On one generalized interpolation polynomial operator

A. F. Galimyanov ; T. Yu. Gorskaya

Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 22-32

Résumé

The article deals with the construction of a generalized polynomial operator necessary for finding approximate solutions of equations with fractional order of integration. Integral equations of fractional order are used in a number of problems related to the study of processes that behave discontinuously, for example, for diffusion problems, economic problems related to the theory of sustainable development and other similar problems. At present, interest in such equations has increased, as evidenced by the publications of recent years in which the processes described by such equations are investigated. In this connection, it becomes relevant to study methods for solving such problems. Since these equations cannot be solved exactly, there is a need to develop and apply approximate methods for their solution. In this article we obtain a form of polynomial operator for some continuous functions on $(0,2\pi)$ expressed through the Lagrange interpolation polynomial on equally spaced knots. The connection of the generalized interpolation operator with the Fourier operator is also established, and the closeness value of these operators is obtained. For the interpolation polynomial operator an estimate of the error of approximation of the exact value by the metric of the space of $(0,2\pi)$ continuous functions is found. This work is a continuation of the research of the authors.

Détail

Keywords: approximate methods, interpolation polynomial operators, error estimation.