Vol 27, No 4 (2025)

The modification of the mother cubic Schoenberg spline is carried out using four cubic Schoenberg splines having finite supports, the sizes of which are smaller compared to the size of the finite support of the mother spline. As a result, eight grid sets of orthogonal cubic Schoenberg splines with real values are constructed. A theorem on the order of approximation of any function of the Sobolev space by linear combinations of constructed orthogonal cubic Schoenberg splines is proved. It is shown that the order of approximation by Schoenberg splines, also modified by Schoenberg splines, is significantly higher than the order of approximation by Schoenberg splines modified by step functions, and coincides with the order of approximation by classical cubic Schoenberg splines. The defect of the modified Schoenberg spline is equal to one, as that of the classical Schoenberg spline. A modified spline is a continuous function in which there are no breaks in the first and second derivatives at the points where the parts of the mother spline and the parts of the splines used for modification meet.

When calculating the strength of structural elements, one of the steps is to study the dynamics of these elements under various force loads. In this paper, based on the classical model of free vibrations of an elastic plate, in contrast to previous numerical and analytical studies, an analytical method for studying the dynamics of a concrete slab pinned at its edges is developed. According to the Galerkin method, an approximate solution of the partial differential equation used in the model is found as a linear combination of basis functions. This results in a system of ordinary differential equations for determining the coefficients of this combination. Based on the construction of a Lyapunov-type functional for the partial differential equation and on a Lyapunov function for the system of ordinary differential equations, several methods for determining the error of obtained approximate solution are proposed. Numerical calculations demonstrate the accuracy of error estimates. For this purpose, plots of the difference between the approximation under study and the higher-order approximation are constructed. The best estimate was shown by the method of error determination using the following basis function, whose coefficient was found from the equation obtained in study of the Lyapunov-type functional for the original partial differential equation.

This paper examines the mathematical modeling of ventilation systems consisting of deformable air ducts through which an air flow is supplied. Using constructed three- dimensional mathematical model described by a system of partial differential equations, the paper investigates dynamic stability of the elastic wall of an air duct where some gas flows. The Lyapunov dynamic stability criterion is used to study the mechanical system’s stability. To study stability in problems of aerohydroelasticity in compressible and incompressible medium models, Lyapunov-type functionals are constructed for deduced systems of differential equations. By studying these functionals stability conditions are obtained. They ensure that the functional is positive and its time derivative is negative. For a compressible medium model, the dependence between the longitudinal force compressing the plate and the air flow velocity is constructed for specific parameters of the mechanical system. Using the plot constructed, a comparison of the stability conditions for compressible and incompressible medium models is made. It is shown that the medium compressibility has negative effect on the stability of the deformable wall of the air duct and leads to decrease of the stability region.