This paper proposes a new approach to the study of direct and inverse problems for a singularly perturbed heat equation with nonlinear temperature-dependent diffusion, based on the further development and use of asymptotic analysis methods in the nonlinear singularly perturbed reactiondiffusion-advection problems. The essence of the approach is presented using the example of a class of one-dimensional stationary problems with nonlinear boundary conditions, for which the case of applicability of asymptotic analysis is highlighted. Sufficient conditions for the existence of classical solutions of the boundary layer type and the type of contrast structures are formulated, asymptotic approximations of an arbitrary order of accuracy of such solutions are constructed, algorithms for constructing formal asymptotics are substantiated, and the Lyapunov asymptotic stability of stationary solutions with boundary and internal layers as solutions to the corresponding parabolic problems is investigated. A class of nonlinear problems that take into account lateral heat exchange with the environment according to Newton’s law is considered. A theorem on the existence and uniqueness of a classical solution with boundary layers in problems of this type is proven. As applications of the study, methods for solving specific direct and inverse problems of nonlinear heat transfer related to increasing the operating efficiency of rectilinear heating elements in the smelting furnaces — heat exchangers are presented: the calculation of thermal fields in the heating elements and the method for restoring the coefficients of thermal diffusion and heat transfer from modeling data.

This paper proposes a new approach to the study of direct and inverse problems for a singularly perturbed heat equation with nonlinear temperature-dependent diffusion, based on the further development and use of asymptotic analysis methods in the nonlinear singularly perturbed reaction-diffusion-advection problems. The essence of the approach is presented using the example of one class of one-dimensional stationary problems with nonlinear boundary conditions, for which the case of applicability of asymptotic analysis is singled out. Sufficient conditions for the existence of classical solutions of the boundary layer type and the type of contrast structures are formulated, asymptotic approximations of an arbitrary order of accuracy to such solutions are constructed, algorithms for constructing formal asymptotics are substantiated, and the Lyapunov asymptotic stability of stationary solutions with boundary and internal layers as solutions of the corresponding parabolic problems is investigated. A class of nonlinear problems that take into account lateral heat exchange with the environment according to Newton's law is considered. A theorem on the existence and uniqueness of a classical solution with boundary layers in problems of this type is proved. As applications of this research, methods for solving specific direct and inverse problems of nonlinear heat transfer related to increasing the operating efficiency of rectilinear heating elements in the smelting furnaces (heat exchangers) are presented, which include the calculation of thermal fields in the heating elements and a method for reconstructing thermal diffusion and heat transfer coefficients based on modeling data.

We study the problem of the existence of stationary, asymptotically Lyapunov-stable solutions with internal transition layers in nonlinear heat conductance problems with a thermal flow containing a negative exponent. We formulate sufficient conditions for the existence of classical solutions with internal layers in such problems. We construct an asymptotic approximation of an arbitrary-order for the solution with a transition layer. We substantiate the algorithm for constructing the formal asymptotics and study the asymptotic Lyapunov stability of the stationary solution with an internal layer as a solution of the corresponding parabolic problem with the description of the local attraction domain of the stable stationary solution. As an application, we present a new effective method for reconstructing the nonlinear thermal conductivity coefficient with a negative exponent using the position of the stationary thermal front in combination with observation data.

In this paper, we consider two reasonable approaches to the problem of numerical simulation of the concentration distribution of a finely dispersed aerosol in spiral vortex structures (rolls) at the atmospheric boundary layer in order to estimate the contribution of vortex structures to the transport of aerosols through the boundary layer. Using the methods of perturbation theory, an approximate solution of a stationary spatially periodic singularly perturbed problem of the reaction–diffusion–advection type, which models the distribution of an aerosol in vortices, is obtained, the residual term is estimated, and a method for numerically solving the zero-approximation problem is proposed. As an alternative approach to the problem of numerical modeling of an aerosol-concentration field in rolls, implementation of the method of evolutionary factorization is considered. Using model data, an estimate of the amount of an aerosol carried by vortex structures is obtained.

A new approach to the problem of determining the density of emission fluxes of anthropogenic impurities from distributed urban sources by the rate of growth of the integral content of impurities in the vertical column of the atmosphere in the morning hours is proposed. The method is based on the use of a singularly perturbed reaction–diffusion model describing the vertical distribution of an admixture (carbon monoxide) over a city, in combination with atmospheric CO measurements over Moscow. The vertical profile of the turbulent diffusion coefficient was calculated from the measurement data at the Ostankino television tower, and the vertical profiles of the CO concentration for different seasons were reconstructed. The average annual CO emissions from the entire territory of Moscow were calculated using model data. The reliability of the obtained emission values is confirmed by comparisons with the emission inventory data.

Recently, there has been a significant increase in the anthropogenic impact on the environment, including on the atmosphere. Therefore, it is very important to understand the mechanisms of transport of pollutants and to have reliable estimates of the impact of various factors on the transport of atmospheric impurities. Ground based measuring stations allow local continuous observations, characterized by high accuracy. The main disadvantage of ground based measurements is the low density of measuring stations, which does not allow reproducing the concentration fields of pollutants. Remote methods include, in particular, satellite observations, the main advantage of which is the ability to cover a large area, but, as a rule, they have rather low spatial resolution. In contrast, this work utilizes new satellite technology providing data with high space resolution. However, for a more detailed description, it is necessary to supplement the measurement data with mathematical modeling of various degrees of complexity. This work is devoted to the construction of a model of NOx transport from local ground sources with high spatial resolution which take into account chemical transformations. To achieve a high spatial resolution, the model uses a numerical solution of a system of three-dimensional reaction-diffusion-advection equations that takes into account the kinetic equations describing chemical reactions. Information on wind speed, temperature and pressure fields are obtained using the HYSPLIT model. The turbulent transport is described using a first-order closure model, where the turbulent diffusion coefficient parameterization is based on data on the friction velocity and the boundary layer height. Validation of the model was carried out by comparing the results of calculations with high-detailed spatial NO2 distributions obtained using measurements of the GSA instrument onboard the Resurs-P satellite.

The process of the formation of a plume of NO2 emitted into the atmosphere by a point source is considered. A new technique is suggested for retrieving NO2 distribution fields in the troposphere from GSA Resource-P satellite data with the use of mathematical simulation. The functional form of the model is determined, which agrees in complexity with the amount of available experimental information; the values of the model parameters are found. The emission power is calculated. The reliability of the simulation results is estimated from the comparison of the calculations with the experimental distribution of the altitude-integratal amount of NO2 in air depending on the horizontal coordinates.

The possibilities of using asymptotic analysis for solving the inverse problem of restoring the parameters of the source of nitrogen oxide industrial emissions into the atmosphere are demonstrated. The asymptotic analysis allows to reduce the subproblem for a three-dimensional singularly perturbed equation of the reaction-diffusion-advection type to a much simpler problem for numerical solution. This allows to significantly increase the efficiency of the numerical solution of the original inverse problem. Numerical experiments demonstrate the effectiveness of the proposed approach.

The paper presents a new approach to estimating of the power of nitrogen oxides emissions from the anthropogenic source. Authors created the model that describes the dynamics of the formation of a plume from the point anthropogenic source, coordinated in the complexity with the amount of the available data. Based on the proposed model, it is become possible to estimate the emission power by the satellite photographs obtained from the Resource-P series satellite. This estimate is the particular interest in the controlling emissions, which is the complex and non-trivial problem. Also authors determined the distribution of the height-integral amount of nitrogen dioxide over the Hebei Province, China, and compared it with the experimental data. The obtained estimates of the emission power can be used to integrate it into the complex chemical transport models.

Experiments for the retrieval of the high-detailed spatial NO2 distribution in the troposphere using measurements of the GSA instrument onboard satellites of Resurs-P series were performed in 2016-2017. The authors developed an algorithm to obtain the tropospheric NO2 2D distribution with the horizontal spatial resolution reaching 2,4 km for the first time at the world level and provided on a grid with a step of 120 m. The high spatial resolution of the NO2 space measurements allowed the identification of local sources of NO2 pollution and their plumes from space observations. To validate the fine structures detected in the NO2 fields of GSA/Resurs-P, we perform comparisons with chemical transport models. The paper presents preliminary results of a comparison with a new model which is based on a numerical-asymptotic approach. The comparison was performed for NO2 observations on September 29, 2016 over Hebei province, the North China Plain. We propose, in particular, a new efficient approach using this model to obtain estimates of emissions from local anthropogenic sources based on GSA/Resurs-P observational data. To validate the coarse structures in the GSA/Resurs-P NO2 field, in this paper, we perform comparisons of our data based on spectral imagery of Tokyo region, Japan, taken in March-April 2017 with observations of OMI/Aura and TROPOMI/Sentinel-5P. The comparison confirmed the reliability of the GSA NO2 fields in general.

A numerical-asymptotic approach is used to solve some coefficient inverse problems of tracer diffusion in the atmosphere. An asymptotic solution of the direct problem for an effective prognostic equation in the near-field zone of the source is obtained via a rigorous asymptotic analysis of a multidimensional singularly perturbed reaction–diffusion–advection problem. This solution is used as a priori information to construct a numerical algorithm for solving the inverse problem of recovering the parameters of an anthropogenic pollution source. The algorithm is implemented using sounding data on the Earth’s atmospheric composition obtained from the Russian Resurs-P satellite with highest available spatial resolution. For the first time, atmospheric pollutant emissions (nitrogen dioxide) from an isolated industrial source have been estimated by applying high-precision space monitoring and mathematical methods.

This paper describes a new approach to calculating the vertical turbulent diffusion coefficient and its variability, based on the use of modern asymptotic analysis in the singularly perturbed reaction–diffusion problems in combination with information obtained at one of the atmospheric monitoring stations. The capabilities of this method are demonstrated by using the diffusion model that describes changes in the vertical distribution of concentrations of anthropogenic impurities due to turbulent diffusion. Field measurements of the carbon monoxide concentration at various altitudes above Moscow are used in order to control the adequacy of the mathematical model and the efficiency of the calculation algorithm. Based on the analytical calculations, taking into account the initial and boundary conditions, which are consistent with the field observations, the vertical profiles of the turbulent diffusion coefficients and their seasonal changes are determined. The estimated reliability of the values recovered confirms the high efficiency of the proposed method and its high potential in the assessment of emissions and in the numerical atmosphere models.

The basis of this work is the use of modern methods of asymptotic analysis in reaction–diffusion–advection problems in order to describe the classical boundary-layer periodic solution of one singularly perturbed problem for the nonlinear diffusion–advection equation. An asymptotic approximation of an arbitrary order of such a solution is constructed, and the formal construction is justified. The uniqueness theorem is proved, the asymptotic Lyapunov stability is established, and the local domain of attraction of the boundary-layer periodic solution is found. One of the applications of this result to atmospheric diffusion problems is discussed, namely, mathematical modeling of the processes of transport and chemical transformation of anthropogenic impurities in the atmospheric boundary layer with allowance for periodic, e.g., daily or seasonal changes. The analytical algorithms developed for this problem as well will form the basis for a new method for calculating daily corrected emission fluxes of anthropogenic impurities from urban sources, which will make it possible to develop improved methods for determining daily integral emissions from the entire territory of a city or a urban agglomeration, based on the use of analytical solutions of model problems in combination with information obtained on a network of atmospheric monitoring stations.

We consider a multidimensional singularly perturbed stationary diffusion model with acubic nonlinearity. For models of this type, a modified asymptotic method of boundary functions,which extends the classical asymptotic analysis methods to the case of multidimensional problems,and the asymptotic method of differential inequalities, which is based on the comparison principle,are used to study the existence of asymptotically Lyapunov stable solutions with internal layers asstationary solutions of the corresponding parabolic problems. Sufficient conditions are establishedfor the existence of such solutions in the form of some conditions on the coefficients of theequation, an asymptotic approximation to the solution of an arbitrary accuracy order withcoefficients is constructed in closed form, and the formal constructions are justified. This resultcan be used for creating efficient numerical algorithms for direct and coefficient inverse problemsfor stationary equations of the reaction–diffusion–advection type as well as for constructing testexamples. Heat and mass transfer problems occurring in chemical industry are pointed out aspossible applications of our results.

In 2016 the authors performed space experiments to obtain the tropospheric NO2 field with the horizontal spatial resolution for the first time at the world level reaching 2.4 km. The NO2 fields were restored based on spectral measurements of GSA instrument installed on board the Russian satellites of the Resurs-P series. For the first time, the high spatial resolution of the new method makes it possible to identify local sources of NO2 pollution and their plumes. Good agreement with OMI NO2 observations with resolution 13 km x 24 km confirmed the reliability of the obtained Resurs-P NO2 fields in general. For the validation of high-detailed structures detected in the NO2 fields of GSA/Resurs-P, we are developing methods based on comparisons with chemical transport models. The comparison is performed for Hebei province, the North China Plain, which is the most NO2 polluted area in the world, using Resurs-P data obtained on September 29, 2016. The paper presents preliminary comparison of the Resurs-P tropospheric NO2 field with simulation based on HYSPLIT transport model. For the solution of the problem a high-detailed chemical transport model based on a solution of the nonlinear heat and mass transfer equation is under development. A theoretical background of the methods of asymptotic analysis of multidimensional singularly perturbed problems for the nonlinear heat and mass transfer equation is proposed.

The periodic problem that arises in the mathematical modeling of the vertical transfer of an anthropogenic impurity in the lower troposphere is considered for the nonlinear diffusion transfer equation. The model problem in dimensionless variables is classified as a nonlinear singularly perturbed reaction—diffusion—advection problem, which is studied by the methods of asymptotic analysis. Using the method of boundary functions and the asymptotic method of differential inequalities based on the principle of comparison, an asymptotic problem solution of arbitrary-order accuracy is constructed with the further substantiation of constructions and the study of this solution for the Lyapunov asymptotic stability property. The results of this work are illustrated using an example that describes the concentration field of a linear substance sink.

The stationary reaction-diffusion-advection problems, modeling the processes of the transport and chemical transformation of active and passive impurities in the surface layer of the atmosphere, to which the asymptotic methods are applicable (to the problems), are considered. We study the multidimensional asymptotically Lyapunov-stable solutions of the boundary layer type and the contrast structures by constructing the formal asymptotic approximations of an arbitrary-order accuracy based on the boundary-function method. To justify the constructed asymptotics, we use an asymptotic method of differential inequalities. The results of the study are illustrated by the example of the two-dimensional boundary value problem with a cubic nonlinearity. They can be used to create a numerical algorithm that uses asymptotic analysis to construct spatially inhomogeneous mashes when describing the internal layer of contrast structure, and also for the purposes of constructing the test examples.

The stationary Lyapunov-stable solutions with internal transition layers (contrast structures) of multidimensional singularly perturbed reaction-diffusion-advection problems are investigated with the use of the modern methods of asymptotic analysis. On the basis of the modified boundary-function method, the asymptotic approximations of such solutions of an arbitrary order of accuracy in the case of a balanced nonlinearity are obtained. We propose a justified and effective algorithm that allows us to define and describe the localization region of the internal layer of contrast structure. We use this result to describe the thermal structures in the homogeneous nonlinear dissipative media.

The authors are developing methods for the determination of the emissions from urban sources of key impurities basing on surface and high-detailed satellite measurements. For the applications in these researches we develop a simplified parameterized model of chemical transformations in the atmosphere. This work is devoted to estimation of the effective lifetimes and the decay rates of nitrogen oxides (NOx) entering the atmosphere as a result of emissions of industrial enterprises basing on chemical-transport simulation. The estimation of effective decay rates, which allows to relatively simply parameterize chemical processes occurring in a plume, is necessary for further use in transport models based on systems of the diffusion-reaction-advection equations and describing the behavior of the plume. The effective decay rates are calculated as the inverse of the times over which the concentrations of the corresponding nitrogen oxides decrease by e times compared to their maximum values. The dependence of their concentrations on time is found by solving a system of kinetic equations describing the reactions occurring in the plume. For the numerical solution of the Cauchy problem, a finite-difference scheme is used that takes into account the structure of the kinetic equations and has the second order of the approximation error.