Integral Error Functions Research Articles

Order Reduction of Discrete Systems using Step Response Matching

A conceptually simple and new method for the model order reduction of linear time invariant discrete system is presented. The proposed method uses the concept of dominant poles retention in the low order model and exact matching of steady-state parts of the transient responses of low and high order models. Zeros of the reduced order system are determined by matching the transient parts at a few selected points of step responses of reduced and high order systems. Selection of points in the step response curve is based on the fact that only the prominent stages like starting point, first undershoot, and peak overshoot are considered. These being well defined points, when chosen, can be applied to most of the systems and give satisfactory results, in terms of minimum integral error function between the original high order system (OHOS) and the reduced low order system (RLOS). When the transient response is without undershoot or overshoot, a different strategy is suggested. Stability of the RLOS is always assured in the proposed method. A number of examples have been tried using the method and a comparative analysis with existing methods highlights its relevance.

On a family of logarithmic and exponential integrals occurring in probability and reliability theory

AbstractWe define an integral function Iμ(α, x; a, b) for non-negative integral values of μ byIt is proved that Iμ(α, x; a, b) satisfies a functional recurrence relation which is exploited to find a closed form evaluation of some incomplete integrals. New integral representations of the exponential integral and complementary error functions are found as special cases.

Approximate solutions of non-linear non-autonomous second-order differential equations†

A method is presented for determining approximate solutions to a class of differential equations characterized by: where resonance phenomena (arising, for example, when ƒ(x, [xdot], t) = x cosω0 t) may be neglected. The approximation is developed from an asymptotic expansion in terms of the amplitude and phase of the solution. Three examples are considered in illustration of the application of the approximation technique, and using an integral error function, solution error is shown graphically for these examples in terms of equation parameters. An expression for the approximate solution is derived which makes it possible to determine solution accuracy for any function f(x, x˙, t) once the approximate amplitude envelope and phase relationships have been derived. Graphical solutions demonstrate the accuracy which can be maintained even up to relatively large values of the parameter µ.

Probabilistic Properties of Bilinear Expansions of Hermite Polynomials

Probabilistic Properties of Bilinear Expansions of Hermite Polynomials

Tables of Integral Error Functions and Hermite Polynomials. By O. S. Berlyand, R. I. Gavrilova & A. P Prudnikov Translated by Prasenjit Basu. Mathematical Tables Series. Vol. 19. Pp. vi, 163. £5. 1962. (Pergamon Press)

Tables of Integral Error Functions and Hermite Polynomials. By O. S. Berlyand, R. I. Gavrilova & A. P Prudnikov Translated by Prasenjit Basu. Mathematical Tables Series. Vol. 19. Pp. vi, 163. £5. 1962. (Pergamon Press) - Volume 49 Issue 367

BOOK REVIEWS

Book reviewed in this article: Cochran, William G. Sampling Techniques. Buckland, William R., and Fox, Ronald A. Bibliography of Basic Texts and Monographs on Statistical Methods. Shilov, G. E. How to Construct Graphs, and Natanson, I. P. Simplest Maxima and Minima Problems. Topics in Mathematics. Whittle, P. Prediction and Regulation by Linear Least Square Method. Beryland, O. S., Gavrilova, R. I., and Prudnikov, A. P. Tables of Integral Error Functions and Hermite Polynomials. Amos, D. E., and Koopmans, L. H. Tables of the Distribution of the Coeficient of Coherence for Stationary Bivariate Gaussian Processes. Vesselo, I. R. How to Read Statistics.

Tables of Integral Error Functions and Hermite Polynomials

Tables of Integral Error Functions and Hermite Polynomials

Tables of Integral Error Functions and Hermite Polynomials.

Tables of Integral Error Functions and Hermite Polynomials.

Mathematical Tables Series. Volume 19 Tables of Integral Error Functions and Hermite Polynomials

By O. S. Berlyand, R. I. Gavrilova and A. P. Prodnikov Oxford: Pergamon Press. 1962. Pp. 163. Price £5 In both cases the functions concerned are given for orders 1 to 30, to 6 significant figures, for arguments at interval 0.01. There is a 5-page introduction.