Article Approximate methods for solving nonlinear initial boundary-value problems based on special constructions of series was published on January 1, 1993 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 8, issue 2).

Most residential buildings and capital structures in the permafrost zone are constructed on the principle of maintaining the frozen state of the foundation soils. The changing climate and the increasing anthropogenic impact on the environment lead to changes in the boundaries of permafrost. These changes are especially relevant in the areas of piling foundations of residential buildings and other engineering structures located in the northern regions since they can lead to serious accidents caused by the degradation of permafrost and decrease the bearing capacity of the soil in such areas. Therefore, organization of temperature monitoring and forecasting of temperature changes in the soil under the buildings is an actual problem. To solve this problem, we use computer simulation methods of three-dimensional nonstationary thermal fields in the soil in combination with real-time monitoring of the temperature of the soil in thermometric wells. The developed approach is verified by using the temperature monitoring data for a specific residential building in the city of Salekhard. Comparison of the results of numerical calculations with experimental data showed good agreement. Using the developed computer software, nonstationary temperature fields under this building are obtained and, on this basis, the bearing capacities of all piles are calculated and a forecast of their changes in the future is given. To avoid decreasing the bearing capacity of piles it is necessary to prevent the degradation of permafrost and to supply the thermal stabilization of the soil. The proposed approach, based on a combination of the soil temperature monitoring and computer modeling methods, can be used to improve geotechnical monitoring methods.

One of the most perspective ways to construct solutions of nonlinear partial differential equations is combination of numerical and analytical methods. In this paper analytical method of special series is used. Essence of this method is in expansion of solutions into a series by the powers of one or several special functions provided the coefficients of the series to be computed recurrently. In contrast to Fourier method, which also used for solving nonlinear partial differential equations, method of special series allows to obtain solutions with preliminary assigned accuracy because the developed approach leads to a sequence of finite systems of ordinary differential equations. The obtained sequence turns out to be linear even for nonlinear initial equation. This fact allows to obtain new results: in a number of cases it is possible to prove global convergence of constructed series including for unbounded domains, where numerical methods occur principal difficulties. This approach allows to obtain constructive solutions in the form of convergent series also for boundary-initial problems for a certain class of nonlinear equations. Some numerical results are presented for nonlinear wave equation.