Solving of some nonlinear ordinary differential equations in the form of power series
I.N. Belyaeva1, I.K. Kirichenko2, N.N. Chekanova3,4
1 Belgorod National Research University
2 Kharkiv National Automobile and Highway University
3 Kharkiv National University named after V. N. Karazin
4 Department of Information Technology and Mathematic Modeling, Karazin Business Schoo
Abstract: In the current scientific literature, a variety of nonlinear ordinary differential equations are widely and successfully used to describe real processes in various fields of natural sciences: optics, elasticity theory, molecular physics, etc. For example, the Ermakov and Riccati equations are used to solve the quantum Schrodinger equation, in electrodynamics. However, unfortunately, there are no well-and reliably developed and generally accepted methods for solving nonlinear differential equations. In addition, most of the Riccati equations are not integrated even in quadratures. In this paper, to construct solutions to the nonlinear Ermakov and Riccati equations, it is proposed to use the corresponding so-called connected linear differential equations, the solutions of the latter are in the form of power series using modern computer systems of analytical calculations.In this paper, solutions for some nonlinear Ermakov and Riccati equations are calculated using this proposed method. It is shown by direct substitution that the obtained solutions in the form of power series satisfy the considered nonlinear equations of Ermakov and Riccati with a known accuracy. Solutions of nonlinear Ermakov and Riccati equations can be used to describe the chemical and physical properties of nanostructures at the quantum level. Besides, solutions of nonlinear Ermakov and Riccati equations can be successfully applied in solving stationary and time-dependent Schrodinger equations.
Keywords: ordinary differential equations, Ermakov equation, Riccati equation, mathematical modeling, power series, Maple computer system
- Irina N. Belyaeva – Ph. D, Docent, Department of Computer Science, Natural Sciences and Teaching Methods, Belgorod National Research University
- Igor K. Kirichenko – Dr. Sc., Professor, Department of higher Mathematics, Kharkiv National Automobile and Highway University
- Natalia N. Chekanova – Ph. D, Docent, Kharkiv National University named after V. N. Karazin, Docent Department of Information Technology and Mathematic Modeling, Karazin Business Schoo
Belyaeva, I.N. Solving of some nonlinear ordinary differential equations in the form of power series / I.N. Belyaeva, I.K. Kirichenko, N.N. Chekanova // Physical and chemical aspects of the study of clusters, nanostructures and nanomaterials. — 2022. — I. 14. — P. 284-291. DOI: 10.26456/pcascnn/2022.14.284. (In Russian).
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