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

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

Optimization of location of injection wells in an open geothermal system

A geothermal open loop system consisting of one production well and several injection wells is considered. The production well serves to raise hot water from a geothermal underground reservoir, which can be used for building heating or for other purposes. As a result, the water is cooled and then pumped through injection wells back into the geothermal reservoir. The cooled water begins to be filtered in the reservoir in the direction of the production well. A mathematical model describing the distribution of a cold water front in such a system is presented. A software package has been developed that makes it possible to evaluate the efficiency of the location of injection wells in a geothermal reservoir in order to increase the operating time of such a system, since the operation of the geothermal system life is over as soon as the cold water front reaches the production well. © 2019 Author(s).Russian Foundation for Basic Research, RFBR: 19–07–00435The work was supported by Russian Foundation for Basic Research 19–07–00435

Rotating range sensor approached for mobile robot obstacle detection and collision avoidance applications

Range finder sensors are widely used in the obstacle detection and collision avoidance applications. In this research, we propose rotating range finder sensor that provides economic and efficient solution for mobile robot applications. Rotating approach is achieved by coupling the range sensors with servomotor. In this article, rotating approach model design, main parameters, equations and limitation are described. In addition, an algorithm is developed to control the rotation angle of the range sensor, extract data from the approach and analyze it. A case study of the rotating approach by implementing ultrasonic sensor is simulated and the results are obtained. Simulation platform Gazebo and ROS are used to simulate the rotating approach. © 2021 Author(s)

A simulation-based study to calculate all the possible trajectories of differential drive mobile robot

Deferential Drive Mobile Robot (DDMR) is being used in many applications as it is easy to be modeled and controlled. This research presents the idea of using DDMR turning motion behavior to develop an algorithm that calculate all the circular trajectories. This can be used to navigate DDMR in a curvature paths instead of linear ones. In this research we design and simulate Differential Drive Mobile Robot (DDMR) model. Then we use the simulated model to calculate all the possible trajectories that DDMR can follow with each left and right wheel velocity configurations. Results are saved in a navigation look-up table that can be implemented in DDMR navigation's approach. © 2021 Author(s)

Application of series with recurrently calculated coefficients for solving initial-boundary value problems for nonlinear wave equations

For one class of nonlinear wave equations with a small parameter, an initial-boundary value problem with zero boundary conditions is considered. The solution of such a problem is constructed with using series with recurrently calculated coefficients in two ways. In the first case, the method of special series is considered, which is based on the choice of some functions (basic functions), by the powers of these functions the solution of the original problem is presented into a series with recurrently calculated coefficients. In the other case to represent solutions of the problem a combination of Fourier and small parameter methods is used. It is shown that both proposed constructions of series with recurrently calculated coefficients converge to the solution of the initial-boundary value problem on a finite time interval. © 2021 Author(s).The work was supported by Russian Foundation for Basic Research 19–07–00435