Development of a method for approximate solution of nonlinear ordinary differential equations using pendulum motion as an example

Abstract

4thermophysical properties, etc. In turn, this causes variation in a wide range of the process parameters subject to control. In such a situation, application of the results obtained by analysis of linear models is very problematic and even unfeasible in most cases.The search for new approaches to analysis of nonlinear models is an important element in solution of the problem of optimal control of power equipment using non-certified fuel of a variable composition. Literature review and problem statementAn effective way to simplify the study of complex models is to make them dimensionless. Reduction of dimensionality of the modeling space makes it possible both to reduce number of experimental (full-scale and numerical) studies by IntroductionWhen studying behavior of control systems, dynamic conditions require special consideration. Apparatus of ordinary differential equations is most often used as a mathematical tool for studying such systems. General methods of solution have only their linear forms. For this reason, linear (linearized) approaches are used to construct models of object and controller. However more exact models are represented by nonlinear differential equations. This situation leads to the fact that the results obtained can only be used within the range of small variation of parameters of the controlled processes nearby the linearization point.The use of fuel of a variable composition [1,2] in power equipment instead of certified fuel with a constant composition brings about a continuous variation of the fuel calorific value, amount of combustion products, their temperature, MATHEMATICS AND CYBERNETICS -APPLIED ASPECTS DEVELOPMENT OF

Constructing a method for solving the riccati equations to describe objects parameters in an analytical form

“…-for 60° -by 13 %; -for 90° -by 9 %. The results reported in [11] show the manifestation of a synergistic effect from combining a «standard» model linearization technology and using a special nondimensionalization method [10]. Moreover, one can talk about the emergence for such a combination of research tools.…”

“…However, the cited paper does not consider the possibility of using the benefits obtained in the process of linearizing the model equations. This possibility is considered in work [11] using an example of studying the motion of a mathematical pendulum. The non-linearity of the model was determined by the possibility of taking into consideration the large initial deviation of the pendulum from the position of equilibrium (by 30°, 60°, and 90°) in the absence and presence of dissipative forces.…”

Development of the method of approximate solution to the nonstationary problem on heat transfer through a flat wall

“…Therefore, comparison of results, obtained based on (16), of the magnitudes, given in [2], is valid. It should only be noted that during cooling the moment the process is over is considered to be a point when temperature reaches magnitude θ = ⋅θ = The comparison was conducted by determining relative error ε of calculating Fo based on (16) relative to Fo 1 . In the process of comparing, we chose the magnitude of k from (13) that ensures minimum magnitude of relative error ε.…”

“…Nevertheless, in addition to it, it makes it possible, with respect to (15), to perform calculations for an infinite cylinder and a sphere. Table 1 Comparison of results of precise [2] and approximate (16) calculations of nondimensionalized time of the end of the process of heating the bodies It follows from results shown in Table 1 that an error of determining the time of the end of heating the bodies using expression (16) compared to the results of precise analytical solution does not exceed 4 % over the entire examined range of change in Bi.…”

“…We shall nondimensionalize time variable according to the method proposed in [16] using a complex included in equation (8) that has a time dimensionality:…”