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Abstract <p>As is known, the asymptotic method is an outstanding achievement of mathematical science, the successful use of this method was pioneered by I. Newton, L. Euler, J.L. Lagrange, and many other brilliant mathematicians. Of these, the world of mathematicians declared Henri Poincaré also the father of the basic technology of asymptotic expansion, this recognition Henri Poincaré received as a result of the publication of the two-part ‘‘monograph’’ New Methods of Celestial Mechanics [<CitationRef CitationID="CR1">1</CitationRef>], along with many other works. Later, the legacy of the outstanding Russian scientist A. Lyapunov was also associated with the technology of asymptotic expansion, and Lyapunov’s name appeared next to Poincaré’s name. If we consider the asymptotic method, when using general basis functions, its use is associated with the issues of absolute and uniform convergence of the series, which is essentially related to the analyticity of the initial function, which is an important restriction on the class of admissible functions. With the approach of orthogonal Fourier expansion, the convergence of the series is proven in terms of differentiability for a much wider class of functions. On the other hand, when using the asymptotic method for qualitative analysis, the solving algorithm is relatively simple and represents a recursive process. In general the use of the Fourier series in the classical way was associated with the difficulty of creating a solving algorithm for the coefficients. The order of the corresponding differential operator system depends on the number of unknown functions in the truncated series and the solution of the approximation problem of the corresponding operator. The essence of what will be presented in this article is the construction of an efficient algorithm for the coefficients, which is analogous to the asymptotic method; The recurrent process operates in cases where the basis system is a system of classical orthogonal polynomials (specifically, Legendre, both of kinds of Chebyshev polynomials, Laguerre, and Hermite) and the explicit generating formulas for the coefficients of the corresponding matrices are given.</p>

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To the Alternative Technology of the Asymptotic Expansion of Poincaré–Lyapunov

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • T. Vashakmadze,
  • G. Buzhghulashvili

摘要

Abstract

As is known, the asymptotic method is an outstanding achievement of mathematical science, the successful use of this method was pioneered by I. Newton, L. Euler, J.L. Lagrange, and many other brilliant mathematicians. Of these, the world of mathematicians declared Henri Poincaré also the father of the basic technology of asymptotic expansion, this recognition Henri Poincaré received as a result of the publication of the two-part ‘‘monograph’’ New Methods of Celestial Mechanics [1], along with many other works. Later, the legacy of the outstanding Russian scientist A. Lyapunov was also associated with the technology of asymptotic expansion, and Lyapunov’s name appeared next to Poincaré’s name. If we consider the asymptotic method, when using general basis functions, its use is associated with the issues of absolute and uniform convergence of the series, which is essentially related to the analyticity of the initial function, which is an important restriction on the class of admissible functions. With the approach of orthogonal Fourier expansion, the convergence of the series is proven in terms of differentiability for a much wider class of functions. On the other hand, when using the asymptotic method for qualitative analysis, the solving algorithm is relatively simple and represents a recursive process. In general the use of the Fourier series in the classical way was associated with the difficulty of creating a solving algorithm for the coefficients. The order of the corresponding differential operator system depends on the number of unknown functions in the truncated series and the solution of the approximation problem of the corresponding operator. The essence of what will be presented in this article is the construction of an efficient algorithm for the coefficients, which is analogous to the asymptotic method; The recurrent process operates in cases where the basis system is a system of classical orthogonal polynomials (specifically, Legendre, both of kinds of Chebyshev polynomials, Laguerre, and Hermite) and the explicit generating formulas for the coefficients of the corresponding matrices are given.