Parallel Splitting and Decomposition Method for Computations of Heat Distribution in Permafrost

A mathematical model, numerical algorithm and program code for simulation and long-term forecasting of changes in permafrost as a result of operation of a multiple well pad of northern oil and gas field are presented. In the model the most significant climatic and physical factors are taken into account such as solar radiation, determined by specific geographical location, heterogeneous structure of frozen soil, thermal stabilization of soil, possible insulation of the objects, seasonal fluctuations in air temperature, and freezing and thawing of the upper soil layer. A parallel algorithm of decomposition with splitting by spatial variables is presented

On application of Fourier method for one class of equations describing nonlinear oscillations

Solutions of nonlinear partial differential equations with a small parameter are constructed as a sum of truncated Fourier series and some additional function. The coefficients of the truncated Fourier series depend on the small parameter and satisfy a nonlinear system of ordinary differential equations, which in turn is determined by nonlinear partial differential equation. A certain class of equations describing nonlinear oscillations was determined, for which the coefficients of the truncated Fourier series are bounded functions of time. This fact makes it possible to estimate the additional function and to justify the applicability of the Fourier method for the constructed class of nonlinear partial differential equations. © 2018 Author(s).The work was supported by Russian Foundation for Basic Research 16–01–00401 and program of scientific research UrB RAS 18–1–1–8

Application of admissible functions in studying partial stability of solutions of a nonlinear system of differential equations

A new approach related to construction of admissible functions that do not coincide with the Lyapunov functions was proposed to investigate partial stability of solutions of systems of ordinary differential equations. An example of using admissible functions for establishing partial stability of solutions for one nonlinear system of differential equations is presented. © 2018 Author(s).The work was supported by Russian Foundation for Basic Research 16–01–00401 and program of scientific research UrB RAS 18–1–1–8

Investigation of gobal stability of the equilibrium position of a system of differential autonomous equations with the help of admissible functions

To study the global stability of the zero solution, which is a single rest point for a nonlinear system of autonomous differential equations, admissible functions are used. These functions are analogous to Lyapunov functions, but do not coincide with them. Admissible functions can also be used for study the global stability of the zero solution of such systems with respect to part of the variables. An example of the application of such functions to the study of the global stability of the equilibrium state for a second-order nonlinear system is presented. © 2019 Author(s)

On new classes of solutions of nonlinear partial differential equations in the form of convergent special series

The method of special series with recursively calculated coefficients is used to solve nonlinear partial differential equations. The recurrence of finding the coefficients of the series is achieved due to a special choice of functions, in powers of which the solution is expanded in a series. We obtain a sequence of linear partial differential equations to find the coefficients of the series constructed. In many cases, one can deal with a sequence of linear ordinary differential equations. We construct classes of solutions in the form of convergent series for a certain class of nonlinear evolution equations. A new class of solutions of generalized Boussinesque equation with an arbitrary function in the form of a convergent series is constructed. © 2017 Author(s).The work was supported by Russian Foundation for Basic RTesearch 16–01–00401

On global asymptotic stability of solutions of a system of ordinary differential equations

To investigate global asymptotic stability in general of an equilibrium position of an autonomous system of ordinary differential equations, considered by V.A. Pliss, a function different from the Lyapunov functions is applied. V.A. Pliss proved that for this system it is impossible to construct the Lyapunov function as a sum of a quadratic form and an integral of some nonlinear function defined by the right-hand side of the system. © 2017 Author(s).Russian Foundation for Basic Research, RFBR: 16–01–00401The work was supported by Russian Foundation for Basic Research 16–01–00401

Formation of phase structure in the system Ti-3Al at the stage of secondary structurization at synthesis in the mode of thermal explosion

Features of phase formation processes in the system Ti-3Al at realization of self-extending synthesis in the mode of thermal explosion have been established with use of technological reactor enabling instant switching-off of the heating source. The analysis of synthesis finished products allows to draw a conclusion that phase structure of charge is abnormally depends on sizes of titanium particles. The single-phase product corresponding to initial stoichiometry is synthesized on fine and large fractions for the induction period. On intermediate fraction the product of synthesis is multiphas

Formation of phase structure in the system Ti-3Al at the stage of secondary structurization at synthesis in the mode of thermal explosion

Features of phase formation processes in the system Ti-3Al at realization of self-extending synthesis in the mode of thermal explosion have been established with use of technological reactor enabling instant switching-off of the heating source. The analysis of synthesis finished products allows to draw a conclusion that phase structure of charge is abnormally depends on sizes of titanium particles. The single-phase product corresponding to initial stoichiometry is synthesized on fine and large fractions for the induction period. On intermediate fraction the product of synthesis is multiphas

Simulation of permafrost changes due to technogenic influences of different ingeneering constructions used in nothern oil and gas fields

Significant amount of oil and gas is producted in Russian Federation on the territories with permafrost soils. Ice-saturated rocks thawing due to global warming or effects of various human activity will be accompanied by termocarst and others dangerous geological processes in permafrost. Design and construction of well pads in permafrost zones have some special features. The main objective is to minimize the influence of different heat sources (engineering objects) inserted into permafrost and accounting long-term forecast of development of permafrost degradation due to different factors in particular generated by human activity. In this work on the basis a mathematical model and numerical algorithms approved on 11 northern oil and gas fields some effects obtained by carrying out numerical simulations for various engineering systems are discussed. © Published under licence by IOP Publishing Ltd.This work was supported by RFBR projects (16-01-00401, 14-01-00155) by contract 02.A03.21.0006 (reg.N211 of RF Government) and Program of UB RAS, project 15-7-1-13