Continuous trigonometric collocation polynomial approximations with geometric and superconvergence analysis for efficiently solving semi-linear highly oscillatory hyperbolic systems

In this paper, based on the continuous collocation polynomial approximations, we derive and analyse a class of trigonometric collocation integrators for solving the highly oscillatory hyperbolic system. The symmetry, convergence and energy conservation of the continuous collocation polynomial approximations are rigorously analysed in details. Moreover, we also proved that the continuous collocation polynomial approximations could achieve at superconvergence by choosing suitable collocation points. Numerical experiments verify our theoretical analysis results, and demonstrate the remarkable superiority in comparison with the traditional temporal integration methods in the literature.

On new approximations for the modified Bessel function of the second kind \(K_0(x)\)

A new series representation of the modified Bessel function of the second kind \(K_0(x)\) in terms of simple elementary functions (Kummer's function) is obtained. The accuracy of different orders in this expansion is analysed and has been shown not to be so good as those of different approximations found in the literature. In the sequel, new polynomial approximations for \(K_0(x)\), in the limits \(0< x\leq 2\) and \(2\leq x < \infty\), are obtained. They are shown to be much more accurate than the two best classical approximations given by the Abramowitz and Stegun's Handbook, for those intervals.

Approximations of the Sixth Order with the Polynomial and Non-polynomial Splines and Variational-difference Method

This paper discusses the approximations with the local basis of the second level and the sixth order. We call it the approximation of the second level because in addition to the function values in the grid nodes it uses the values of the function, and the first and the second derivatives of the function. Here the polynomial approximations and the non-polynomial approximations of a special form are discussed. The non-polynomial approximation has the properties of polynomial and trigonometric functions. The approximations are twice continuously differentiable. Approximation theorems are given. These approximations use the values of the function at the nodes, the values of the first and the second derivatives of the function at the nodes, and the local basis splines. These basis splines are used for constructing variational-difference schemes for solving boundary value problems for differential equations. Numerical examples are given

PROBABILISTIC CHARACTERISTICS OF WIRELESS NETWORKS WITH INFRASTRUCTURE TOPOLOGY

Based on the analysis of the operation of networks IEEE 802.11 DCF, a function is proposed for determining the probability of frame transmission to a central node depending on the number of stations operating in saturation mode. The probabilities of collisions are calculated. Using a polynomial approximations an expression is obtained for the network throughput, which explicitly depends on the number of the simultaneously operating stations.