Abstract
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.
Highlights
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
In this paper we have shown that for linear ordinary differential equations of second order with constant coefficients, there exists a discretization which simulates properly all features of the differential equation
The solutions of this discrete equation exactly coincide with the solutions of the corresponding differential equation evaluated at a discrete lattice
Summary
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]. Where x = x(t) and the dot means the t-derivative This is a linear equation and its general solution is well known. Discretization procedures are not so important (but sometimes are applied, see [3]) This example allows us to show and illustrate some more general ideas. The continuum limit consists in replacing xn by x(tn) = x(t) and the neighboring values are computed from the Taylor expansion of the function x(t) at t = tn: xn+k = x tn + kε. Substituting these expansions into the difference equation and leaving only the leading term we should obtain the considered differential equation.
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