No abstract

For a quadratic equation containing a small parameter regularly, and for the Duffing equation with small nonlinearity, their asymptotic solutions are constructed in the form of asymptotic Poincare series for a small parameter. The properties of the obtained asymptotic solutions when a small parameter tends to zero are analyzed. The sense of the theorem on the continuous dependence of the solution on the parameter for systems with regular perturbation is demonstrated. A Taylor series for the exact solution of the quadratic equation with small parameter is compared with its obtained asymptotic expansion. For the Duffing equation with small nonlinearity, we compare the graphs of the exact and asymptotic solutions under the same initial conditions.

Main purpose of the present work is development of qualitative theory of difference equations in the space of bounded numeric sequences.Main result is the establishment of necessary conditions of the existence of invariant toroidal manifolds for countable systems of differential and difference equations. In order to solve this problem, observed spaces are constructed in a special way. Necessary conditions of the existence of invariant tori for countable systems of differential and difference equations are derived.A concept of a continuous periodic in each variable function with period 2Pi , values of which lie in l2 , is introduced. Spaces, in which observations are made, are constructed in a special way. A theorem on approximation of a function from the corresponding space bytrigonometric polynomials is proven.

The article deals with some interpolation representations of random processes with non-equidistance interpolation knots. Research is based on observations of the process and its derivatives of the first IntroductionInterpolation representations of a class of random process with non-equidistance interpolation knots are investigated. The necessary results from the theory of entire functions of complex variable are formulated. The function bounded on any bounded region of the complex plane is considered. The estimation of the residual of the interpolation series is obtained. The interpolation formula that uses the value of the process and its derivatives at the knots of interpolation is proved. Considering the separability of the process and the convergence of a row, it is obtained that the interpolation row converges to the random process uniformly over in any bounded area of parameter changing. The convergence with probability 1 of the corresponding interpolation series to a random process in any bounded domain of parameter changes is proved.

The article deals with some interpolation representations of stochastic processes with non-equidistance interpolation knots. Research is based on observations of the process and its derivatives of the first and second orders at some types of knots and observations of the process and its derivatives of the first orders at other types of knots. The necessary results from the theory of entire functions of complex variable are formulated. The function bounded on any bounded region of the complex plane is considered. The estimate of the residual of the interpolation series is obtained. The interpolation formula that uses the value of the process and its derivatives at the knots of interpolation is proved. Considering the separability of the process and the convergence of a row that the interpolation row converges to the stochastic process uniformly over in any bounded area of changing of parameter is obtained. The main purpose of this article is the obtained convergence with probability 1 of the corresponding interpolation series to a stochastic process in any bounded domain of changes of parameter. Obtained results may be applied in the modern theory of information transmission.