: We investigate the continuation problem for the elliptic equation. The continuation problem is formulated in operator form . The singular values of the operator A are presented and analyzed for the continuation problem for the Helmholtz equation. Results of numerical experiments are presented.

We consider an ill-posed initial boundary value problem for the Helmholtz equation. This problem is reduced to the inverse continuation problem for the Helmholtz equation. We prove the well-posedness of the direct problem and obtain a stability estimate of its solution. We solve numerically the inverse problem using the Tikhonov regularization, Godunov approach, and the Landweber iteration. Comparative analysis of these methods is presented.

In the world, air pollution ranks among the primary sources of risk to human health and the environment. To assess the risk of impact of atmospheric pollution, a comprehensive research cycle was designed to develop a unified ecosystem for monitoring air pollution in industrial cities in Kazakhstan. Research involves analyzing data for the winter period from 20 automated monitoring stations (AMS) located in Almaty and conducting chemical-analytical studies of snowmelt water samples from 22 points to identify such pollutants as fine particulate matters, petroleum products, and heavy metals. Research includes a bio-experiment involving the cultivation of watercress on samples of melt water collected from snow cover to examine the effects of pollution on plants. In the framework of this research, we determined API based on data obtained from AMS. In order to determine the influence of atmospheric pollution on the environment, a multiple regression model was developed using machine learning algorithms to reveal the relationship between the bio-experiment data and data on pollutants of chemical-analytical research. The results revealed a wide spread of pollutants in the snow cover of the urban environment, a correlation between pollutants in the snow cover and the airspace of the city, and their negative impact on flora.

This study focuses on assessing the level of morbidity among the population of Almaty, Kazakhstan, and investigating its connection with atmospheric air pollution using machine learning algorithms. The use of these algorithms is aimed at analyzing the relationship between air pollution levels and the state of public health, as well as the correlations between COVID-19 infection and the development of respiratory diseases. This study analyzes the respiratory diseases of the population of Almaty and the level of air pollution as a result of suspended particles for the period of 2017–2022. The study includes recommendations to reduce harmful emissions into the atmosphere using machine learning methods. The results of the study show that air pollution is a critical factor affecting the increase in the number of diseases of the respiratory system. The study recommends taking measures to reduce air pollution and improve air quality in order to prevent the development of chronic respiratory diseases. The study offers recommendations to industrial enterprises, traffic management organizations, thermal power plants, the Department of Environmental Protection, and local executive bodies in order to reduce respiratory diseases among the population.

The paper considers the problem of continuation of solutions of hyperbolic equations from a part of the domain boundary. These problems include the Cauchy problem for a hyperbolic equation with data on a timelike surface. In the inverse problems, the inhomogeneities are located at some depth under the medium layer, the parameters of which are known. In this case, an important tool for practitioners are the problems of continuation of geophysical fields from the Earth's surface towards the lay of inhomogeneities. In equations of mathematical physics, solution of the continuation problem from part of the boundary is in many cases strongly ill-posed problems in classes of functions of finite smoothness. The ill-posedness of this problem is considered, that is, the example of Hadamard, a Cauchy problem for a hyperbolic equation, is given. The physical formulation of the continuation problem is considered and reduced to the inverse problem. The definition of the generalized solution is formulated and the correctness of the direct problem is presented in the form of a theorem. The inverse problem is reduced to the problem of minimizing the objective functional. The objective functional is minimized by the Landweber method. By the increment of the functional, we consider the perturbed problem for the direct problem. We multiply the equation of the perturbed problem by some function and integrate by parts, we obtain the formulation of the conjugate problem. After that, we get the gradient of the functional. The algorithm for solving the inverse problem is listed. A finite-difference algorithm for the numerical solution of the problem is presented. The numerical solution of the direct problem is performed by the method of inversion of difference schemes. The results of numerical calculations are presented