Difference Equations, An Introduction

Introduction

Although calculus is not a requirement for this topic, an understanding of differentiation, and ordinary differential equations, is recommended. As you will see, there are similarities between difference equations (presented here), and differential equations (described under calculus). The former is a quantized version of the latter, or if you prefer, the latter is a continuous version of the former.

Difference equations are easier to manipulate and solve, because you don't need an underlying theory of limits and continuity. The functions are discrete sequences, with no δ's or ε's in sight. However, the methods employed to solve these equations often come directly from calculus, and some of the intuition may be lost if you have never solved a differential equation before.

The Difference Sequence

Let y(x) be a sequence of complex numbers. (Actually you can use any field, but the complex plane handles most applications.) Although x looks like a real variable, it takes on the values 0, 1, 2, 3, 4, and so on, building the sequence.

As you recall from calculus, y′ is the derivative of the continuous function y(x). When x is discrete, y′(x) = y(x+1)-y(x). This builds the "difference sequence", which is analogous to the derivative, hence the borrowed notation.

There are higher order difference sequences, just as there are higher order derivatives. The sequence y′′(x) = y′(x+1)-y′(x). This can be expanded to y(x+2)-2y(x+1)+y(x). A recursive procedure builds the n^th order difference sequence. The n^th difference, evaluated at x, looks like the n^th row of pascal's triangle, although the coefficients alternate in sign. For example, the fourth difference sequence, evaluated at x, looks like this.

y(x+4) - 4y(x+3) + 6y(x+2) - 4y(x+1) + y(x)

Linear Operator

Like differentiation, taking the difference sequence is a linear operator. If y and z are two sequences of x, and c is any constant in the range, (y+z)′ = y′+z′, and (cy)′ = cy′.

Difference Equation

Let f be a multi-argument function that is evaluated at x, y(x), y′(x), y′′(x), and so on. This function must equal 0 for all x. The difference equation is "solved" when y(x) is found, such that f(x,y,y′,y′′,…) = 0 for all x ≥ 0.

In most situations, the n^th order difference stands alone. It is set to a function of x, y, y′, and the other differences up to n-1. This is similar to a differential equation that sets the n^th derivative of y equal to a function of x, y, and the lower derivatives of y.