Analytical Approximation of Periodic Ateb-Functions via Elementary Functions

Abstract

Abstract We consider the problem of analytic approximation of periodic Ateb- functions, widely used in nonlinear dynamics. Ateb-functions are the result of the following procedure. Initial ODE contains only the inertial and non-linear terms. It can be integrated, which leads to an implicit solution. To obtain explicit solutions we are led to necessity to inverse incomplete Beta functions. As a result of this inversion we obtain the special Atebfunctions. Their properties are well known, but the use of Ateb- functions is difficult in practice. In this regard, the problem arises of the Ateb functions approximation by smooth elementary functions. For this purpose in the present article the asymptotic method is used with a quantity 1 / (a + 1) as a small parameter, were a > 1 — exponent of nonlinearity. We also investigated the analytical approximation of Ate-b functions' period. Comparison of simulation results, obtained by the approximate expression, with the results of numerical solution of the corresponding Cauchy problem shows their sufficient accuracy for practical purposes, even for a = 1.