Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: ON SIMULATIONS OF THE CLASSICAL HARMONIC OSCILLATOR EQUATION BY DIFFERENCE EQUATIONS | ON SIMULATIONS OF THE CLASSICAL HARMONIC OSCILLATOR EQUATION BY DIFFERENCE EQUATIONS JAN L. CIESliNsKI AND BOGUSLAW RATKIEWICZ Received 29 October 2005 Accepted 10 January 2006 We discuss the discretizations of the second-order linear ordinary diffrential equations with constant coefficients. Special attention is given to the exact discretization because there exists a difference equation whose solutions exactly coincide with solutions of the corresponding differential equation evaluated at a discrete sequence of points. Such exact discretization can be found for an arbitrary lattice spacing. Copyright 2006 J. L. Cieslihski and B. Ratkiewicz. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction The motivation for writing this paper is an observation that small and apparently not very important changes in the discretization of a differential equation lead to difference equations with completely different properties. By the discretization we mean a simulation of the differential equation by a difference equation 5 . In this paper we consider the damped harmonic oscillator equation x 2yx uịx 0 where x x t and the dot means the t-derivative. This is a linear equation and its general solution is well known. Therefore discretization procedures are not so important but sometimes are applied see 3 . However this example allows us to show and illustrate some more general ideas. The most natural discretization known as the Euler method Appendix B cf. 5 10 consists in replacing x by xn x by the difference ratio xn 1 - xn e x by the difference ratio of difference ratios that is Xn 2 - Xn 1 Xn 1 - Xn e x 1 e Xn 2 - 2Xn 1 Xn J e2 Hindawi Publishing Corporation Advances in Difference Equations Volume 2006 Article ID40171 Pages 1-17 DOI ADE 2006 40171 2 Simulations of the harmonic oscillator .
