TABLE OF CONTENTS This chapter is an introduction to difference equations based on finite differences. The discussion is limited to linear difference equations with constant coefficients. The Fibonacci numbers are defined, and a practical example in electric circuit theory is given at the end of this chapter.
In mathematics, a recurrence relation is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. A difference equation is a specific type of recurrence relation, and this type is discussed in this chapter. Difference equations as used with discrete type systems, are discussed in Appendix A.
The difference equations discussed in this chapter, are used in numerous applications such as engineering, mathematics, physics, and other sciences.
The general form of a linear, constant coefficient difference equation has the form
where a k represents a constant coefficient and E is an operator similar to the D operator in ordinary differential equations. The E operator increases the argument of a function by one interval h, and r is a positive integer that denotes the order of the difference equation.
In terms of the interval h, the difference operator E is
The interval h is usually unity, i.e., h = 1, and the subscript k is normally omitted. Thus, (11.2) is written as
If, in (11.3), we increase the argument of f by another unit, we obtain the second order operator E 2, that is,
and...
