MATHEMATICALLY MODELING PERIODICHNYE PROCESSES FOR ADDITIONAL SERIES FURIE

Abstract

The article is dedicate to mathematical modeling of periodic processes using Fourier series. The subject of the studying in this research was the analysis of various periodic, cyclical, oscillating processes (exchange rate fluctuations, forecasting the index of industrial products, demand for jewelry depending on the season, deviation of manufacturing, passenger transport, demand for products and services and other).

What are the requirements for functions that are decomposed in the Fourier series? How to check these functions for convergence? What are the types of Fourier clans? An analysis of the literature show that there is a convergence of the Fourier series at a point, uniform convergence, and convergence of the Fourier series in space L₂. There is no necessary condition for convergence for these series, but there are sufficient conditions: the Dirichlet sign and the Dini sign, which are quite sufficient for decomposition in the Fourier series.

Examples illustrate the decomposition of functions into Fourier series. The aim was to show the convergence of the series to the selected function. The Dirichlet conditions are checked first, and then the Fourier series coefficients are searched. This takes into account both the functions themselves and their properties (trigonometric or other; even, odd). The number of members of decomposition is unlimited and can be chosen arbitrarily. We also took  and  that is quite enough to assess convergence. The results of the calculations show that the convergence of the series increases  with increasing, i.e. the difference between the Fourier series and the decomposable function of the series decreases, as can be seen from Figures 1-4.

The conclusions confirm the value of the issues considered. Mathematical modeling of various periodic processes using Fourier series is useful to analyze the impact of changes on various process elements.