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




deletion of dimensions of a mathematical model, group methods of solution, second order inertial link


A method of deletion of dimensions for a mathematical model which gives the number of variables less than that prescribed by the π-theorem was proposed. In a number of cases, it is possible to exclude from consideration all similarity criteria or, in other words, achieve self-similarity in them. In the framework of deletion of dimensions, this is expressed in a transition from criteria to similarity numbers. Thus, information is reduced without its loss.

The limit reduction of the number of variables in the mathematical model makes it possible to use analytical approximation dependences as approximate solutions. Such dependences are obtained from the initial solutions by using the coefficients of stretch, which corresponds to the group methods for solving equations. It was proposed to use solutions of the linearized forms of the original nonlinear equations as initial solutions. Such approach makes it possible to take into account physical character of the change in the studied quantities in the approximation dependences when solution of nonlinear equations cannot be realized using standard functions.

Efficiency of the method was illustrated by the example of study of the pendulum motion, which is a counterpart of inertial link of the second order in the theory of automatic control. Solutions were obtained for the cases of presence and absence of environmental resistance. The last variant is interesting by the feasibility of comparing the proposed and available analytical solutions in terms of elliptic integrals.

Author Biographies

Olexander Brunetkin, Odessa National Polytechnic University Shevchenko ave., 1, Odessa, Ukraine, 65044

PhD, Associate Professor

Department of thermal power automation processes

Maksym Maksymov, Odessa National Polytechnic University Shevchenko ave., 1, Odessa, Ukraine, 65044

Doctor of Technical Sciences, Professor, Head of Department

Department of thermal power automation processes

Oksana Maksymova, Odessa National Academy of Food Technologies Kanatna str., 112, Odessa, Ukraine, 65039


Department of the Computer Systems and Business Process Management 

Anton Zosymchuk, Odessa National Polytechnic University Shevchenko ave., 1, Odessa, Ukraine, 65044

Department of thermal power automation processes


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How to Cite

Brunetkin, O., Maksymov, M., Maksymova, O., & Zosymchuk, A. (2017). Development of a method for approximate solution of nonlinear ordinary differential equations using pendulum motion as an example. Eastern-European Journal of Enterprise Technologies, 5(4 (89), 4–11.



Mathematics and Cybernetics - applied aspects