Числове Дослідження Базисних Систем При Розв’язуванні Аналітико-Числовими Методами Крайових Задач На Відрізку

Abstract

Spectral methods have indisputable advantages over numerical methods in solving various problems of mathematical physics. The advantages are the high convergence and accuracy of approximate solutions, which is most relevant for calculating the strength of aerospace machinery. The problem of choosing basic functions inevitably arises when applying spectral methods. The problem is that, in addition to providing exponential convergence, the system of basic functions has to satisfy some other requirements: stability of approximate solutions and procedures for their obtaining, reduction of calculations, convenience, and some more. This paper compares eleven basic systems: the systems constructed in the form of linear combinations of Legendre polynomials that satisfy either only the main boundary conditions or the main and natural ones, similar systems constructed using Chebyshev polynomials, and the functions proposed by Khalilov S. A., systems of Lagrange – Lobatto interpolation polynomials using the Legendre and Chebyshev interpolation points, system of trigonometric functions, exponentiation, and system of finite functions of the finite element method. The convergence speed of the approximate solution to the exact one, the error in the equations of boundary value problems, and the condition numbers of matrices of linear algebraic equation systems, which arise when using variational, projection and collocation methods, were compared. The study performed on three test problems modeling beam bending: classic beam bending under unevenly distributed load, bending of additionally stretched beam on the elastic basis and geometrically nonlinear bending. The impact of the Gibbs effect on the approximate solution convergence is investigated. Among the considered basic systems, the system of basis functions in the form of linear combinations of Legendre polynomials has proved to be the best, as they satisfy all boundary conditions. This basis leads to the highest speed at which approximate solution approaches the exact one, the error in the equations approaches to zero, and also it has the smallest increase in the condition number with the increase in the order of SLE matrices, which appear due to variational and projection methods. The finite functions of the finite element method have proved to be the worst in terms of accuracy and convergence.