On Linear Differential Equation Discretization

A. O. Egorshin

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 14 (2012), pp. 59-72

Résumé

Some problems of obtaining the discrete description of the first order differential system (DS) on the uniform lattice have been considered. These DS are regarded in the form of system $n$ of the first order ordinary linear differential equations with constant coefficients or as one $n$-order equation for the observed functional of the DS state. The problems under consideration are of some importance for the problems of the variational identification and approximation of the dynamic processes by means of that type models on the finite interval. There are compared the analytic uniform method of discretization (based on Cayley–Hamilton theorem) and that of the local one on the basis of the interpolation of the samples of $n+1$ counting by Taylor polynomials to the power $n$. There have been obtained the general formula of the local discretization that makes it possible to compare its difference and interpolarization methods. It has been shown by using Vandermond inverse matrices that in the obtained general formula of the local discretization $n+1$ Taylor matrices (from Taylor polynomial coefficients) correspond to its interpolational method while $n+1$ Pascal matrices (from Pascal triangle numbers) correspond to the difference method. It has been shown that matrix nondegeneracy of the DS observability on the lattice is a necessary and sufficient condition both for analytic discretizability and for reducing the discete system (of the DS description of the lattice) to Frobenius canonical form. It is equivalent to one ordinary difference equation for the observed variable with constant coefficients. This equation is a basis of the well-known variational method of identification. It has been shown that interpolation method of the local discretization is the first order linear approximation of the uniform analytic discretization formula. It has been demonstrated that its zero order approximation does not depend on the DS coefficients and is a vector of the coefficients of the $n$-th difference. We conclude that zero order approximation of the observability matrix of DS and of the observability matrix of the polynomial system $y^{(n)} = 0$ on the lattice is Taylor $n$-matrix.

Détail

Keywords: variational approximation and identification, discretization of differential equation, analytical discretization, linear approximation, Cayley–Hamilton theorem, local discretization, Vandermond matrices
Mots-clés : Teylor polynomial, Pascal triangle.