Dynamic solutions of linear matrix differential equations

Abstract

Abstract. We discuss the mth-order linear differential equation with matrix coefficients in terms of a particular matrix solution that enjoys properties similar to the exponential of first-order equations. A new formula for the exponential matrix is established with dynamical solutions related to the generalized Lucas polynomials.1. Introduction. The study of higher-order linear differential equations with square matrix coefficients is usually left aside for being equivalent to a first-order equation having the companion matrix as its coefficient. The handling of this matrix is not an amenable task for obtaining results proper of higher-order equations which are shadowed or simply unknown. The companion matrix, although sparse, does not preserve any common property that the involved matrix coefficients might share.In this paper, we shall present a direct study of higher-order equations in terms of a particular square matrix solution, which is characterized by zero initial "displacement" values and a unit initial "impulse" value, and that we shall refer to in the sequel as the dynamical solution. It will be shown that this solution enjoys intrinsic properties not necessarily shared by the complementary basis solutions, and that in a certain way the dynamical solution should do for higher-order equations what the

Operator Identities and the Solution of Linear Matrix Difference and Differential Equations

“…The articles by Ruiz Claeyssen, Gallicchio, and Vilhena [9], Ruiz Claeyssen and Tsukazan [8], and Runekel and Pittelkow [10] are closely related to the present work. These papers contain some propositions that can be used to extend our methods in order to solve nonhomogeneous equations, to invert matrix polynomials, and to compute matrix functions.…”

“…We call the h k the Horner polynomials of w. It is clear that they form a basis for the complex vector space .9 of polynomials of degree less than or equal to n. Formulas (2.20) and (2.21) are obtained from (2.15) and (2.16), and they are special cases of the general interpolation formula for matrix polynomials, as we will show in the following sections. • The function g of the previous theorem is called the dynamic solution of (2.17) [8,9].…”

On linear matrix differential equations

“…The idea of using the dynamic solution to construct solutions of matrix equations was used in [7,8]. For the numerical aspects of the computation of the matrix exponential see [5].…”

Respostas Dinâmicas em Sistemas Discretos Matriciais de Ordem Arbitrária

“…De acordo com resultados obtidos para ordem arbitrária de equações em diferenças lineares com coeficientes matriciais, conforme [2] e [5], entre outros; tem-se que o estudo do sistema (2.1) depende da solução h k do problema de valor inicial…”

“…A resposta impulso discreta que corresponde a solução do problema de valor inicial (2.2) pode ser calculada através da fórmula desenvolvida por Claeyssen, veja-se em [2] e [3], sendo expressa por…”