Finite Difference Equations

In this chapter, the exact analytical solution of linear finite difference equations is discussed. The main purpose is to identify the similarities and differences between solutions of differential equations and finite difference equations. Attention is drawn to the intrinsic problems of using a high-order finite difference equation to approx­imate a partial differential equation. Since exact analytical solutions are used, the conclusions of this chapter are not subjected to numerical errors.

1.1. Order of Finite Difference Equations: Concept of Solution

Domain: In this chapter the domain considered consists of the set of integers k = 0,

±1, ±2, ±3,____ The general member of the sequence…, y-2, y_1, y0, y1, y2,… will

be denoted by yk.

An ordinary difference equation is an algorithm relating the values of different members of the sequence yk. In general, a finite difference equation can be written in the form

yk+n = F^k+n-V yk+n-2’ •••> yк k)> (L1)

where F is a general function.

The order of a difference equation is the difference between the highest and lowest indices appearing in the equation. For linear difference equations, the number of linearly independent solutions is equal to the order of the equation.

A difference equation is linear if it can be put in the following form:

yk+n + a1 (k) yk+n-1 + a2 (k) yk+n-2 + •••+ an-1 (k) yk+1 + an (k) yk = Rk> (1.2)

where a(k), i = 1, 2, 3,…, n and Rk are given functions of k.

EXAMPLES

(a) yk+1 – 3yk + yk-1 = 6e-k (second-order, linear)

(b) yk+1 = y2k (first-order, nonlinear)

(c) yk+2 = sin(yk) (second-order, nonlinear)

The solution of a difference equation is a function yk = ф(^ that reduces the equation to an identity.

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