# To understand the original motivation write the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: t violations to [email protected] y (k + 1) = αm y (k ) + αm−1 y (k − 1) · · · + α1 y (k − m + 1) + α0 (7.10.1) in which α0 , α1 , . . . , αm along with initial conditions y (0), y (1), . . . , y (m − 1) are known constants, and y (m), y (m + 1), y (m + 2) . . . are unknown. Diﬀerence equations are the discrete analogs of diﬀerential equations, and, among other ways, they arise by discretizing diﬀerential equations. For example, discretizing a second-order linear diﬀerential equation results in a system of second-order diﬀerence equations as illustrated in Example 1.4.1, p 19. The theory of linear diﬀerence equations parallels the theory for linear diﬀerential equations, and a technique similar to the one used to solve linear diﬀerential equations with constant coeﬃcients produces the solution of (7.10.1) as D E T H m y (k ) = α0 + βi λk , i 1 − α1 − · · · − αm i=1 IG R for k = 0, 1, . . . (7.10.2) in which the λi ’s are the roots of λm − αm λm−1 − · · · − α0 = 0, and the βi ’s are constants determined by the initial conditions y (0), y (1), . . . , y (m − 1). The ﬁrst term on the right-hand side of (7.10.2) is a particular solution of (7.10.1), and the summation term in (7.10.2) is the general solution of the associated homogeneous equation deﬁned by setting α0 = 0. This section focuses on systems of ﬁrst-order linear diﬀerence equations with constant coeﬃcients, and such systems can be written in matrix form as Y P O C x(k + 1) = Ax(k ) (a homogeneous system) x(k + 1) = Ax(k ) + b(k ) or (a nonhomogeneous system), (7.10.3) where matrix An×n , the initial vector x(0), and vectors b(k ), k = 0, 1, . . . , are known. The problem is to determine the unknown vectors x(k ), k = 1, 2, . . . , along with an expression for the limiting vector limk→∞ x(k ). Such systems are used to model linear discrete-time evolutionary processes, and the goal is usually to predict how (or to where) the pr...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online