired; forecasts for the third and fourth waves in Moscow. Discrepancies that manifested after 01.09.2021 and possible further modifications of the model are discussed

We obtain an analytic description of the excitation front propagation in a medium during nonstationary superdiffusive (nonlocal) transport in the case of a finite fixed velocity of perturbation carriers (so-called Lévy walks with stops). This problem embraces phenomena such as resonance radiation transport in astrophysical gases and plasma, biological migration, and energy transfer by waves in a plasma. In this approach, the result obtained by integrating the exact solution to the kinetic equation for the Green function is independent of the coordinate space dimensionality. The results are compared with data obtained using another more exact method of determining the front and with the results of numerical calculations of the statistics of trajectories using the Monte Carlo method. Comparison demonstrates the applicability of our results in a wide range of parameters of the problem. We propose a universal description of the perturbation front dynamics in the medium for an arbitrary (including infinitely high) fixed velocity of perturbation carriers. This corresponds to the combination of expressions for the front in the case of transport by Lévy flights and by Lévy walks. We consider the criteria of transition between these regimes of the superdiffusive transport, which corresponds, in particular, to the account for the finite velocity of light in the superdiffusive transport of resonance radiation in gases and plasmas. For Lévy walks, we have obtained a relation between the integral characteristic of perturbation of the medium and its carriers.

This work is aimed at numerical studies of inverse problems of experiment processing (identification of unknown parameters of mathematical models from experimental data) based on the balanced identification technology. Such problems are inverse in their nature and often turn out to be ill-posed. To solve them, various regularization methods are used, which differ in regularizing additions and methods for choosing the values of the regularization parameters. Balanced identification technology uses the cross-validation root-mean-square error to select the values of the regularization parameters. Its minimization leads to an optimally balanced solution, and the obtained value is used as a quantitative criterion for the correspondence of the model and the regularization method to the data. The approach is illustrated by the problem of identifying the heat-conduction coefficient on temperature. A mixed one-dimensional nonlinear heat conduction problem was chosen as a model. The one-dimensional problem was chosen based on the convenience of the graphical presentation of the results. The experimental data are synthetic data obtained on the basis of a known exact solution with added random errors. In total, nine problems (some original) were considered, differing in data sets and criteria for choosing solutions. This is the first time such a comprehensive study with error analysis has been carried out. Various estimates of the modeling errors are given and show a good agreement with the characteristics of the synthetic data errors. The effectiveness of the technology is confirmed by comparing numerical solutions with exact ones.