A direct method for exact discretization of ordinary differential equations

Differential equations are frequently used to analyze and design analog circuits and numerically solved from computation of difference equations. The equivalent difference equations are obtained by using the solutions or the Laplace transform of the corresponding differential equations for exact discretization. Here, we propose a direct method for exact discretization obtaining the equivalent difference equation whose solution is equal to the solution of a differential equation at any discrete periodic point. The direct method greatly differs from the existing method in needing neither the solutions nor the Laplace transform of differential equations. The equivalent difference equations are readily produced by direct transform of replacing (d/dt-α)x(t) with (1-e^αTE^-1)x_n in factorized differential equations (where x_n equal to by definition x(nT), α is a constant, T the sampling period, and E the shift operator defined by E^-1x_n=x_n-1). The method is applied to some examples including ordinary differential equations (ODEs) and nonlinear one of logistic equation which represents the chaos behavior. The proposed method is the simplest, easiest and fastest method ever, and very important and useful for generating exact numerical solutions of ODEs.